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(oo), then J=
oo
/
/(^(t))^(t)dt,
(3.2.10)
-oo
(x) xl>(x) Taylor's theorem, i = 1, • • •, m. The monographs [23] and [132] discussed the error estimations of the fi nite element method. When considering the flexibility of the finite element method, one must note that the accuracy depends directly on the partition. For example, in order to guarantee the convergence, the partition usually has to be regular and quasi-uniform^. In order to guarantee that the er rors at the nodes have asymptotic expansions, the partition is required to be piecewise uniform^ 1 '. Recent research shows that an "optimal partition" should use as many rectangular elements or piecewise regular quadrilateral elements as possible (for details, see Q. Lin and Q.D. Zhu^'''). In the fol lowing, we discuss the SEM for finite element method by using the integral identity of rectangular elements. We assume that our readers are already familiar with the basic knowledge of the finite element method and only the relevant references are given. - ' i i ; =f = =0 (0{n~ n "- 221 n ¥>" ')-).
(3-2.13)
i=-N
with h = 4/N. When t tends to infinity, all the derivatives of the integrand tend to zero as fast as a double exponential function does. This ensures that the trapezoidal rule has a better accuracy. Merit 94 ! proved that J — Jl ~
exp(-CN/\ogN). There are many double exponential transforms, such as x = y>i(i) 1
A"
= - [ t a n h ( - sinh(t)) + l] : (0,1) <- (-co, oo), ^ 2 7T
x —
/
f(x)dx,
(3.2.14)
-oo
if f(x) is a function which converges very slowly to zero as x —► ±oo, it can also be converted to a function, which varies to zero at a double exponential speed as x —► ±oo by the double exponential transform x = sinh(Jsinh(*)).
(3.2.15)
For details see [94]. The double exponential transform can be used in multidimensional sin gular integrals as well, e.g., for 75 =
/
'"/
/(Xl»"'"'x«)dxi'""rfx»'
(3.2.16)
III. Application
82
1
let Xi = MU) = o [
tanh
of SEM to multidimensional
IT sinh(
(2
numerical
integration
-TTT T 7 * Ccosh(£,)
^) + ^ A(U)
=
1
cosh2(-7T sinh(t,-))
i = 1, • • •, s. One can consider r4
/5« /
• /
r4
f(Mh)r''^s(ts))t[
(3.2.17)
which can be computed, for example, by the trapezoidal rule. When the double exponential transform is applied to multidimensional integrals, in order to prevent the computer from overflowing or underflowing, it is neces sary to choose appropriate ai,a2 and as in (3.2.11). In fact, our experience shows that this method is not suitable for high dimensional cases, but is an effective method for lower-dimensional cases. E x a m p l e 1.5. Consider a double integral -dxdy. Jo Jo
y/xy
Firstly we convert it into the form (3.2.17) by a double exponential trans form, then evaluate by the trapezoidal rule with splitting extrapolation. The results are listed in Table 1.5. T a b l e 1.5 N u m e r i c a l r e s u l t s o f a p p l y i n g S E M and double exponential transform in evaluating a singular integral number
Type 1
Type 2
Type 3
of splits
error
CPU(sec)
error
CPU(sec)
error
CPU(sec)
1
7.40E-1
0.17
7.40E-1
0.18
7.40E-1
0.14
2
2.60E-1
0.54
2.59E-1
0.49
2.59E-1
0.47
3
2.20E-1
1.52
2.24E-1
1.11
2.31E-1
1.15
4
3.23E-2
4.11
3.51E-2
2.21
3.52E-1
2.54
5
1.28E-3
10.26
3.11E-2
4.07
7.34E-4
5.24
6
6.18E-5
24.33
1.98E-3
6.76
7.71E-4
10.61
7
4.41E-6
57.58
2.51E-4
10.55
6.77E-5
20.51
8
7.24E-9
131.75
1.68E-5
15.90
1.40E-6
38.24
The table shows that the SEM acceleration is almost invalid for the small number of splits, but performs better where the number of splits becomes larger. The reason is that under the double exponential trans form, the terms with lower order powers in the asymptotic expansion of the trapezoidal rule are close to zero.
3. Duffy transform for multidimensional
improper integrals
83
Remark. M. Beckers and A. Haegemans compared numerical values of various transformations and found that the TMT transform is better than the DE transform for multiple integral (cf. [29], pp. 329-340). 3. Duffy transform for multidimensional improper integrals Consider the multidimensional improper integral Hf) = I • • • / f(xu...,xs)dxx...dx„ Jo
(3.3.1)
Jo
with integrand / ( * i , • • •, xs) = gfa, • •, xs)h(xly g(xu • '• •, xs) = x*1, • • •, x*'gp(xu
• • •, x,), • • •, x 5 ),
(3.3.2) (3.3.3)
where h(x\, • • •, xs) is a smooth function on [0, l] 5 , <7/i(xi, • • •, xs) is a ho mogeneous function of degree /i. Suppose A,- > - 1 , (i = 1, • • • , * ) , and /i + ^ A , - > - s . As an example, one can consider
2
5
, or gn(xir-,xs)
= (^Pa.xf)",
with
h
«i > 0,/? > 0 and a/3 = /i.
*=i
Many practical problems can be brought into the framework of improper integral of the type (3.3.1)-(3.3.3). For this type of improper integrals, Duffy^'! proposed the so-called Duffy transform which can eliminate the singularity at the origin caused by the homogeneous function #(xi, • • •, xs) of the form (3.3.3). Further, by some suitable transforms, it is often con verted into the proper integral which can be computed by the product-type Gauss-Jacobi formula. The Duffy transform considers integrals over an s-dimensional pyramid Vi = { ( s i , - - - , * , ) : 0 < * i < l , x i > X i , V i ^ l } ,
h(f)=
I I*1---!'1 Jo Jo
f{xlr..,x9)d*i~'d**
Jo
Jo Jo Jo •h(xi, - - • ,xs)dxi
• • dxs.
(3.3.4)
84
HI. Application
of SEM to multidimensional
numerical
integration
Taking the transform: x\ = 2/1, x 2 = t/it/2? * • •, z* = 2/i2/*> we have h{f)
= / • • • / 2/i 1 (yiy2) A 2 ---(2/i2/5) A '^(2/i,yi2/2, Jo Jo h(yliy1y2l • • •,yiVsWi' 1 dyi • • • dy,
= I
-,2/iy*)
M2-~v}'gr(i,V2,---,y*)
"1
Jo Jo •^(yi, 2/12/2, • • •, 2/12/5)^2/1 "dy5,
(3.3.5) s a
where 77 = Ai H h A, + /1 + s - 1, ^ ( 1 , 2 / 2 , ••, 2/«) * regular function, and h(yi, 2/12/2, • • •, 2/i2/«) is a smooth function of the variables 2/1,2/2, * • •, 2/«It follows from (3.3.4) t h a t 77 > — 1 , A,- > — 1 for 1 = 2, • • •, s, and hence the integral (3.3.5) is meaningful. If A, = Pi/qi, i = 1, • • •, s are fractions, then by a transform tA; = z\{, i = 1, • • •, s, the integral can be converted into a proper integral of type (3.2.3). If the integration domain is V — [0, l ] 5 , then V has the decomposition (3.3.6)
V=(jVi, »= 1
where V% is the closure of V*, and Vi is an s-dimensional pyramid Vi = { ( x i , . . . , x , ) :0<Xi
<
l,xi>xj)Vj^i}1
i = l , ••-,*.
(3.3.7)
It is easy to verify t h a t Vi fl Vj = 0, for i ^ j . In fact, if x £ Vi D Vj, then x £ Vi implies x,- > Xj. Again x £ Vj implies Xj > x,-, and this is a contradiction. On the other hand, Vi C V, and the volume of every Vi satisfies measV; = - , which leads to t h a t {Vi} is a decomposition of V, and (3.3.6) holds. Therefore, the integral (3.3.1) can be described as
1(f) =
/ • • • /
Jo
/(xii-
",x8)dxi-"dx8
Jo
= f f(x)dx Jv
,■=1
= /
- ^
[ ■■[ F(xi,-;x,)dxv-dx„
Jo Jo
Jo
(3.3.8)
4. Multivariate
asymptotic
expansion
of multidimensional
improper
integrals
85
with s
F{xu - . . , * , ) = £ / , ( * ! , . . . , *,),
(3.3.9)
and • -,x 5 ) = /(x,-,x t + 1 , • • • , x 5 , x i , - - - , x t _ i ) ,
fi(xlr
i = l, •••,*.
(3.3.10)
Since the singularity of F(x\, • • •, xs) comes from the function f(x\, • • •, xs) when the Duffy transform is applied to (3.3.8), the singularity can be soft ened, or even eliminated. If g(x\) • • •, xs) in (3.3.3) has logarithmic structure: ra lnr, then Duffy transform remains effective. The singularity is transferred to the form 1 /9
Xj*(l + x\ + • • • + xj) l n x i . In this case, (3.3.1) cannot be converted into a proper integral through transforms, but using the product form of the Euler-Maclaurin expansion (1.2.45) with singularity at end points, the splitting extrapolation method described in (2.5.25) still works. 4. M u l t i v a r i a t e a s y m p t o t i c expansion of multidimensional im p r o p e r integrals Consider the improper integral
Q(/)= f ■■■[ n(*rinA-xo^(«i.---.«.)d«i•••<'«.. Jo
Jo
i=1
- 1 < m < 1, A, = 0 or 1.
(3.4.1)
This type of integral not only occurs in practice, but is also obtained from applying the Duffy transform to the integral of the form (3.3.1), where H(xlr--,xs) e C 2 m + 1 ([0, l] 5 ). If m is an irrational number or A,- ^ 0, one cannot convert (3.4.1) into a proper integral, but can apply splitting extrapolation methods by means of multivariate Euler-Maclaurin expansion Qh(f)
where
Ni-l
JV.-l
»1=0
t,=0
hi = l/Ni,
0<9i < 1, i = l, • • . , « .
86
III. Application
of SEM to multidimensional
iintegration
numerical
If Oi = - , this is the rectangular rule. In (1.2.38) and (1.2.45), the onedimensional asymptotic expansion with singularities at end points is given. For the general case 0 < Q\ < 1, Navot (cf. [25]) proposed the following L e m m a 4 . 1 . If G(x) - xpg(x)
with - 1 < p < 0, g(x) € C 2 m [0,1],
and h = —, then the following asymptotic expansion holds [1
E(h)=
=
n-l
G(x)dx-h^2G((i
+ e1)h)
+P+I _ ^ 1 W ) G 0 . 1 ) ( 1 ) f c f .-E'""'?'"."')')"^ j=l
;=0
i=l i! m ++ 00 (( // ii 22 m )) ,, 3'
i=0
-7*
^*
(3.4.3) (3.4.3)
where (.(t^e) is the generalized Riemann zeta function defined as oo OO
C(M) = £ > + *)"'•
(3.4.4) (3.4.4)
Re * > 1,
fc=0 k=0
anrf (3.4.5)
C(-i,e) = - 5 i + 1 ( £ ) / ( i + 1), j = 0 , 1 , - . . , where Bj(x) are the Bernoulli polynomials. L e m m a 4.2. Under the assumptions of Lemma Ji.l} g(x)xp In x, — 1 < p < 0, then 2m-l
G{x) =
2m-l
Ajhj +lnh^2
£(/*) = £
if
Bjhj+P+1 j=0
j=l 2m-l
+ 5 3 C'i&i+|,+1+0(ft2m)>
-Kp<0.
(3.4.6)
£j^'+1+0(/i2m),
(3.4.7)
/ / G(x) = p ( x ) l n x , Men 2m-l
£(A) = ^
2m-l
AjW+logfc Yl
where Aj,Bj and Cj are constants independent of h, and depend on the derivative of g(x) at the end points.
4. Multivariate
asymptotic
expansion of multidimensional
improper integrals
87
Theorem 4.1. Suppose that Q(f) is an integral defined in (3.4.1), Qh(f) is the quadrature rule defined in (3.4.2), and H(xi, • • • , x 5 ) G C 2 m + 1 ([0, l] 5 ), then there exist constants Aa)Ba and Ca which are inde pendent of h = (fti, • • • }hs), and depend on H(x\y • • • , x 5 ) and its deriva tives such that the following multivariate asymptotic expansion holds
Q(f)-Qh(f)=
A ha
X
°
+
E
Ca^
+/i+1
l<|a|<2m
+
0<|a|<2m
Yl
5a/»a+"+i(in/lr
+ 0 ( ^ m + 1 ( l n / i ) A ) , (3.4.8)
0<\a\<2m
here (ln/i) A = (ln/ii) A l • • (ln/i 5 ) A », A = (Ai, • • •, A,),/i = (/ii, • - , / / , ) , ot = (ai, • • •, a,), and fc«+"+i = ft«i+^+1 . . . ft«.+<*.+i. Proof. We use induction with respect to the dimension. For s = 1, (3.4.8) follows from Lemmas 4.1 and 4.2. Now suppose that for (s — 1)dimensional case the integral asymptotic expansion (3.4.8) holds, we shall prove that it also holds for the s-dimensional case. For simplicity, let x = (x 2 , • • •, x 5 ), A = (A2, • • •, XS),~P= (/*2, • • • ,/**), dx = dx2 - - - dxs, and consider F(x1)=
I • • • / x^{\nx)XH{xux)dx. Jo
(3.4.9)
Jo
This is an (s — l)-dimensional integral with parametric variable x\. By the inductive assumption, there exist functions AQT(XI), B-a(x\) and C-a{x\), which are independent of h = (/12, • • •, hs), but depend on xi, such that F(Xl) N3-l
N.-l
5
» 2 =0
:,=0 j = 2
=h2...h, Y: ••• E n{[(ii+^)^]"i[in(^+^)^)] 0 •^(*i,(»'a + tfa)A2,---,(t. + ^ ) M
+
E
^ * i ) ^+
l<|«|<2m
+
E
E
^(xx^^'Onft) 1
0<|cF|<2m
^o(*i)A5+5r+1+0(^m+1(lnA))X.
(3.4.10)
0<|a|<2m
By the asymptotic expansion of one-dimensional integral and (3.4.10), Q(f) 1
Al
x5' (lnx 1 ) ^(x 1 )rfxi / Jo
88 88
III. Application Application of of SEM SEM to to multidimensional multidimensional III. N2-l
N.-l
s
numerical numerical
integration integration
A
l
= h2...h. ..-h, £ £ . .. ... ££ n{ife+«i)*ir[»«(( >[ln((ij+ej)hj)]!Ai'} =h } 2 t =0 ».=0 j = 2 t 22 =0
».=0 j = 2
•• // Jo Jo
x^(lnx1)1Xl )XlH(x H{x1,(i +e2)h2,---,(i, 1,(i z?(l*z + 02)h2r..,(i9 22
+ +
£ J2
IT h°
ll<|«l<2m <|o|<2m
+
Xl ff11x^(lnx )dx11 x^(\nx11))XlA^{x Ao{x11)dx
J
°°
J
^+Jr+10nA)X
J2 0<|o|<2m |«l<2m
+ +
/'x^(\nx1)XlB^{x1)dx1
■'O "/o +1I+1
Xl f1x^(lnx 1) C^{x1)dx1 fx^ilnx^C^xJdx!
h"+1I+1 T
££
+ 009a)h,)dx )hs)dx11 +
0<|a|<2m ^° 0< 2 m+1 A +O{h2 , m0 + 1 (ln/l) (\nhf) +0(A ) m+1 2 m+1 J X ), = /
where Ki Ki ,K ,K22,Kz,K± ,Kz,K± where of (3.4.11). (3.4.11). Since Since of
(3.4.11) (3.4.11)
stand for for the the four four summations summations on on the the right-hand right-hand side side stand
Xl Xl (\nXl H(xltlt Xl)) H(x // ** jj "" (\n Jo Jo
(ta + + 0022)h )h22,, •■■-,((,+ (ta •,(*, +
)htt)dxt )dxt eett)h
JV,-I JV,-I
= &i E
[(''i + ^i)/«i]'"[Jn((n + ^ 1 )/, 1 )] Al
*i=0
•//((«! + «!)&!, •••,(«, + 0.)h.) 2m 2m
2m 2m +/il+1 A1 +E E 66 ii (( (( ii 22 + + ^)/i2,--,(i, ^)/i2,--,(i, + + ^0MM ^0MM+/il+1 (lnAi) + (lnAi)Al
i=o i=o 2m 2m
1+1 E Cc>> ((*'» *.)ft.)M ++""1+1 +E (('2 ++ '»)*»• ^ 2 , •••,(*,+ •••,(!•.+ ff.)*.)M 7=0 J'=0
+0(fc*mm++11 lnA liiA11), ), +0(ftg
(3.4.12) (3.4.12)
where aj-(x a,(x 22,, •• •• ••,,xxa5),bj{x ), 6j(x • • •, xs) and • • •,>*») xs) are where andCj(x c ; (x2,2 ,-are coefficients coefficients of of the the 2,2> ■ ■• ,x,) expansion (3.4.6) (3.4.6) with with parameters parameters xx22,, ■ • •■•,■,x sx. sLet expansion . Let Q(g) = JJo ■■[ 9(x2,--,xa)dx2-dx, Jo
(3.4.13) (3.4.13)
4. Multivariate
asymptotic
expansion of multidimensional
improper integrals
89
be an (s — l)-tuple integral, and N2-l
AT.-l
Qh(g) = h2-.-hs 53
• 53 g((i2 +
» 2 =0
02)h2r~,(i.+0.)ha)
»t=0
(3.4.14) be the corresponding quadrature rule. Let , *,) = J J [XV(InXj)x'].
w(x) = w(x2,
(3.4.15)
i=2
It follows from (3.4.12) that in the expansion (3.4.11) 2m
2m
Ki = Qh(f) + ^ZQh(waj)h{
+
j=l
^Q^wb^hi+^ilnh^ j=0
2m
+E^(^)M + M l + 1 2m
2m
= Qh(f) + 53Q(^)/i{+53Q(^)M + / i i + 1 an/ii) A i 2m
2m
+ li+1
+ £ o(^i)M ' i=o 2m
+ £ {^(^i) - «(^i)}M j=i
+ £ {Q»M>;) - Q(^;)}ft{+/il+1(in fti)^ 2m
+ 53{Q,(^c i )-Q( U ;c i )}/i{ + ^ +1 .
(3.4.16)
i=o Applying the inductive assumption to the terms with braces, and noting that ahbj and Cj G C 2 m _ i ( [ 0 , l] 5 " 1 ), (3.4.16) becomes
Ki = Qh(f) +
53
*ahx
l<|a|<2m
+
53
^a^a+/i+1(ln/i)A+
0<jor|<2m
+0(ft^ m + 1 (lnft) A ),
53
5«^+Ai+10n/i)A
0<|cr|<2m
(3.4.17)
where AajBa and C a are constants independent of ft. Combining (3.4.17) with the terms K2lKs and K±y the proof of (3.4.8) is complete. □
90
HI. Application
of SEM to multidimensional
numerical
integration
Corollary 1. J/0, = - , i = 1, • • •, s, then the asymptotic expansion is
Q(f)-Qh(f)=
A2ah2°+
£
J2 Baha+»+\\nh)x
l<ja|<2m
+
0
Ca/i
]T
a+
^ + 1 - f O ( / i ' m + 1 ( l n / i ) A ) . (3.4.18)
0<|a|<2m
Corollary 2. If the integrand
= fl K't
1
- ^rOn^)^(ln(l -
Xj))^}H(xu
• • •, * , ) , (3.4.19)
where \j,r)j take value 0 or 1, AajBajCa and Da such that Q(f)-Qk(f)=
HJ^UJJ
£
> —1, then there exist constants
Aah"+^
0<\a\<2m
+
Yl
Baha+^l+
Y,
0<|a|<2m
+
Y
Caha^+1(lnh)x
0<|a|<2m Q
Dah +»+\\nhy
+ 0(hlm+1(\rih)x+ri).
(3.4.20)
0<|or|<2m
5. Integration on multidimensional simplexes The multidimensional simplex is the simplest one among various multi dimensional polyhedrons. Many complicated domains can be divided into simplexes. In order to apply the SEM, we suggest a method, similar to the Duffy transform mentioned above, to map the simplex onto a cube. Such a mapping has two advantages: first, a multidimensional simplex is mapped onto a multidimensional cube, and remains the smoothness of the integrand, which facilitates the use of the SEM; second, if the integrand has a singularity at the origin or on the surface of simplex, then the mapping can "soften" or even eliminate the singularity. Consider an integral I = / f(zir-,2s)dxi JT
= / / Jo Jo
••/ Jo
"dx3 / ( * i , - - - , * f ) d * i - -dx,,
(3.5.1)
5. Integration on multidimensional
simplexes
91
where T = {On, • • •, xs) : 0 < X! < 1, xx > x 2 > • • • > xs}
(3.5.2)
is a simplex in Ms. Applying the integral transform *i = i/i, xi = xi_1yi,
i = 2, ••-,*,
(3.5.3a)
xi = 2/i, Xi = yi---y,-, i = 2, •••,*,
(3.5.3b)
or equivalently
it is easy to prove that the transform (3.5.3a) maps T onto [0, l] 5 , and the Jacobian of the transform is J = ys1ys2-1--ys-i.
(3.5.4)
Hence,
= J f(x)dx vly^1-'y*-if(yi>yiy2r-,yi'-ys)dyi'-'dys.
= / • • / Jo
Jo
(3.5.5) Obviously, if / ( x i , • • •, x 5 ) is a smooth function on T, then the new inte grand is smooth on [0, l] 5 . If / is a singular function of the form /(xi,---,x5) = x f • • • x?'gi(xi, • • •, x,)/i(xi, • •, x 5 ),
(3.5.6)
where /(xi, • • •, x 5 ) is a homogeneous function of degree /, and h(x\, • • •, x 5 ) is a smooth function, then one can prove the following T h e o r e m 5.1. If /i! + s + / > - 1 ,
/it- + s + 1 - i > - 1 (i = 2, • • •, s),
(3.5.7)
Men Me following improper integral on an s-dimensional simplex T I=[
x^gi(x)h(x)dx,
(3.5.8)
JT
converges, where x** = x^1 • • -x%9, and
Vo ./o -9i(l, *2, z 2 x 3 , • • •, x 2 • • • x5)/i(xi, x x x 2 , • • •, xi •..x5)dx. (3.5.9)
HI. Application Application of of SEM SEM to to multidimensional multidimensional HI.
92 92
numerical numerical
integration integration
Theorem 5.1 5.1 shows shows that that for for an an improper improper integral integral of of the the form form (3.5.8) (3.5.8) Theorem denned on a simplex, if /ii, • • •,//, and / are fractional numbers, then this denned on a simplex, if /ii, • • •,//, and / are fractional numbers, then this integral is of type (3.2.1), and can be converted into a proper integral of integral is of type (3.2.1), and can be converted into a proper integral of type (3.2.2) and computed by SEM. If/ has a logarithmic singularity, e.g., type (3.2.2) and computed by SEM. If/ has a logarithmic singularity, e.g., //(a?i, ( a ? i , •• •• •• ,, ** ,, ))
p/(xi • • • i xs)Kxli.. •, xB), (3.5.10) 3 5 10 == nn [*ron*o [*r(inx0AiAi]]#(*!» ' • • •' *• w*i. • • • * *•)■ ( - - ) s
»=1 »=i
with At- = 1 or 0, then through the transform (3.5.1), with At- = 1 or 0, then through the transform (3.5.1),
I = / f(x)dx
I = /
J J°
°
f(x)dx J J°
j=2 ° j=2 jj == ll (3.5.11) •^/(Ij xX2, " ' ' , *2 ' ' ' Xs)h(xi,XiX • • Xs)dx.(3.5.11) •0/(1, • •'■' ,•*, !Xi • • •• x,)dx. 2 , • • •, x 2 • • • xs)h(x llx1x2, 2j
Thus, under under the the assumptions assumptions of of Theorem Theorem 5.1, 5.1, the the improper improper integral integral of of aa Thus, singular function defined in (3.5.10) on the simplex T converges. Applying singular function defined in (3.5.10) on the simplex T converges. Applying the quadrature quadrature formula, formula, the the error error expansions expansions of of integrals integrals of of this this type type are are the described by by (3.4.8) (3.4.8) and and (3.4.15). (3.4.15). One One can can obtain obtain the the approximation approximation by by described the successive successive recurrence recurrence algorithm algorithm 1.2 1.2 of of SEM. SEM. the Consider the the following following multiple multiple integral integral over over aa simplex simplex Consider
h = = /f f(*)d* f(x)dx
M where where
== / / Jo Jo Jo JO
••••/ ••/ Jo Jo
(3.5.12) f(xi r"i1x..8)dxi'"dx / ( * i , - ",x dxS)8 , (3.5.12) s)dx
T11 == J| (( xx 11 ,, .. .. .. ,, xx O O :: ^^ xx ,, << ll ,, xx ll >> 00 ,, ii == l ,l ., ,. 5, 5l .l . T
(3.5.13) (3.5.13)
Obviously, the the transform transform Obviously, tt -- ii
X\ = 2/i, 2/1, s.Xi = y .i -((ll - ]^^3Xj j) ), ,(( ** = 2,---,s), 2,-.-,s), *i i=i i=i maps Ti Ti onto onto [0,1]*. [0,1]*. Besides, Besides, (3.5.14) (3.5.14) has has an an explicit explicit expression expression maps «-i
*i = l/i, Xi = yiY[(1-"yj)>(i i=i
= 2
>'-,s),
(3.5.14) (3.5.14)
(3.5.15) (3.5.15)
multidimensional 5. Integration on multidimensional
simplexes simplexes
93 93
and one obtains »—1
» — l
= U(1-VJU*
i-Y,*
(3.5.16) (3-5.16)
= 2>---,').
it follows immediately from (3.5.14) that for ti = = 22 In fact, by induction, it (3.5.15) and (3.5.16) obviously hold. Now assume that for ii — 1, (3.5.16) holds, then by using »-i
»
i=i
i=i
=n( -w)-wii( -w)=ii (!-»)' (3-5-i?) tt - -l l
i
xx --l 1
j=l
i
i
ii ==l l
JJ== l
i, (3.5.16) holds. Again, by (3.5.14) and (3.5.17), we obtain that for i, t
t
= w+ill( 1 -w)'
< 3 - 5 - 18 )
it is proved that for i + 1 , (3.5.15) holds. Denote by * the contents not used in the evaluation of the determinant. Since ' 0, 0, dxj
i i>j, > h
(( ll -- t»/ ii )) -- -- -- ((ll--t M / ; -_ il )) ,,
dyi
, *> which means that the Jacobian is
(3.5.19) (35.19)
ti = J, = i,
* i
JJ = = (1 - 0i)*(l 2/i)*(l - 0i fae))"- 1 • -• •((1I - y ., - i ) . (3.5.20) (3-5.20) 5 TAe transform (3.5.15) maps 7\ T\ on [0,l] [0,1]*,, and the T h e o r e m 5.2. Tfce s integral (3.5.12) is converted into the following integral defined on [0, l ] 5 .
h = f f{x)dx ix
5 >+i * f(yi>y2(i-yi),---,y f[(i-yi))dy ---f ll^-yiY~ == /1--Y1n([i-%) " /^'^(i-yi)r--,y/na 1
1
i 1
t
*/o
Jo
i=1
j=1
1
= Jo Jo
Jo j ==11 Jo J —•*■
yj))^y
[ -rf[z'j-
i+1
f(l-z1,z1(l-z2),...,(l-zs)Sf[zj)dz. , i== 11 (3.5.21)
94
HI. Application
of SEM to multidimensional
numerical
integration
P r o o f . Under the transform (3.5.15), it follows from (3.5 .20) t h a t
Jo
Jo
=1
5-1
(3.5.22)
•/(2/i, 2/2(1 - y i ) , •• • , y , [ | ( l --yj))dyi--dy.. Again, applying the following transform i = !>••• >*,
Z
J = (i-Vj)'
(3.5.22) becomes
«/o
./o
=1
• / ( l - z i , z i ( l - z 2 ), • • - , ( ! - 2,)*i .■ • • z , _ i ) d z i - - • ^ 5 , which completes the proof of (3.5.21).
D
C o r o l l a r y 1. If gi(x\} • • • , x 5 ) is an homogeneous with a unique singular point at the origin, and / ( x i , • • •, xs) = x^1 • • • x*'gi(l
functuon of degree I
- a?i, x2} • ••,**)M*i>--
where ft(a?i, • • •, ar5) is a smooth function,
and
s
fij > - 1 , / + * + ^ ^ .
•>**)i
s
> —1,^ —i-hl-h
i=2
^
/i, > - l ,
t=j+i
(3.5.23) //ten Me integral of f(x)
on T\ converges,
f f(x)dx= / 1 .../ 1 *j+'+*»+
+
and
" ' { f l ( l - -'i)"'}
•{n*;-' + 1 + w + i + - + "-}„(i >
1 - * 2 | . - - , ( 1 - -2*)*2 • • • * * )
i=2 • A ( l - * i , • • • , ( ! - - * * ) / i i •••/**-_i)cfc.
(3.5.24)
C o r o l l a r y 2 . Tf/ii, • • •, JI 5 and / are fractions satisfying the condition (3.5.23) ; Men Mis integral is of type (3.2.2) and can be computed by SEM.
5. Integration on multidimensional
simplexes
95
Moreover, consider the integral over T\ with the integrand
/(x 1 ,...,x 5 )=nK j ( i n ^) A j i •0/(1 - a?!, x 2 , • • •, xs)h{xu
• • •, x a ) ,
which has a logarithmic singularity with Aj = 0 or 1. W h e n the integral is converted into the type (3.4.13) on [0, l ] a , the asymptotic expansion (3.4.20) shows t h a t it can be computed by SEM. We have discussed the simplexes T defined by (3.5.2) and T\ defined by (3.5.13). By a suitable affine transform one can m a p an arbitrary sdimensional simplex onto one of these two standard forms. Let T3 be an 5-dimensional simplex with vertexes Ai, • • •, As+i, the base corresponds to the vertex A\ satisfies the plane equation Lx(x)
= a i * i + ■ • • + asxs - 6 = 0.
(3.5.25)
Consider the following integral /
w(x)g(x)dx,
(3.5.26)
JT3
where g(x) is a smooth function on T3, and s
wx=
s
aiXi
()
\Yl
-H^(M]C a * x , '~H)
»=i
1
»A«I
> -iiAi = o,i,
i=i
is a function with singularity on the base corresponding to A\. Map Z3 onto the standard simplex T\ by an affine transform, and suppose ^^(1,0,--.,0),-.-, A , - . ( 0 , 0 , ■■•',!), i 4 , + i - ( 0 , . . . , 0 ) . Under this mapping, the new base corresponding to A\ satisfies 2/1 = 0. Thus, / JT3
w(x)g(x)dx=
I
y^(lnyi)^g(y)dy.
(3.5.27)
JT2
Applying the transform (3.5.15), it is converted into the form (3.5.21). If the integral on a cube has a singularity on the diagonal plane, it can also be decomposed so t h a t the singularity will be eliminated. For example, the integral \x-y\fil(\Ti\x-y\)Xlg(xiy)dxdy,fi 1 > - 1
h= Jo Jo
III. Application
96
of SEM to multidimensional
numerical
integration
has a singularity on the diagonal line x = y, then [0, l ] 2 can be decomposed into two triangles with this diagonal line as hypotenuse. For details, see A. Sidi's w o r k t 1 1 ' } . This article discusses t h e Euler-Maclaurin expansion for an integral over the triangle a n d t h e integrand h a s an algebraic a n d logarithmic singularity. T h e transforms discussed in this section n o t only can m a p s-dimensional simplexes onto [0,1]*, b u t also reduce t h e m t o [0, l ] 5 so t h a t SEM can b e used. 6. I n t e g r a t i o n o n m u l t i d i m e n s i o n a l d o m a i n s w i t h c u r v e d b o u n d aries Many integrals on the domains with curved boundaries can be converted into integrals on [0, l ] 5 by using suitable transforms. For example, let
ft = {(*!,-••,*,) : 0 < z i < 1,0 <ar,- ^ f l i - i f a i , . . . ,*,•-!), 2 < t < * } ,
(3.6.1)
and 1=
I I Jo Jo
■■ Jo
f(x1,---,x,)dx1---dxs.
(3.6.2)
Consider the transform * i = 2/i, *2 = 2/2#i(zi), • • •, x9 = ys0s-i{xly
• • •, x , _ i ) .
(3.6.3)
Obviously, such a transform m a p s ft onto [0, l ] 5 . In order t o express ex plicitly this transform, determine recursively * i = 2/1,
*2 = 2 / 2 # i ( z i ) = y20i(2/i) = 2^2(2/1,2/2),
and * 3 = 2/302(*i, x2) = 2/302(2/1,2/201 (2/1)) = £3(2/1,2/2,2/3).
In general, if x, = £,(2/1, • • •, y : ), i < s, have been defined, then x,+i = yi+16i(xu
••-,x t -)
= 2/t>i0.- (yi, ^2(2/1, y2), • • •, * . ( y i , • • •, y,)) = £»>i(yi,---,y«+i).
(3.6.4)
Now define 0»(yi,---,y»)
= 0i(yi,x2(yi,y2),-;*i-i(yw-,yi)),
i = i,-••,*-!,
(3.6.5)
6. Integration on multidimensional
domains with curved boundaries
97
the transform can hence be expressed explicitly as xi =yi,x.-+i = y,+i0*(yi,---,y,), t = 1, • • • , * - 1.
(3.6.6)
Denote by * the contents not used in the evaluation of the determinant. Since the elements of the Jacobi matrix are 0, dxj — =^ <9y«-
j
i = t,
the Jacobian is .(3.6.7)
J = e1{y1)02(yuy2)---6s-1{ylr..,ys-1).
For the integral on a more complicated domain with curved boundary,
h = [ f(x)dx rl
rV?i(ari)
/»V.-i(a?i,---,a?.-i)
••/
= »/0 Jxpi{xx)
/(xi,---,x5)dxi---dx5,
^I-I(*I,-,*I-I)
(3.6.8) where fii = { ( x i , - - . , * , ) : 0 < a ? i < l , ^.•-i(xi,---,x.--i) < xt- < ^•_i(a?i,--- > a:i_i),
i = 2, •.,*},
(3.6.9)
one can use the transform x i = yi,
xi = yi - ^ t - i ( x i , - - - , X i _ i ) , i = 2 , - - - , «
(3.6.10)
to convert it into one of the types (3.6.2). Express recursively the transform (3.6.10) by xi = x i ( y i ) = yi,
Xi = yi - tl>i-i(xi(yi)r = Xi(yi,---,yi),
• - ,Xi-i(yir
-,yi-i))
i = 2, • • • , « .
Let 0i{yi, • • •, yt) =
i = ! , • • • , * - 1,
(3.6.11)
98
III. Application
of SEM to multidimensional
numerical
integration
(3.6.8) becomes ri
rOi(yi)
r*i(yi,—,y.-i)
/
h= Jo Jo
g(yi,-~,ys)dyi~-dy„
(3.6.12)
Jo
with g(yi,
• • •, y*) = / ( ^ i ( 2 / i ) , • • •, * . ( y i , • • •, 2/5)).
Integral (3.6.12) can now be converted by the transform (3.6.6) into an integral on [0, l] 5 . E x a m p l e 6.1. Let
fi2 = | (*ir • •, x,) : J2 ( ^ ) m i ^ L XL- • •>«» > 0 > ,
(3-6.13)
with ai,rrii > 0 (i = 1, • • •, s). The transform yi = &)mi,i=l,-,'
(3.6.14)
a,-
maps Q2 into the standard simplex T\ and / f(x)dx= Jn2
m\rri2 • • rns
• / ny;1+1/mV(*iy^^
(3-6-15)
This can be converted into an integral on [0, l] 5 by using (3.5.15), its sin gularity may then be eliminated by the Duffy transform, and hence may be evaluated by SEM.
Chapter IV SEM FOR INTEGRAL EQUATIONS
1. General theory of integral operators Suppose that X is a Banach space and K : X —* X is a bounded linear operator. Consider the approximate solution of the operator equation u - Ku = / ,
(4.1.1)
where / G X, 1 G p(K), and p(K) is the resolvent set of K. The operator K is called an integral operator, if Ku = / k(x,y)u(y)dy,
V* G
ft,
(4.1.2)
where ft is a bounded domain in Ms. Equations of the form (4.1.1) are called the Fredholm integral equations of the second kind. k(x,y) is called the kernel of the integral operator K. If k(x, y) G C(ft x ft), then Ar(x, y) is a continuous kernel. If fc(x, y) is continuous and fc(z, y) G C r (ft x ft), r > 0, then k{x) y) is a smooth kernel. If there exists a(x, y) G C(ft x ft) such that a discontinuous kernel is of one of the following forms, k{xyy) = a(x,y)\x-y\-vJ
0
(4.1.3)
or k(x, y) = a(x, y) In |x - y|,
(4.1.4)
then it is called a weakly-singular kernel. It is noteworthy to study the condition under which an integral operator K is compact. Graham and I.H. Sloan (cf. [20]) proved the following L e m m a 1.1. Suppose that for 1 < p, q < oo, p~x + q"1 = 1, K is a compact operator mapping i/(ft) (q < r < oo) onto C(ft) if and only if the function kx{y) = k(x,y) satisfies (i) sup \\kx\\p < oo, sen
and 99
100
IV. SEM for integral
equations
(it) lim ||i c . - i,||p = 0, Vx',z G fi. x'—*x
Due to the following inclusion relation C(Q) C L°°(n) C Lr(Q) C L^Cl),
(4.1.5)
the conditions in Lemma 1.1 imply that K : 27 (ft) —* I/(£2) and L°°{Q) —► C(Q). The latter is more important in application. Similar to Lemma 1.1, the condition that K is a compact operator mapping C(Q) onto itself can be given as follows: Lemma 1.2. If the kernel k{x,y) (i) sup / \k(x, y)\dy
satisfies
and
(ii) lim / \k(x'iy)-k(x,y)\dy
= 0, V * ' , * G O ,
/Aen K is a compact operator mapping C(Q) onto C(Q). Proof. Setting p = 1 in Lemma 1.1, it follows that K is a compact operator mapping L°°(Ct) onto C(ft). Due to the fact that C(Q) C £°°(Q), it is clear that K is a compact operator mapping C(Q) onto C(Q). Lemma 1.2 can also be proved directly by the Ascoli-Arzela Theorem. Corollary 1. If the kernel k(x,y) operator mapping C(Q) onto C(Q.).
£ C(Q, x Q), then K is a compact
The condition under which an integral operator K : L2(£l) —► L2(Q) is a compact operator is as follows: Lemma 1.3. If the kernel k(x1y) M2=
satisfies
[ f \k(x,y)\2dxdy<
oo,
(4.1.6)
JVLJil
then K is a compact operator mapping L2(Q) onto L2(Q) satisfying \\K\\ = M. Proof. First, for any u 6 X 2 (ft),
PHI 2 < / \K*,y)?dy I \u{y)\2dy,
(4.1.7)
hence under the L 2 (fi) norm, \\Ku\\2 < M 2 ||«|| 2 , V« € £ 2 (fi).
(4.1.8)
1. General theory of integral operators
101
Hence it follows immediately that \\K\\ = M. Second, since k(x, y) G X 2 (fix Q), and {
*(*>y) = £ ^wWw(»).
( 4L9 )
•j=i
where {^>»(x)} is a complete orthonormal system of £ 2 (Q), and ilii= /
I k(xiy)(pi(x)(Pj(y)dxdy.
(4.1.10)
Let n
M * i y ) = ^2 AijWWwiy)
(4.1.11)
be a degenerate kernel, i.e., the range of the corresponding integral operator Kn is finite-dimensional and hence is a compact operator. Since \\K-Kn\\2
= /
\k{x,y)-kn{x,y)\2dxdy
I oo
= £
4
- 0 (n - oo),
(4.1.12)
i,j=n
then if, as the limit of the uniformly convergent sequence {-Kn}, is a com pact operator. □ In the following, we shall prove that the integral operator corresponding to a weakly-singular kernel is also a compact operator. Define *(2)(*,y)= / k(x,z)k(z,y)dz
(4.1.13)
as the iterated kernel of k(x, y), and fc(n)(x,y)=
/
fc(x,2:)fc(n_i)(z,y)(f2:
(4.1.14)
Jet as the n-th iterated kernel of k(x, y). Letfc(,-)(ar,y) be the kernel of If*. The weakly-singular kernel has the following well-known properties: L e m m a 1 . 4 . ^ Suppose that P{x,y) and Q(x,y) are two weaklysingular kernels, and there exist constants A\ and A2, such that |P(z,y)|<% ra
\Q(x,y)\<^,
(4.1.15) rp
102
IV. SEM for integral
equations
s
with 0 < a < s, 0 < /? < s, r = | x - y | = ( ^ ( * t - Vi)2)1 *, ihen for the t=i
kernel R(x, y) = / P(x, z)Q(z, y)dz,
(4.1.16)
the following estimate holds:
\R(x,y)\<{
C,
for a + $ < s,
C | l n r | + Ci,
for a + /? = 5,
[ C/ra+?—,
(4.1.17)
/ora + /?>s,
where C and C\ are positive constants. L e m m a 1.5. Ifk(x,y) is a weakly-singular kernel, then the correspond ing operator K is a compact operator mapping L2(Q) onto L2(Q) and C(Q) onto C(Q). Proof. If Jb(x,r/) is weakly-singular, so is k(y1x). kernel
By Lemma 1.4, the
t(x, y) = / t(x, z)k(y, z)dz
(4.1.18)
Jfi
is also weakly-singular, which is the kernel of the integral operator T = K*K. Repeated applications of Lemma 1.4 show that there exists a suf ficiently large integer n such that t( 2 n)(x,y), the 2 n -th iterated kernel of t(x,y), is a bounded function. By Lemma 1.3, the corresponding operator T 2 is a compact operator. It is easy to prove that if A*A is a compact operator, so is A. It follows that T = K* K is a compact operator, hence K is a compact operator mapping L 2 (Q) onto L 2 (Q). The fact that K is a compact operator mapping C(ft) onto C(Q) can be easily proved by the Ascoli-Arzela Theorem. D Consider the convergence of the operator sequence {l^n}Definition 1.1. a) {Kn} is said to be pointwise convergent to K, denoted by Kn -^> K, if and only if for any u G X, Knu —> Ku. b) {Kn} is said to be uniformly convergent to K, if and only if \\K — #„||->0(n-oo). c) \I^n} is said to be collectively compact convergent to K denoted by Kn —► K, if and only if
1. General theory of integral operators
(%) Kn A K,
103
and oo
(ii) the set S := (J (K — Kn)B
is a relatively compact set in X, where
n= l
B is the unit sphere in X. Using the fact that Kn approaches K, construct the following approxi mate equation of (4.1.1): un-Knun
(4.1.19)
= fny
where fn is an approximation of / . The following problems are of great importance: Pi) The existence, uniqueness, convergence and stability of the approx imate equation (4.1.19). P2) The complexity in solving the approximate equation (4.1.19). and P3) The existence of methods for accelerating the convergence of the approximate solution un. For the problem Pi), we have the following theorem: T h e o r e m 1.1. If (I — K)'1
exists and is bounded, and Kn satisfies
\\K-Kn\\<\\(I-K)-l\\-\
(4.1.20)
then (4.1.19) has a unique solution un and the following error estimate:
llu
u
II < IKJ-^)- 1 H(l|Jg-^n|||H| + ||/-/nH)
, 4 ^
Proof. By I-Kn
= (I= (I-
K) - (Kn - K) K)[I - (I - K)~\Kn
- Kj\,
let Q = (I - K)-\Kn
- K).
It follows from the assumption (4.1.20) that
WQWKWil-K^UK-KnWKl, hence, (I — Q)"
1
exists and oo
IKJ-Qm^EllGH^i/U-IIQII)n=0
(4.1.22)
104
IV. SEM for integral
Substituting into (4.1.22), (7 - Kn)'1
equations
exists and satisfies
(J-iQ-i = (I-Q)-i(/-K)-\ and
Ha-**)- 1 !! <
i-IIQII
,
iie-^ni
, 4123)
which proves the existence, uniqueness and stability of the solution of (4.1.19). For the error estimate, since (I-Kn)(u-un) = u - Knu - u„ + Knun = Ku + f - Knu - Knun - /„ + Knun
=
(K-Kn)u+(f-fn),
then u - un = (I - Kn)'1 [(K - Kn)u + ( / - / „ ) ] .
(4.1.24)
By the estimate (4.1.23), < | | ( I - Kn)-l\\]\\K
- Kn\\ || U || + | | / - fn\
1
^IIU-^)- ll[ll«-^n|||HI + ||/-/„||; l-\\(I-K)-i\\\\K-Kn\) which completes the proof of (4.1.21). Corollary. If (I — Kn)~l
(4.1.25)
D
exists and
(I - Kn)-1-^ Kn±K,
(I - K)~\ fn->f,
(4.1.26a) (4.1.26b)
then un —+ u (n —► oo). Proof. By the Banach-Steinhaus Theorem, (4.1.26a) implies that the operator sequence {(/ - Kn)'1} is uniformly bounded. Therefore by the condition (4.1.26b) and the identity (4.1.24), the proof is complete. D
2. Approximate
quadrature
methods
105
As far as the problem P2) is concerned, the complexity in solving the approximate equation depends on the construction of the approximate op erator Kn. Usually, Kn is constructed as a finite dimensional operator, i.e., the range of Kn is in a finite dimensional subspace. There are several methods to construct approximate operators of an integral operator K: a) b) c) and d)
the approximate quadrature method, the collocation method, the Galerkin method, the degenerate kernel method.
With the exception of a), all these methods can be brought into the framework of projection methods. From the computational point of view, a) is the simplest, followed by b). In method a), no integral evaluation is needed, while in method b), each element in the discrete matrix requires the evaluation of an integral on Cl. The most complicated one is c), in which each element in the discrete matrix requires the evaluation of an integral on Q x Q. Since most of the CPU time is used in the integral evaluation, the Galerkin method has little practical value. However, different methods require different conditions on the smoothness of the integral kernel: the condition required by the approximate quadrature method is stronger, by the collocation method is weaker, and the Galerkin method requires a much weaker condition. For the problem P3), in this book, we concentrate on the method of splitting extrapolation. For details of other methods, please refer to the work of ChatelW20! and I.H. Sloan and V. Thomeet 122 '. 2. Approximate quadrature methods Consider the integral on fi / ( / ) = / f(x)dx
(4.2.1)
/„(/) = 5>i B) /(»0-
(4-2.2)
and the n-point quadrature
*=1
Here, the basis points {y{\ and the weights {w\n'} conditions: Ai) V/ € C(J1), / „ ( / ) - / ( / ) (n - 00);
and
satisfy the following
106
IV. SEM for integral
equations
A2) there exists a constant C independent of n, such that
t=i
Using the quadrature rule / n (/)> one can construct n
Knu = ^ u ; { n ) i ( x > W ) t i ( W ) ,
(4.2.3)
i=i
which is an approximation of the integral operator
Ku= I k{x,y)u(y)dy,
(4.2.4)
and Kn is called the Nystrom approximation of K. Concerning the rela tionship between Kn and i£, AnseloneW proved the following: T h e o r e m 2.1. / / the integral operator K has a continuous kernel fc(x, y) on £2 x fi, anrf /Ae quadrature rule satisfies the conditions K\) and A 2 j, then inC{Q), Knc-$K. Proof. First, by the condition Ai), one can easily obtain (4.2.5)
Kn £ K. OO
Hence it is sufficient to prove that the set 5 = |J (K — Kn)B is relatively n= l
compact, where B is the unit sphere in C(Q). By the Ascoli-Arzela Theo rem, in order to prove that S is relatively compact, we need only to prove that the functions in S are uniformly bounded and equicontinuous. In fact, VuE5,
|A>|<^K( n) ||^,2/i)IK2/OI t=l n
< maxli^.yJIIHIoo^KI t=l
< Cmax|Jb(z,y)|,
(4.2.6)
x,y£Cl
i.e., the sequence {/^n^} is uniformly bounded. Furthermore, it is equicon tinuous, since for arbitrarily x, x' £ fl, \Knv(x) -
Knv(x')\
< C\\v\\oo max \K(x, y) - K(x', y)\.
(4.2.7)
2. Approximate
quadrature methods
107
When \x — x'\ —► 0, the right-hand side tends to zero independently of n. Thus, {/
(J KnB is a relatively compact set. Since K is a compact operator, then n= l oo
S = \J (K — Kn)B is relatively compact. According to Definition 3.1, n=l
Kn " K.
D
In general, the Nystrom approximate method is different from other ap proximate methods (e.g., the collocation method, the Galerkin method and etc.), Kn cannot converge uniformly to K. Theorem 2.1, which guarantees that Kn —+ K, is a theoretical foundation of the approximate quadrature method, because Kn —► K guarantees the following theorem: T h e o r e m 2.2. If K is a compact operator on the Banach space X, 1 is not an eigenvalue of K, and suppose Kn -+ K, then \\K2 - KnK\\ -► 0, / o r n - 4 o o , \\K2-KKn\\-+^ /orn-^oo, and when n is sufficiently large, (I—Kn)~l
(4.2.8) (4.2.9)
exists and is uniformly bounded. oo
Proof. We shall now prove (4.2.9). Since Kn -$ K, then Si = (J KnB n= l
is a relatively compact set, and for arbitrarily small e > 0, there exists a finite e-system: {<7i, • • •,flfm},such that for any Knf £ Si, one can always find <7i, such that (4.2.10)
\\9i ~ Knf\\ < e. However, Kn A K, then, for sufficiently large n, there are \\(Kn-K)gj\\
i = l,--.,m,
(4.2.11)
which imply
\\(Kl-KKn)f\\ =
\\{Kn-K)Knf\\
< \\(Kn - K)(Knf
- 9i)\\ + \\(Kn -
K)9i\\
<(||iC„|| + ||A:||)c + £, V / € B . (4.2.12) Since e > 0 can be arbitrarily small, (4.2.9) is proved. Similarly, we can prove (4.2.8). Finally, by the identity (I-K
+ Kn){I -Kn) = I-K
= (I - K) [/ + ( / - K)-\KKn = ( / - K)[I -(I-
KY^Kl
+ (KKn - Kl) - Kl)) - KKn)},
(4.2.13)
108
IV. SEM for integral
equations
and (4.2.9), for sufficiently large n, there is
WV-Kr^KZ-KKJWKl.
(4.2.14)
This implies that the right-hand side of (4.2.13) is invertible, and so is (J — Kn). Furthermore, (I-Kn)-1 = [l-{I-K)-x{Kl-KKn)}-\l-K)-\l-K The uniform boundedness of the operator {(I—Kn)'1}
+ Kn). (4.2.15) follows from (4.2.14).
D
In the following theorem, by using Theorem 2.2, we prove the conver gence of un to u} where un is the solution of the Nystrom approximation (4.2.16)
(I-Kn)un=f, and u is the solution of the integral equation (Z - K)u = f.
(4.2.17)
T h e o r e m 2.3. Under the assumptions of Theorem 2.1, the Nystrom approximation u„ converges to u. Proof. By (4.1.24), ll« " «„|| < ||(/ - Kn)-^\\
\\{K - Kn)u\\.
(4.2.18)
By Kn —► K and the uniform boundedness of ( / — J £ n ) _ 1 , the proof is complete. □ In practice, the numerical solution of (4.2.16) is divided into two parts. Firstly, solve the linear system n u
n(yi) - ^w$n)k(yi>yj)un(yj)
= f(yt),
i = l,- -,m,
(4.2.19)
to obtain the values of un(x) at the nodes {t/i} t _ 1 ; secondly, set n
un(x) = f(x) + 2 u ; 5 n ) t ( * > y i K ( y i ) ,
(4.2.20)
i=i
which is the required numerical solution. {un(yi)}^_1 of (4.2.19) is also called the Fredholm approximation of u, and (4.2.20) is called the Nystrom interpolation.
3. SEM for multidimensional
integral equations with smooth kernels
109
3. SEM for multidimensional integral equations with smooth ker nels Consider the 5-dimensional integral equation u(x) - / jb(x, y)u{y)dy = g(x), V = [0,1]*. (4.3.1) Jv Suppose that g(x) and k(xyy) are smooth functions on V and V x V re spectively. Divide V by the mesh width h = (/ii, • •, h$) into the union of n cuboids, the widths of its sides are /ii, • • •, /i, respectively: U Vj.
V=
(4.3.2)
7= 1
Let Mj be the center of V}, using the rectangular rule (3.1.2) we construct the Nystrom approximation n h
K u = ] T meas(^)ib(a:,M i )t/(M i )
(4.3.3)
i=i
of the operator i f t i = / k(x,y)u(y)dy,
(4.3.4)
and the corresponding approximate equation uh-Khuh = g.
(4.3.5)
The following theorem shows that the accuracy of the numerical solution of (4.3.5) can be improved by the SEM. Theorem 3.1. If I is not an eigenvalue of K, g E C 2 m + 2 (V r ), k e C ( V x V), then there is a multivariate asymptotic expansion 2m+2
u(x)-uh(x)=
£
h2°wa(x)
+ 0{hlm+2),
(4.3.6)
l<|a|<m
where wa(x) are functions independent
ofh.
Proof. By the assumptions, u G C2m+2(V). It follows from the asymp totic expansion (3.1.4) of numerical integration that there exist functions wa E C 2 m + 2 - 2 l a '(V r ), such that Ku - Khu =
^ l<|or|<m
h2awa(x)
+ O(h20m+2).
(4.3.7)
HO
IV. SEM for integral
equations
By (4.2.18), for m = 0, u-uh
=
(4.3.8)
0(hl).
For m = 1, we shall prove that there exists the expansion u(x)-uh(x)
= J2 h2awa(x)
+
0(ht).
(4.:J.9)
|a| = l
In fact, let eh(x) = u(x) - uh{x). By using (4.3.7), we have (I-Kh)eh
= (K-Kh)u=
^
h2awa{x)^-0{ht),
|a| = l
with wa(x) e C2(V) and efc(x) =
u(x)-uh(x)
= ^^(/-/^r^^+o^) |a| = l
|a| = l
+E
fc2a[(/«^fcr1-(/-^)-i]t5tt(x)+ o(hi).
Letting u; a (x) = (I - K^w^x), and by (4.2.18), (4.3.9) holds. Now suppose that for /, 1 < / < m, there already exist the functions wa(x) independent of A, such that u(x)-uh(x)=
£
h2awa(x)
+
0(h2l+2)
(4.3 .10)
i<M<J
holds. We shall prove that (4.3.10) holds also for replacing / by / + 1. In fact, by (4.3.7), (I-Kh)eh
=(K-Kh)u
=
£
h*°wa(z) + 0{hll+*).
(4.3 .11)
i<M<'+i Construct an auxiliary integral equation (7 - K)va = wa, 1 < |or| < / + 1
(4.3 .12)
and the approximate equation {I-Kh)vha
= wa, 1 < H < / + 1 .
(4.3 .13)
3. SEM for multidimensional
integral equations with smooth
kernels
ill
By the inductive assumption, there exist functions wap, such that
vvaa-v -v\= = a
h^wa, + 0(hll+'-2^),
Y,
i
(4.3 .14)
Substituting (4.3.13) into (4.3.11), one obtains
(I-Kh)(eh-
h2av
J2
°)
i<M+i
= (I-Kh)(ek-
£
h
*"v«
l<|a|+l
+
£
h*°+Vw2a+o(hi<+<))
£
l<|a|<7+ll<|/?|+l-|a|
= (I-Kh)(u-uh-
J2
h2a™a + 0(hll+*))
l<|a|+l
(4.3.15)
= 0(hl<+*). Using Theorem 2.2, the proof of (4.3.6) is complete.
□
Example 3.1. Consider the solution of the two-dimensional integral equation u{x1,x2)
= exp(-xi — x2) -
Jo Jo
x1x2exv(y1+y2)u(yl,y2)dy1dy2.
Its exact solution is u = exp(—x\ —x2)—-xy.
Let the initial mesh width be
h = ( - , - ) , using the Richardson extrapolation method and the Type 1 and Type 2 methods of splitting extrapolation, the numerical results obtained are illustrated in Table 3.1. For h = ( —, — ) , without acceleration by extrapolation methods, the maximum error of the Nystrom approximate solution is 0.00128, the con vergence is very slow. Although the Richardson extrapolation converges faster than the SEM, the CPU time and storage increase rapidly with the number of extrapolations. The SEM not only reduces the CPU time and storage but also has higher order of error.
IV. SEM for integral
112
equations
Table 3.1. The comparison between numerical results of the Richardson and SEM for Example 3.1. number of
Richardson
Type 1
Type 2
extrapolations max error CPU(sec) max error CPU(sec) max error CPU(sec) 2.0E-2
0
2.0E-2
1
1.2E-4
0.18
2.0E-2 1.2E-4
0.11
1.3E-4
0.14
2
5.3E-8
6.67
5.3E-7
0.65
1.6E-6
0.46
2.1E-8
1.64
3
Fig. 3 . 1 .
P l o t of log ( m a x i m u m a b s o l u t e error)
-*— Richardson's extrapolation -*- Type 2 Splitting extrapolation TTl = ?7l-step/split extrapolations 771 = Nystrom method with step size /i/2* 1
Type 1 Splitting extrapolation Nystrom method
E x a m p l e 3.2. Consider the solution of the two-dimensional integral equation u(xx, x2) = exp(xi
+x2)
-[(exp(xi + 1) - l)(exp(x 2 - h i ) - l ) ] / [ ( * i + 1)(* 2 + 1)] + / / Jo Jo
z*v{xiyi+x2y2)u(yl,y2)dyldy2)
with the exact solution u(xly x2) = exp(xi + x 2 ). The equation is solved by the Nystrom approximate method, with an initial mesh width h = ( - , - ) ,
3. SEM for multidimensional
integral equations with smooth
kernels
113
and then is accelerated by the Richardson extrapolation method and the SEM. The numerical results obtained are illustrated in Table 3.2: Table 3.2. The comparison between numerical results of the Richardson and SEM for Example 3.2. number of Richardson Type 1 Type 2 extrapolations max error CPU(sec) max error CPU(sec) max error CPU(sec) 3.6E-1
0
3.6E-1
1
3.0E-3
0.2
4.3E-3
0.14
4.3E-3
0.12
2
1.9E-5
6.56
7.4E-5
0.69
8.6E-5
0.49
6.5E-7
5.36
1.2E-6
1.80
3
3.6E-1
Fig. 3.2. Plot of the number of kernel evaluations
For h =
(T^>7^)>
the maximum error of the Nystrom approximate
solution without extrapolation is 0.02278. The SEM not only improves the accuracy but also reduces the amount of function evaluation and storage. Figure 3.1 gives the accuracy of the four methods, Fig. 3.2 compares the amount of work of the three extrapolation methods, and the amount of storage of these methods are given in Fig. 3.3.
114
IV. SEM for integral Fig. 3.3.
equations
P l o t of s t o r a g e
4. SEM for integral equations in polygonal domains The SEM can also be applied to the numerical solution of integral equa tions on complex domains such as plane polygonal domains. Since a polygon can always be divided into finite number of triangles, we shall discuss firstly the extrapolation of numerical integration on triangular domains. Suppose that A is a given triangle. Consider the integral on this trian gular domain J w i t h y = (2/1,2/2), f(y)
I f(v)dy,
= /(2/1,2/2), dy =
(4.4.1)
dyidy2.
In order to establish the quadrature formula for (4.4.1), one can divide A by successive refinements, that is, the k-ih refinement is obtained by joining the midpoints of the sides of each triangle of the (k — l)-th refinement. After / refinements, A is divided into 4' congruent triangles: A = (J A*i=i hl
Let A
= — measA. One can construct two kinds of numerical integral
formulae: the first one is the rectangular rule 7*1
J
R
=
Ah^f(Mj),
(4.4.2)
4.
SEMforinitegral equations in polygonal
domains
115
where Mj is the barycenter of A j , the other one is the trapezoidal rule 1
*'
(4.4.3) i=l
where Aj, Bj and C, are the three vertices of Ay. For both (4.4 .2) and (4.4.3), we have the following theorem: T h e o r e m 4.1.( 65 ) Iffe
C 2 m + 2 ( A ) , then m
ER(f) = J-JhR>= 5>,-(/)A?< + 0(fcf"+2),
(4.4.4)
t= l
and m
Er(f) = J-Jfr=
£>,(/)/*?* + 0(hlm+2),
(4.4.5)
1= 1
where Wi(f) and Wi(f) are constants which are independent of hi = 1/2' and depend on f. This theorem will be proved later. Consider the evaluation of the following integral equation of the second
kind
u(x)=
/ k(x,y)u(y)dy Jn
+ /(x),
(4.4.6)
where Q is a polygonal domain. According to the Nystrom method , (4.4.6) can be computed by the following steps: S t e p 1. Subdivide the polygonal domain Cl into an initial triangulation such that
ft = U Qt. S t e p 2. For each triangle, join the midpoints of the sides and thus obtain the next refinement, after /, refinements, ft,- = (J A , j , n, = <*'',(»• = 1
\
1, •••,*•)• S t e p 3. Solve the linear system r
^(Mij) -J2Y,
nk
&hkHMij,Mkt)uh(Mki) = /(Ma),
i= 1, ••,»*, j = ! , - • • , «i
IV. SEM for integral
116
equations
and thus obtain {uh (My), i = 1, •••, r, j = 1, • •, n , } , where A^ = meas(ft,-)/4*S hi = 2 -'*, i = l , - - , r , fc = (fci,---,ftr)> ^o == max/i,-. S t e p 4. Compute the Nystrom approximate solution uh(x) z Jb = l
* = 1
For simplicity, (4.4.6) can be expressed in the operator form u — Ku = / ,
(4.4 •7)
and the corresponding Nystrom approximate equation is u h - J^t/*1 = / ,
(4.4 .8)
with Rhv
= E E Aji(x,Mfct)t;(Mfc0.
Now we prove the following expansion theorem of the Nystrom approx imate solution: T h e o r e m 4.2. If f E C 2 m + 2 (ft), *Ae ifcerne/ k(x,y) G C 2 m + 2 (ft x Q), and 1 is no/ an eigenvalue of K, then there exist functions wa(x) G C 2 m + 2 ~ 2 l a l(Q), which are independent of h = (/i1} • ••, hr), such that u(x)-uh(x)=
h2awa(x)
J2
+ 0(hlm+2).
(4.4.9)
l<|a|<m
Proof. Under the assumptions, u G C 2 m + 2 (ft). By Theorem 2.1, Kh ^ J£. Let e,l(ar) = tx(ar) — wh(x) be the error of the Nystrom approximate solution. Hence by Theorem 4.1, it follows that
(I-Kh)eh
(K-Kh)u
= = f r
h(x9y)u{y)dy^^f^A^k(xiMij)u(Mij) m
'EE^w+o^.
(4-41°)
»=ii=i
with fity G C 2 m + 2 - 2 i ( f i ) . Similar to that in Theorem 2.3, we shall prove inductively the asymptotic expansion (4.4.9). For the case m — 1, it is easy
4. SEM for integral equations in polygonal
domains
117
to prove that there exist functions Wi G C 2 (Q), which are independent of ft, such that r
«(*) - uh(x) = £&?«;,•(*) + 0(ht). »= 1
Suppose that for / < m, the asymptotic expansion u(x)-uh(x)=
h*awa(x) +
]T
0(hl'+2)
(4.4.11)
l<\a\
holds. Now we shall prove that it also holds for replacing / by / + 1. Con struct an auxiliary problem (I - K)vij =Wij, t = 1, • • •, r; j = 1, • • •, / + 1,
(4.4.12)
and the corresponding Nystrom approximation is
(I - K ^
= wih i = l,...,r; i = l , - " , / + l .
(4.4.13)
By the inductive assumption, we have aa +4 w +4 aa +4- w w-)- )-v^= J2 J2 **aa *"y-+o(*? "y-+o(*? -v^= -v^= J2 "y-+o(*? )1<|<*|<M-1-J 1<|<*|<M-1-J
Vij Vij Vij --
(4.4.14)
l<M
Substituting (4.4.13) and (4.4.14) into (4.4.10), we obtain
(I-Kh)(eh
rrr
-r
rl+l (+1 r/+1 /+1J+l l+l
t : = 1 ::=1 t : = 1 =1
ii
hVv\ hVv\
= (I-Kh)(u-u»-J2!£h?vij 1= 1 >= 1
/+1 rr r/+1 /+1
hfh w , +, o(hl'+*)) + o(hl'+*)) hfh £ hfh w w , + o(hl'+*)) £ «=i i = n < | o | < / + i - i
l 1< |o|
= 0(ft*+ ).
2a 2a 2a ijc ijc ijc
(4.4.15)
Because tty-, u>tja are independent of ft, and Kh —» K guarantees that (I—Kh)~l exists and is uniformly bounded, the proof of (4.4.9) is completed by grouping the terms with the same power. □ Note that the number of the mesh parameters ftt- (i = 1, • • •, r) men tioned above is not the dimension but the number of triangles in the initial triangulation. This shows that lower dimensional problems can also have many independent mesh parameters. Because only one mesh parameter is
IV. SEM for integral
118
equations
refined at each time, the more the independent mesh parameters, the less the amount of work and storage needed by the SEM, and the higher the degree of parallelism. For example, let ti§ = u(hi} • • •, hr) be the solution with the initial triangulation, and u1* = ti(Ai, • • •, hi/2, • •, hr) be the solu tion with one refinement in ft,-, then the error with only one split of SEM is
* " ^ 4 X > ? - (*" - *)<] = °C*o),
(4.4.16)
where tig, uf (i = 1 , - -,r) can be evaluated in parallel, which are more economical than the Richardson extrapolation. We now go back to the proof of Theorem 4.1. The asymptotic expansions (4.4.4) and (4.4.5) are very useful not only in the SEM but also in the evaluation of three-dimensional boundary integral equations. Suppose that a triangle A is divided into 4' elements after / refinements. Let $ = {1, • • •, 4'}. A,-, i G $ is called an element with positive direction if it can be obtained by a parallel and similarity transformation from A. The numbering set of these A,- is denoted by ^"^. Let 3>~ = $ \ 3>+, then each A,- (i G $ " ) is called an element with negative direction. Consider a fixed side of A, say AB, for each element Aj (j G $ " ) , there exists a unique consecutive A,-, i G $ + , such that Aj and A,- form a parallelogram /^7j and their common side is parallel to AB. Let 0 be the set of all numberings of the elements with positive direction without pairing element. These elements have a side on AB. In the following, denote by rrij (j G $"") the center of /Z7j and denote by my (j G $°) the midpoint of the side on AB of A ; (j G $°). In order to prove Theorem 4.1, the following lemmas are needed. L e m m a 4 . 1 . There exist constants Ca and Da independent of hi = 1/2', satisfying f (y-Mi)ady
= Cah[alAh\
ie$+,
[ (y-Mi)ady
= (-l)°Cah[alAh\
j (y-mi)ady
= DahiflAh*,ie90,
(4.4.17a) i e r ,
(4.4.17b) (4.4.17c)
'Ai
ft
/
Jcyi
VM J °' a
(y - mi) dy = i I 2Dah[a]Ah>,
2 W
f '
2||a|.
i€$~.(4.4.17d)
4. SEM for integral
equations
in polygonal
domains
119
Proof. If |a| = 1, it is trivial that Ca — 0. Suppose that i G $ + , i.e., A,- is an element with positive direction. Take a transform y — Mi = hi(z — M) : At- —» A, where M is the barycenter of A, then [ (y- MiYdy
= h[al+2 I (z - M)adz,
\*\ > 1.
JA
JAi
Set h\ Ca = ^ A"
1
r
(z- M)adz,
(4.4.18)
JA
which is a constant independent of hi. Therefore, (4.4.17a) holds. Further, for i E $ ~ , take the transform y—M{ = —hi(z-M) : A,- —> A, f (y- Mi)°dy = (-irh[a{+2
f (z-
M)°dz.
JA
JAi
It follows from (4.4.18) that (4.4.17b) holds. (4.4.17c) can be proved in a similar way. Because rrij is the symmetric center of CJj, (4.4.17d) also holds. D L e m m a 4.2. If f E C 2 m + 2 ( A ) , then there exist constants &,(/) (i = l,---,ra) independent of hi, such that
Ei(f) = f f(y)dy-2
£
/(m^A** - £
/(m^A^
m
= 5>(/)^+0(A?m+2).
(4-4-19)
*=1
Proof. It can be proved by induction. For the case m = 0, by a simple estimate, (4.4.19) holds. Suppose that (4.4.19) holds for m < k - 1, we shall prove that it also holds for m = k. By Lemma 4.1, £
i(/)= E / (/(»)-/K)) d w+E / (/(y)-/^))^
jLL{y-mj)2adyf{2a){mi)
=£ E +E
E
-i / (»- -i) a ^/ (a) K)+o(Af +2 )
120 120
IV. SEM SEM for for integral integral equations equations IV.
= EE EE(2a)! ^ ' •gzt^W^) W'-W = je*~
i<\<*\
je*~ i
——I v ; *
+ EE EE ^ | a^hr^f^imj) W 2 «>(m,-) + jG*°l<|a|
att*
'
a| +a + EE EE ^^ AV->(wj)+o(*? )) + ^/w^j+o^ je$°\<x\eik je$°\<x\eik
= -1<M<* E ^^l''MMf™)+ + E
E 1<M<* ^
2a) f \y)dy ^*l/ wy {2a
''
A
^ l A ^ / ( « > ( m i ) + 0(/if+2),
E
(4.4.20) (4.4.20)
jj€^°|a|€/fc €*°|o|€/fc
where I* I* — — {1, {1, 3, 3, •• •• •, •, 2k 2k 44- 1} 1} stands stands for for aa subset subset with with odd odd numbers. numbers. By By where 2a the inductive inductive assumptions, assumptions, Ei(f( Ei(f(2a ^) (1 (1 < < ||a| < k) k) possesses possesses the the following following the ^) a| < expansion with with even even powers powers only only expansion
£?!(/('«>) = J2 b2aJh? + 0(hfk-l«'+1>),
(4.4.21) (4.4.21)
where &2a,j &2a,j are are constants constants independent independent of of hi. hi. On On the the other other hand, hand, by by the where the Euler-Maclaurin expansion, Euler-Maclaurin expansion, ^—' a! a!
= =
x
£
E E Ai/^i) >*i/(a)(™;)
a!
j€$°
ftW-^^/Ja ' , ••'
,
a
/
f(a)(vi(t),
V2(t))dt - d2hl
, 2 i b - | a j - l , ^ / , 2Jb-|<*|-l\l
, rf jk-|a|-lfti , 2 i b - | a j - l +, 0^( f/f tt, !2 J b - | a | - l)\ lj , 2 <*2fc-|a|-lfci + 0( l )j, VMEJfc, VHG4,
(4.4.22) (4.4.22)
where cf, cf, are are constants constants which which are are independent independent of of hi hi but but depend depend on on // and and where a, and (yi(t) y (t)) is a point on the line AB. Substituting (4.4.21) and 1 2 a, and (yi(t)1y2(t)) is a point on the line AB. Substituting (4.4.21) and (4.4.22) into (4.4.20), the proof of (4.4.19) is complete. D (4.4.22) into (4.4.20), the proof of (4.4.19) is complete. D Proof Proof expansion expansion
of Theorem T h e o r e m 4.1. 4 . 1 . First, First, we we shall shall prove prove that that the the asymptotic of asymptotic (4.4.4) of the rectangular rule holds. By induction, for m m = = 00 (4.4.4) of the rectangular rule holds. By induction, for
4. SEM SEM for for integral integral equations equations in in polygonal polygonal domains domains 4.
121
(4.4.4) obviously holds. Suppose that (4.4.4) holds for m = k -— 1, then for
£ / (f) == J2f
E E(f)
(f(y)-f(Mj))dy (m-f(Mj))dy
E == E
E E
2 (» -- M^dyf^Mj) + 0{h?») i3 // (y M^dyf^Mj)+o(^*+ ) JA > ' k+2 "^i
i £ * l<|«l<2Jb+l je* l<|a|<2fc+l
where where
(4.4,23) (4.4.23)
= Ji + J2 + 0(hl ), = / i + J 2 + 0(/i?*+ 2 ),
= EE EE h = ^
j€*l<\a\
7£y/jLfiy-M^dyf^iM,) (y-Mj^dyf^Mj)
2o) = EE EE % ^ ^ / ((2o) w = %^^/ w))
2 ! l [/(2^)-/(2a)W)]^ = - E E %^T/
~ k s f e , < °> ^ + i <£M < *^*l' M [ 1 1
; ;
''
fi^wv-
A A
4 4L24) ((4.4 - - 24 )
i<M<* ' By the inductive assumptions, Ji can be expanded in even powers; of h\. By the inductive assumptions, J2 can be expanded in even powers of h\. Second, Second, r
/■'"'A**
0) ** = = EE EE ^ i ^ / (^-^m (^) ii€€$$||aa||G G//ffcc
E E E == E
L {a) ^!^(M ^!^-[fL[f{a) (M j)-&HMp} j)-&HMp}
je*-\a\eik je*-\a\eik
+ ^E ^E
+
je*o\<*\ei je*o\<*\eikk = J3 J3 + + J4, ^4, =
^ f^w) ^^-^m
a!
(4.4.25) (4.4.25) 1. Hence, Mf is the bary barycenter of the the pairing pairing element element of of Aj Aj (.; (j G G $~). $~) Hence, where Mj" center of there exists a vector a, which is independent of /ii, such that Mj = = rrij m;- + ftia, Mj M;* = = rrij raj — — /iia. hia. Mj Since
^a\Mj)-^(M;)
122
IV. SEM for integral
equations
= E s^/C^KO + o^r2"1^), Me/* We/*
|/?|<2fc-a \p\<2k-a
then
* - E EE
j € * - | ao r | € / *
E
|/?|€/fc W\<2k-\a\ |/?|<2*-|«l
^ -b^r■^/*"'♦- ^ " o" -^. )
+0(/i + 0 ( / l2f*+ + 22))..
(4.4.26)
On the other hand, Mj = m, + /ii
*= E E ^ A ^ / W ^ + M)
= r ^ / 1( /^'^) "+^) =EEEE EE ^^V^ \6\<2k-\<*\ |/?|<2fc-|a|
+E E je*°\a\6ik
E \p\eik |/?|<2*-|a|+l \(3\<2k-\a\+l
r
/i|a+/?la/J
„,,, ^f^\rnj) 'p'
= JJ5 ++ /6 ( / ik^++2).2 ) . = J6 ++ 00(hl
+ 0(hl^) (4.4.27)
By Lemma 4.2, JZ + JQ can be expanded in even powers of h\. By the EulerMaclaurin expansion of integral on AB, J$ has an asymptotic expansion with even powers of h\. In summary, J\ = J 3 4- J$ + «/6 has an asymptotic expansion with even powers of h\. Therefore, the proof of (4.4.4) is complete. Concerning the proof of (4.4.5), we note that
£ETr{f) (/)
= £ / (/M ./(A) + W)+/(a))dv^ =£/ im.m±m±m j-l^A, 3 j=iJ±i 4'1 4' , 4
3
4'1 , 4'
4 (f(v)-f(Mj))dy-,,£\[
= =E E //
(f(v)-f(Mj))dy- £\[
4 ! , 4' ' i r ~ E (f(Bj)-f(M j))dy - E ^W/ (f(Bj)-f(Mj))dy i=i
3
^ i
,
(/(A^-fiMjfidy (/(A^-fiMjfidy
4. SEM for integral equations in polygonal domains 4
' 1 /
' 1 r -- £EW W 3 3
j=l i=i
j
123
(m)-m))dy. (fm-fMDdy.
A] ±i
J
(4.4.28)
(4.4.28)
By (4.4.4), the first term possesses an asymptotic expansion with even powers of hi. For the second term, by the Taylor expansion,
//
JA,
hl = =A Ahl
( /(/(A^-fiM^dy (^-/(M,))^ 1 m 2 m+2 /f(a^\(MM ).). (4.4.29) j)(A, J) ^ - -- M,-)*/" Mj)a/a\ ++ 0(hl 0(&J+ (4.4.29)
5Yl3
l<|a|<2m+l l<|«|<2m+l
There exists a vector a independent of j and hi, such that
Aj-Mj=ahu
VjE*+, Vje*+,
A,- - M,- = -—afti, a/ii,
Vj E $ ~ ,
and when |a| is an even number, we have 4'
i=i 41
a , , aa,, hhll ((a) = 53a ^ a a / /i i 11 A ( M ii )/a! )/a! = A // a ) (M
= ^
E
/
[/(O)(M,)-/<«>()]*/
f_^_ y/ /W( + +—h f(a)(y))rfdy-
(4.4.30) ( 4A3 °)
By (4.4.4), in this case there exists an asymptotic expansion with even powers of hi. When \a\ is an odd number, 4'
Y^A^f^iMjKAj-Mjr/al , a,
-= 5J2 3 a^ , 11a , A^(/W(M jj)~/W(M;))/a! + 53 aafc|a|AfcV(a)(JW;)/a! = J7 J7 + + // 88 ,, =
(4.4.31)
where J7 and Jg have asymptotic expansions with even powers of hi, the proofs are the same as those for J3 and J4. Using the similar argument,
124
IV. SEM for integral
equations
one can show that the asymptotic expansion (4.4.5) holds for the third and fourth terms on the right-hand side of (4.4.28). D Corollary. Define the Simpson rule of numerical integration on trian gular domains as follows: J^ = ljRl
+ \ 4 \
(4-4.32)
then the error satisfies the asymptotic expansion Es{f) = w2h\ + w3hl + ... + wmhlm
+ 0(hlm+2).
(4.4.33)
Proof. By direct verification, J^1 (/) has an algebraic precision of de gree 2, hence the h\ term in the asymptotic expansion of Es(f) vanishes, that is, there exist constants Wi (i = 2, • • •, m) independent of Ai, such that (4.4.33) holds. D The following asymptotic expansion was given in the monograph by H. Engelst28^ ET(f)
= C2h2 + C3h3 + CAhA + • • •.
(4.4.34)
In our proof, this result is improved to one which includes even powers only, and so the extrapolation results will be more accurate. E x a m p l e 4 . 1 . Evaluate the following integral: ex+ydxdy,
/ JA
where A is a triangle with vertices (0,0), (0,1) and (1,0), and its exact value is 1. The extrapolation results obtained from the rectangular rule, trapezoidal rule and the Simpson rule are as follows: 1° The rectangular rule 0.97386027 0.97316288
0.99959403
0.99827062 0.99997340
0.99999869
5. SEM for eigenvalues and eigen functions
125
2° The trapezoidal rule 1.07276061 1.02015570
1.00262078
1.00516571
1.00016905
1.00000560
3° The Simpson rule 0.99859042 0.99991064
0.99999865
0.99999439
0.99999998
0.99999999
According to (4.4.34), /^-extrapolation error of Ex{f) will be 2 x 10~ 4 , but the error obtained in this example is just 5 x 10~ 6 , which is only about l/40 of the former. In addition, the rectangular rule and the trapezoidal rule have the same order of error, but the latter needs only (2l + 1)(2* + 2)/2 nodes, while the former needs 4l nodes. Hence, the latter is superior to the former, and this superiority will be more obvious in the evaluation of integral equations. 5. SEM for eigenvalues and eigenfunctions Consider the eigenvalue problem of the integral equation / k(x,y)u(y)dy = Au(x), (4.5.1) Jn where Q is a bounded domain in Ms, k(x, y) is a smooth function o n f i x f i , and so the integral operator I k{x,y)u{y)dy (4.5.2) Jn is a compact operator mapping C(Q,) onto C(fi). Let In(f) be the quadra ture formula defined by (4.2.2), satisfying the conditions Ai) and A2), Kn be the Nystrom approximation of K, which is defined by (4.2.3). Construct the approximation eigenvalue problem Ku=
(4.5.3) and consider the evaluation of the first eigenvalue (i.e., the one with the largest absolute value) of K.
126
IV. SEM for integral
equations
Suppose the first eigenvalue A of K is simple, since Kn -^ K, the first eigenvalue An of Kn is also simple, and An —* A (cf. [20]). It is impossible to compare the approximation of the eigenfunctions without standardization. Let ij) be the eigenfunction of K*, i.e., (\-K*)i/>
(4.5.4)
= 0,
where A is the conjugate complex of A and K* is_the conjugate operator of K. By the properties of the compact operators, A is the simple eigenvalue of K*, hence ip is unique except for a constant factor. Take a fixed %j>y the standardization conditions for u and un are chosen as <«,V> = 1, («n,^) = l,
(4.5.5)
where (•, •) stands for the pairing of elements in X — C(Q) and its conjugate space. We now estimate |A — A n | and \\u — un\\ under the condition (4.5.5). By using (A - K){u - «„) = -Xu„ + Kun = (A„ - A)«„ + (Kun - Knun),
(4.5.6)
one obtains 0 =
{(X-K)(u-un),i,)
= (A„ - A)(u„, ip) + {Kun - Knun,rl>) =
\n-\+j{K(K-Kn)un,4>),
hence |A„ - A| = 0(\\K(K
- Kn)\\ I K H ) .
(4.5.7)
We shall now prove that {||txn||} is bounded. Note that the restriction of A — K in the subspace V={v:{v^)
(4.5.8)
= 0}
is a bounded reversible operator, and u — un G V, hence, by using (4.5.6),
II«-«„II < C(|A - A„| ||u n || + ±-\\{K - Kn)Kn\\
IKH),
where C is a positive constant independent of n. Due to |A-A„|^0,
\\{K-Kn)Kn\\-+Q,
(4.5.9)
5. SEM for eigenvalues and eigen functions
127
and
ll«»ll
- Kn)\\), - Kn)\\).
(4.5.10a) (4.5.10b)
Theorem 2.3 guarantees the convergence of the Nystrom approximate solu tion of the eigenvalue problem. If the kernel is symmetric: Ar(x, y) = k(y, x), then the corresponding integral operator K is self-conjugate, which implies that the eigenvalues of K are real numbers, and the set of eigenfunctions forms a complete orthogonal system in L2(Q). In order to apply the SEM to the eigenvalue problems, suppose that Cl = [0, l] 5 , and the Nystrom approximation Kh of K is defined by (4.3.3), then the approximation eigenvalue problem is Khuh
= Xhuh.
(4.5.11)
The following theorem gives SEM for eigenvalue problems of integral oper ator with symmetric smooth kernel. T h e o r e m 5.1. Suppose that ft = [0,1] 5 , the kernel k(x,y) of the in tegral operator defined by (4-5.2) is symmetric, and Ar(x,t/) = k{y,x) G C 2 m + 2 (fl x ft). Again, suppose that X is a simple eigenvalue of K, and u, uh are eigenfunctions corresponding respectively to X and Xh, then there exist constants aa and functions va(x)y which are independent of h — (/ii, • • •, hs), such that the following asymptotic expansions hold: Xh = X+
Yl
a«h2a + O(h20m+2)
(4.5.12a)
l<|a|<m
and uh = u+
£
a
2 a
+ 0(C+2).
(4.5.12b)
l<|a|<m
Proof. First, when / = 0, the estimate (4.5.10) implies that \Xh - X\ = 0{hl) and \\u - uh\\ = 0(hl). Second, suppose that for / < m, there exists an asymptotic expansion up to h2a (\a\ — I). Now we shall prove
128
IV. SEM for integral
equations
that for the case / + 1, there also exists an asymptotic expansion up to h2a (|a| = / + l ) . Consider (A* - Kh)(u - uh) = Xhu - Kh u
\u + {\h = (K-Kh)u
-X)u-Khu + (\h-\)u
=
=
wa(u)h2°+
£
£
a*fta«ti + 0(ftg , + 4 ).
(4.5.13)
Here we use the asymptotic expansion (4.3.7), and assume
\h-X=
«aft2or + 0(/»o'+4).
J2
(4-5!4)
1<|«|<J+1
where {a£} are undetermined constants. Construct an operator equation tofind{5*,o*}, (Ah - Kh)v* = wa(u)+Z*u,
(4.5.15a)
(wa(u) + ahau, uh) = 0.
(4.5.15b)
{Ah,w'1} is the simple eigenpair of Kh, which guarantees that the system (4.5.15) has a unique solution. It follows that a* = - ( « ? « ( « ) , « ) .
(4.5.16)
Now construct another operator equation for finding { v a , a a } , (A - K)va = wa(u) + aau,
(4.5.17a)
(iZa(u) + aau,uh)
(4.5.17b)
= 0.
The condition (4.5.17b) implies a a = -(ffi a (ti),ti),
(4.5.18)
and the solution va exists uniquely. By the inductive assumption and wa(u) £ C 2 m + 2 ~ 2 ' a '(fi), there exist constants bap, which are independent of h, such that a*-aa=
^ E l<|0|+l-a
- J 2/l . 2 /+ ? _0(fc? i . n ^ +4 + --2|tt| H ). »«/ift
h
2/
4
2
(4.5.19)
5. SEM for eigenvalues and eigen functions
129
By (4.5.19) and Theorem 3.1, there exist functions wap{x) independent of h such that
?*-*« =
^ f t w + o(fc?+4-aw).
£ l<|/?|<J+l-<*
Substituting these results into (4.5.13), it becomes
(\h-Kh)(u-uh-
vhah2a)
]T l<|a|
h
h
h
= (\ -K )(u-u -
Yl
^h2a
l<|a|+l
-
E
E
l<|a|+l l<|/?|+l-|a|
»<*Aa(fl,+')+o(Ay+4))
= 0(fc* + 4 ).
(4.5.20)
Noting that
which implies that the left-hand side of (4.5.20) belongs to Vh — {vh : (vh,uh) = 0}, so that the restriction of (Xh — Kh) on Vh is invertible, and the inverse operator is uniformly bounded. Hence, by (4.5.20),
u - uh -
v°h2a
Yl 1<|<*|<J+1
- E
E ».**<*»
l<|a|+ll<|/?|+l-|a|
Grouping the terms with the same power offt,we obtain
uh - u -
Yl
l<|a|-fl
v h2a
°
= °(hll+4)-
(4.5.21a)
In addition, substituting (4.5.19) into (4.5.14), it follows that \
h
- \ -
]T
aah2a = 0(hll+4).
(4.5.21b)
1<M+1
Therefore, the asymptotic expansion holds for / + 1.
□
The symmetry assumption on k(x,y) in Theorem 5.1 is unessential. It can be proved that the asymptotic expansion (4.5.14) also holds for
130
IV. SEM for integral
equations
nonsymmetrical kernels. But the standardization condition (4.5.5) makes the proof more complicated. In the following, only the expansion containing the h2 terms is proved. Theorem 5.2. If ft = [0,1]* and the nonsymmetric kernel k(xiy) £ C 4 (Q x Q), then there exist constants a,- and functions Vj(x) (i = 1, • • •, s), which are independent of h, such that 5
XAh - X T a ; / i ? + O(ftJ), A A = ]53««*? O(^),
(4.5.22a)
ti — = i1
and 5s
uh-u- u = Y^vihl
+ °(ht)-
(4.5.22b)
t=i
Proof. Apply the method of undetermined coefficients. First assume that (4.5.22) holds. Evaluate 55
5 5
5 5
K\u + J2 viht) vih}) ~(\ + J2aih?)(u + J 3 vihf) Vihf) K\U+53 - (A+53 aih?)(u+53 «= 1 «=i
1= 1 »=i
5
1= 1 »=i
5
5
2 iKhVi - ^ Y,hfcLiU X]Th v{ ++ O(Aj) = Khuu - Xu + JY^h T &?#**,ft?a,-ti - A ] T2ft?t;,O(Aj)
i=i
*=i
t=i
55
fc 2 = ti -- Ku) (KhhVi iU = ifti Ku -- Ati Xu + + (# (Khu Ku) + + Y Y hh1(K Vi -- a am - Xv Xv{{)) + + O(Aj) 0(h*) 55
5 5
i=i
»=i
ii = = ii
h h = Yh ViVi -- aiauiU- - Xvi) X )2iWi{u) h2iWi{u) ++ ]Thf{K ] T h2(K At;,-)++ 0(ht) OfaS) 5*
5 5
i=i '=1
»=i
= 53 ] T A? (Xi;,. -- A Xvi T fc?(lif*«,iU ++ffii(ti)) = *?(#«< w<- -aa,« £,(«))++] 53 *?(#*«<- - lift;,-) #»i) +0(ht). +o(K).
(4.5.23)
Choose Vi and at- (i = 1, •• •, s), satisfying the operator equations Kvi — Xvi=—a,u aiU Wi(u), - Xvi -— Wi{u), (atu Wi(u), V > ) = 0, i ==l ,1,. .• -•, •, (aiu u?i(ti), t/>) « .s.
(4.5.24a) (4.5.24b)
Since A is the simple eigenvalue of Ky then (4.5.24b) guarantees that the solution Vi satisfying (viy V>) = 0 exists uniquely, and a, (wi(u),i>). ai = {wi(u),xl>).
(4.5.25)
5. SEM for eigenvalues and eigeniunctions
131
Due to u e C 4 (ft), it follows that Wi(u) £ C 2 (Q), and so v.- G C 2 (fi), hence \\Kvi-Khvi\\
(4.15.26)
t = l , . " s,s.
= 0{hl),
It follows from (4.5.24) and (4.5.26) that 5
5
h
K (u + J2 vihf) - (A + j ; o,-fc?)(u 1= 1
t=l
1= 1
(4- 5.27)
= o(^). 1
1
Note that A' and tx' satisfy Khuh -\huh
= 0.
(4- 5.28)
Subtracting (4.5.27) from (4.5.28), we have s
Kh(uh -u-J2
vih?) - \h(uh - u »=i
_(A*_A-X)a,-fc?)(« + E i«^?) 1=1
1= 1
(4- 5.29)
= 0(h*). Let V'/l he the standardized eigenfunction of the eigenproblem
(&-(Khy)*ph = o. Acting the operation (-,il>h) on (4.5.29), one obtains s
s
1=1
1=1
O(fcJ).
(4-.5.30)
Since V,/l —► Vs then 3
(u + 5 3 Vih2i^H)
"" (w' ^) = 1, as ^
-o,
i=i
which gives 5
A " - A - £ a , / i ? = 0(/i4). 1= 1
Substituting into (4.5.27), it leads to s
{Kh - \h)(uh - ti - 5>?«.-) = C ( ^ ) . 1= 1
(4 .5.31)
IV. SEM for integral equations
132
Note that Kh - Xh in the invariant subspace Vh = {vh : (vh, ^h) = 0} is invertible as well as uniformly bounded. Although s
uh-u-y52h?vi
= g<£Vh,
«=i
the spectral project of g on Vh satisfies 9-
,
h
lhu
€Vh.
And it follows from (4.5.31) that
9-r^
u h
=0 ^ -
(4 5 32)
--
By using the standardization condition (4.5.5) and (vt, V>) — 0, one obtains <<7,V>h) (uh,i/>h)
=
i + ^ i (u\i>») n
-I {uhAh) = o(ht).
Substituting into (4.5.32), it becomes s
g=
Uh-U-Y,hiVi=0(hi). 1=1
The proof of (4.5.22) is complete.
D
6. SEM for integral equations with nonsmooth kernels Theoretically, the extrapolation methods and the SEM for integral equa tions with nonsmooth kernels are far from perfect. Q. Lin and J.Q. Liu''"' have discussed the extrapolation algorithms of the one-dimensional Green's kernel, which is now introduced as follows: Consider the one-dimensional integral equation «(:<)- ■ f k(t,T)u(r)dT Jo
= g(t).
(4.6.1)
6. SEM for integral equations with nonsmooth kernels
133
Suppose that 1 is not an eigenvalue of the integral operator, &(£, r ) is a Green's kernel, i.e., (4.6.2)
*(t,r) = { k2(tir))
r>t,
where fci(*,r) and k2{t,T) are smooth functions, and ki(t,t) = k2(t,t). These facts show that fc(£,r) is a continuous function on [0,1] 2 , but its derivatives are discontinuous at t = r. Rewrite (4.6.1) as u(t)-
I i i ( t , r ) t i ( r ) d r - I k2(t,T)u(T)dr
= g(t).
(4.6.3)
Take the mesh width h\ = 1/n and the nodes tj — Tj = jhi. Construct, by using the trapezoidal rule, the Nystrom approximation of the integral operator K
»" *»/(*) = - £ ^ . ^ / ( T J ) ,
(4.6.4)
with //
£ *(*.*i-)/fo) j=0
+k(i, r i ) / ( n ) + • • • + *(t, r n . i ) / ( r n _ i ) + - i ( t , r n ) / ( r n ) and the approximate equation ti„ - i £ n u n =flf.
(4.6.5)
The values {u n (£,)} of un(x) at the nodes are obtained from the linear equations >
Un(U)
hfc,
Tj)un(Tj)
Tj
Yl Tj>ti
k2(ti,Tj)un(Tj)
= g(ti), i = Or--,n.
(4.6.6)
134
IV. SEM for integral equations
Now we shall prove that there exists a function w(x) independent of n, such that «»(*<) = «(**) + HU)/n ™(ti)/n2 2 + 0(n-4),
(4.6.7)
i.e., at the nodes <,• (i = 0, • • •, n), the approximate solution can reach the order of error 0(n~4) by extrapolation. Note that 1 n" " "(*«)- ~nrrE HU,Tj)u(Tj) nrr n r,11 j = 0 I! n ""
T T = 9(ti) = // tk(ti,T)u(r)dT--J2 ( « i , r ) « ( r ) d r - - V tHti, f t . }M ^ u fi)o ) ++ $(*,•)
= { //
fci(^,r)u(r)drfcl(ti,T)«(T)dT-
+{ /
- ]T Tj )U(T,-)| T fc^tj, fci(tj,Tj)«(Tj))
*k22(U,T)u(T)d,T (*i,r)«(T)dT
- r V^ M*« h(ti,Tj)u(Tj)\ Y) > i)«(rj)i
+g(ti), •••,!», +$(*•)> «'t = 0, ••-,«,
(4.6.8)
and the Euler-Maclaurin expansion, U f/"*' 11 vv-^" ^" r fci(ti,r)w(r)
=
T -ii^^^'^))|;:%o =-is?^^' ^))i;ro+o((Bn-), -4), Z"1 / k2(ti,T)u(T)dT /
1 v-^ " >
fc2(t,-,r)u(r)dr--
=
1
d
-Y^TM^
> T)U{T))\
k2{U,Tj)u(Tj) k (ti,TJ)U(TJ) |T =2 1
+ 0(n-- 4 ) .
let let let w(t) = - ^ [ s : ( M ' * , r M r ) ) | ; % | : ( * a ( « * , r ) t , ( r ) ) | ^ 1 ] >
5(0=
~iyi^ i( '*' r)u(r) C + ^ (fc2(
substituting into (4.6 .8), one obtains
1 "" substituting into«(4.6.8), ( * * ) - one - £ obtains *:(<»> T;)u(r.z) nrrn 1 w(tt) " = (<«)+ - ^ 2 i + 0 ( n - 4 ) , t = 0, •••,!».
(4.6.9)
= *(*<) + - ^ 2 i + 0 ( n - 4 ) , t = 0 , - - . , n .
(4.6.9)
j=o
6. SEM for integral equations with nonsmooth kernels
135
Since w(x) is a smooth function, the solution w of the integral equation (I-K)w = w
(4.6.10)
(K-Kn)w = 0(n-2).
(4.6.11)
is also smooth, and so
Hence, (I-Kn)(un = 9-(I= 9-(I= g-(I-
-u + n~2w) Kn)u + n-\l - Kn)w Kn)u + n~2(I - K)w + n-2(Kn Kn)u + n~2w + 0 ( n " 4 ) .
- K)w (4.6.12)
Let t — ti (i = 0, • • •, n) in (4.6.12), which forms a linear system with un knowns v(U) = un(ti)-u(ti)-n~2w(ti) (i = 0, • • •, n). Kn ^ /^ guarantees that the inverse of the coefficient matrix of (4.6.12) exists and is uniformly bounded and (4.6.9) shows that the right-hand side of the system is of 0(ji~4)y hence, un{U) - u(U) + n~2w(ti) = 0{n~A),
i = 0, • • •, n.
(4.6.13)
T h e o r e m 6.1. If the kernel of the integral equation (4.6.1) is the Green kernel defined by (4.6.2), and one is not an eigenvalue, then un(ti) (i = 0 , - - , n ) , the solution of the linear system (4.6.6), have the asymptotic ex pansion (4.6.13). Theorem 6.1 guarantees that the nodal values can be extrapolated, while it may not be true for the non-nodal values determined by the Nystrom interpolation (4.2.20). A detailed analysis can further prove that (4.6.13) has a higher order provided ki(x}y) and £2(2,2/) are smooth enough. The proof is omitted here. The disadvantage of the global extrapolation mentioned above is the large amount of work involved in obtaining the approximation u-in* The SEM reduces the amount of computation and storage, and it can be im plemented on multiprocessor computers in parallel. For this aim, divide r
the interval [0,1] into r subintervals: [0,1] = (J /,-, U = [Tv_i,rt-], where »=i
0 = ro < Ti < • • < Tr = 1, i = 1, • • •, r, and I,- are not necessary isometric. In /,-, insert (nt- — 1) isometric nodes: TtJ- = T,_I -fj/i, (j = 0, • • •, n,), hi = (T,- — T,-i)/n t -, i = l , - - , r are r independent mesh widths. Define the
136
IV. SEM for integral
equations
Nystrom approximate integral operator as rr m ni "" Kh*f{t) * /« = E 5 3E 5 3 WKt,7*i)/(7y). *.-*(*, T ^ / t e ) .
(4-6.14) (4.6.14)
t=lj=0
tij = r , j , (i = 1, • • •, r; j = 0, • • •, n t ), and {ti {uhfc(ti(*j Let ty m)} m )} satisfy the linear system // -- ll ni ni 'I a
h U hik1(tim,Tij)uhhfaj) (Tij) Uh(Um)-^2J2 (*Jm)-53X3 Ml(*lm,^>
»=1 j=0
m n, m" " m "" t J 3 ^ iM m^, i^^) f^a^j ) - 5 3 ft'*2( Im,^j)lifc(77J-) -5 f a^m l*2(*lm,T>j)u*fa) j—0 j=0 j=m j=m r
ni n< a "
- 5 3 1 3 ftf*2(tIm,nj)wfc(rO')=^(t/m), =00 t = /J++lUj =
/ = 1, •■•,!•; m = 0,---,n/.
(4.6.15)
Noting that {u($jm)}> the values of the solution u(x) at the nodes, satisfy rr
ufam)
m ni II" fc
- 53X3 tt = = ll jj = = 00 JJ -- ll
,,
" " ii " "
fc fc i(^m,^)w(r)c?r-53
= = 5 53 3^ ^ // t=l
«'M*fm,T,j)utej)
i(^m,^)w(r)c?r-53
^
A A «'*i(*im,njMrij)}
«'*i(^m,ni)ti(rij)}
i=0
r)u(r)dr - 5 3 &i(
/hlkl l/ 1
T ^ (*/m' ^i V)} ^ ))UU((^J)}
i=o 0
**2(*Jm,r)ti(r)dr-53 2 ( * J m , r ) t i ( r ) d r - 5 3 ft/fcft2(^m,rzj)ix(r/j)} /fc2(^m,rzj)ix(r/j)}
+{ / r
.
m»
+ 5 3 { / M*Jm,r)ii(r)dr-53 i=J+l
^
A
»'*2(*Jm,njMr,-,-)}
j=0
+0(*/m), / = 1, • • •, r; m = 0, • • •, n,.
(4.6.16)
Applying the Euler-Maclaurin asymptotic expansion on every term on the right of (4.6.16), we have r w
ni a
(^m)~53X3 t = l j=0
^K^mjTij^Tij)
6. SEM for integral equations with nonsmooth kernels ,_1
=
137
1 d ft ?
E-i2 « ^N*'m'rH'-)]|
+
tim T)u
-^'{^^^^t., i^ ' ^0 r
1
fi
tt
- E ^ ^ V N ^ . ^ M ] ' +^/m) + 0(^), (4.6.17) •=/+i
i Z
a r
u<
~1
with ho = maxft,-.Let l
-j2^[klit'T)uiT)] l t i - i >
'
for t > t{,
~^Mt,rHr)} l t i - l > for t < U.lt
Wi(t) = <
(4-6.18)
-B0^ [ i b l ( t ' T ) u ( T ) C- 1 + | ^ < ' r ) u ( r C } for tj_i < £ < t{,
z = l , - -,r.
i
Obviously, W{(t) are functions independent of h. Therefore, by (4.6.17), I
ni "
»=1J=0 r
= ^«!.(<;m)ft, ? + ff(tim) + O(ftJ).
(4.6.19)
t= l
Using the argument similar to that used in proving Theorem 6.1, we can prove the following asymptotic expansion for splitting extrapolation. T h e o r e m 6.2. If the assumptions of Theorem 6.1 are satisfied, and [0,1] = U I,-, /,- = [r t _i,r t ], where 0 = r 0 < 7i < • • • < r r = 1
(4.6.20)
is a fixed partition, hi = (r,- — r,_i)/n,- (i = 1, • • • ,r) are isometric refine ment mesh widtf 15 of I{, and {uh(tim) : / = 1, • •, r; m = 0, • • •, n/} is the
138
IV. SEM for integral
equations
solution of the Nystrom equation (4.6.15), then the nodal values of uh(t) have the asymptotic expansion
uh(tlm) - u(tlm) = J2wi(tlm)h? + 0(h*),
(4.6.21)
»= 1
where Wi(t) (i = 1,• • •, r) are functions independent of h. Furthermore, if i i ( t , r ) G C 2 m + 2 ( Q i ) , k2(tyr) G C 2 m + 2 (ft 2 ), ^ i = {(*, T) : 0 < t < 1, 0 < r < t < 1} and ft2 = {(*, r ) : 0 < t < 1, 0 < * < r < 1}, one can then obtain the asymptotic expansion of higher order: Uh(tlm)-U(tlm)=
]T
^ ( ^ m ) f t 2 a + O(/i 2 0 m + 2 ),
l<|<*|<m
where wa are functions independent of h. If ii(-, •) and k2(-,-) are nonsmooth on ^ i and Q2l for instance, k(t, r) = \t- r|"tf(t, r ) , 1 > /i > 0,
(4.6.22)
where H(t, r) is a smooth function on [0,1] x [0,1], then *!(*, r ) = (t - r)"ff (t, r ) , 1 > * > r > 0,
(4.6.23a)
k2(t, T) = (T- t)»H(t, r ) , 1 > r > * > 0.
(4.6.23b)
and
Obviously, the derivatives of ki(t, r) and k2(t) r ) are not continuous. How ever by using the Euler-Maclaurin expansion (1.2.38) with the singularity at the end points, one can still obtain the following: T h e o r e m 6.3. If the kernel of the integral equation (4.6.1) is defined by (4.6.22), 1 is not an eigenvalue, and (u n (i,)} is the Nystrom approx imation determined by the equation (4.6.6), then there exist functions w^{t) and Wi(i), which are independent of n, such that un(U) - u(U) + w^n-v-1 i = 0,
-,n.
+ w^n-2
=
0(n"2^^), (4.6.24)
6. SEM for integral equations with nonsmooth
kernels
139
Proof. By (4.6.8), 1 " "
"(*'•)-~y) *(**> Ti)u(ri) = { /
(*,- - T)»H{h, r)u(r)dr
- V
n-^U
,i
- r^HiU, T3■Mrj)}
» "
+ { / ( r - < , - ) " f f ( * i , r ) u ( T ) d r - 5 3 n " 1 ^ - - *,)"#(*i > T j) u ( r i)} 7 —*
+9(U) (4.6.25)
t = 0, • -,n.
= Ji + J2+9{ti),
By using (1.2.39), we have Ji = n - " - 1 C ( - | i ) / r ( t i ,<,>(<,) - n - 2 C ( - l ) | - [ ( * i - r)"i/(<,-, r)ti(r)]| + 0 ( n "2 " " ) , at/ lr=o /2 = - n - " - 1 C ( - p ) f f (<<,«.■)«(«.•) + n - 2 C ( - l ) ^ - [ ( r - ti)"H(tit oy
T)U(T))\
lr=i
+ 0{n~ 2 - / A
Let fi^t)
=
-2<;(-f*)H(t,t)u(t),
wtf)
= -C(-l){|-[(*-ry'tf(*,r)U(r)]|T=o
_-![(<-,-)"#(*, r)U(r)]|
}.
Substituting into (4.6.25), we obtain «(*.•) -
=
1 1
n ^
nn
"
k(U, Tj)u(jj)
Wfx{U) +
1
i? m(U) + 9(U)
+ 0 ( n - 2 - ^ ) , i = 0.--.,n.
(4.6.26)
Let u;^ and wi satisfy respectively the following equation: (I - K)w^ - uip, (I - K)wi = w\.
(4.6.27)
IV. SEM for integral
140
equations
It is easy to prove that (K - Kn)wp = O f a - 1 - " ) , (K - Kn)w1 = C ^ n " 1 - " ) . Hence, (I - Kn)(un - u + n " 1 - ' 4 ^ + n" 2 tx;i) = < / - ( / - tfn)u + n - 1 - ^ / - X ) ^ + n " 2 ( / 1
=
g
K)w1
i
+n- ~> (K - X n ) ^ + rT\K - 2Cn)wi - (/ - 7
(4.6.28)
On both sides of (4.6.28), let t = U (i = 0 , - - , n ) . By (4.6.26), each component of the column vector on the right is 0 ( n ~ 2 - / i ) , but Kn —+ K guarantees that the inverse of the coefficient matrix on the left exists and is uniformly bounded, which implies that (4.6.24) holds. □ Corollary. If the kernel function is k(t, T) = \t - r|" In \t - r\H(t, r ) , 0 < \i < 1, then there exist functions w^^^w^t) n} such that
(4.6.29)
and w(t), which are independent of
un(ti) - u{U) + Wnitfin-"-1 Inn + wpL(ti)n'fA"1 + ^ i ( ^ ) n " 2 = 0 ( n - ^ 2 ) , i = 0, -,n.
(4.6.30)
Proof. It follows from the expansion (1.2.58) and the proof is similar to that of Theorem 6.3. D By means of induction which has been used repeatedly, the asymptotic expansion up to a higher order can also be obtained, provided that H{t,r) is sufficiently smooth. Further, Theorem 6.3 can be extended to higherdimensional integral equations, provided that the kernel can be expressed as
*(*.*) = n; = i {l*i - ViHlnte -% I)*'}#(*,),
(4.6.31)
where x = (x x , • • •, x,), y = (y1} • • •, y 5 ), 0 < /xj < 1, Xj = 0 or 1. By using the asymptotic expansion (3.4.8) of the quadrature formulae, one can obtain the multivariate asymptotic expansion of the Nystrom approximation at nodes. Obviously, the SEM can improve the accuracy of the approximate solution. For the one-dimensional nonsmooth integral equations, Theorems 6.2 and 6.3 can be generalized to the following:
6. SEM for integral equations with nonsmooth
kernels
141
Theorem 6.4. Suppose that the kernel of the integral equation (4.6.1) has the form of (4.6.22), and one is not an eigenvalue of the integral op erator. Again, suppose that the nodes {tim : / = 1, ••-,?•; m = 0, — , n r } and the mesh width h = (/ii, • • •, hr) satisfy the assumptions of Theorem 6.2, then there exist functions W{(t) and w t/i (t), i = 1, • • •, r, such that the values of the Nystrom approximate solution uh(x) at the nodes have the asymptotic expansions uh(Um) - u{tlm) =
w^(t1m)h}+ti
•=1
for 1 < / < r, 0 < m < nu
(4.6.32a)
and uh(tj) - u(U) = wi-ltr(ti)ti±»
+
w^(U)h^
r
+ Y^Wi(tl)h? + 0(hl+>i)) / = l,..-,r. (4.6.32b) t=i
Proof. Since tim £ Ii (i / /), then fc(J/m,r) is smooth on Jt- (t ^ /), and nonsmooth on //. On the right-hand side of (4.6.16), using the EulerMaclaurin expansion for the error of the numerical integration on 7t- (i ^ /), while using the expansion with singularity at the end points for the error of the numerical integration on I\ = [2/,2/m] U[*'m>*i+i]j (4.6.32a) follows from the same argument used in proving Theorem 6.3. Similarly, since tj £ 1^ (i z£l — 1, /), k(ti, T) is smooth on 2, (i ^ I — 1, /) and nonsmooth at the end points of J/_i and //. Hence, on the right-hand side of (4.6.16), using the Euler-Maclaurin expansion for the error of the numerical integration on Ii (i z£ /, /— 1), while using the expansion with singularity at the end points of //_i and /,, (4.6.32b) holds. D The previous analysis shows that the Nystrom approximate solutions of the integral equations with nonsmooth kernel defined by (4.6.22) or (4.6.29) has a lower order of error, which is 0(/iJ + / i ) or 0(/iJ + / i | ln/i 0 |) respectively; and after extrapolation is 0{h2^^) or 0 ( / I Q + / 1 | ln/i 0 |) respectively. Kantorovich and Krylov^ proposed an algorithm with higher order of error, called the modified quadrature method. Its basic idea is to describe the integral equation (4.6.1) in the equivalent form [l - / Jb(t, r)dr\ u(t) + / .*(*, r) \u(t) - ti(r)] dr = g(t), (4.6.33)
142
IV. SEM for integral
equations
and to construct, by the quadrature rule, an approximate equation ,1
1
n »
[1 - / k(t, T)dT]u(t) + -J2 Kt, Tj)[un(t) Jo nfT'o
«B(TJ)]
=
9(t),
or equivalently, 1 " " [1 + - V k(t,TJ)-
f1
k(t,r)dr]«B(0
1 n " — Y,Kt,Ti)un(Tj)
= g{t).
(4.6 .34)
Take J = r,- (i = 0, • • •, n), then (4.6.34) is converted into linear equafrions of the nodal values {w n (r,), i = 0, - , n } . After getting {u n (r,)} and substituting into (4.6.34), one can obtain
^ (f\ —
1 n " 9(f) + -^2 Ht>Tj)Un(Tj) i5° n
j=o
(4 fi.35)
J
°
The modified quadrature method was studied by Baker'"!, Anselone' ' and Chatelint 20 !. Intuitively, the approximate equation based on (4.6.33) has higher order of error. In fact, if the solution u(t) is smooth, then the expansion of the quadrature formula is /
k(t, r)[u(t) - u(r)]dr - -J2 =
-iS?|:W''r)W«)-tt(T))]r;i + o ( ^ 7 ) ,
because —[k(t,T)(u(t) <9T
*(*, Tj)[u(t) - ufo)] (4-6 .36)
— u(r))] is not only continuous but is also of the
form \t-r\»H{t,T). In order to discuss the convergence and stability of the approximate equation (4.6.34), (4.6.35) can be expressed in the operator form (4.61.37)
(I-Kn)un=g, with Kn = Kn - (Knv - Kv)I,
v e C[0,1], v = 1.
(4.61.38)
7. SEM for ID weakly-singular integral
equations
143
Kn are denned by (4.6.4). Since fc(x, y) is continuous on [0,1] x [0,1], then Kn ^ K, which implies that Kn -^ K, and \\Kn - Kn\\ = \\Knv - Kv\\ - 0, (n -> oo).
(4.6.39)
Consequently, iCn —► I<\ Hence for sufficiently large n, (I — Kn)'1 exists and is uniformly bounded, which proves the existence and uniqueness of the solution of the modified quadrature formula (4.6.34). Repeating the argument used in the proof of Theorem 6.4, one can derive the SEM for the approximation obtained from the modified quadrature method. T h e o r e m 6.5. Under the assumptions of Theorem 6.J^, the solution of the modified quadrature method, at the node tjm has the following asymp totic expansion: r
h
u (tlm)
- u(tlm) = £ > ( t i m ) f t ? + O(h30+^).
(4.6.40)
»=i
/ / the kernel has a logarithmic singularity, described in (4.6.29), then the remainder of (4.6.40) is of 0(/io + / i |ln/io|)7. S E M for collocation-method solutions of one-dimensional weak ly-singular integral equations Consider the one-dimensional weakly-singular integral equation u(t) - / Jb(*, r)u(r)dr Jo
= g(t),
(4.7.1)
where the kernel k(t> r) is a function of the form jb(t, r ) = |i - r\aH{t, r ) , - 1 < a < 0,
(4.7.2a)
k(t, r) = \t - r\a In \t - r|, - 1 < a < 0.
(4.7.2b)
or
Since fc(t,r) is discontinuous on [0,1] x [0,1], the approximation operator {i£ n } derived from the quadrature method does not collectively compact converge to K. The numerical results show also that the approximate solutions obtained by the quadrature method diverge. At present, the main methods for solving the weakly-singular integral equation are as follows: a) the collocation method, b) the product integration method,
IV. SEM for integral
144
equations
c) the modified quadrature method, and d) the Galerkin method. The Galerkin method has a higher order of error. However, the gener ation of the elements in the coefficients matrix involves the evaluation of multiple integrals, it leads to a larger amount of work and so may make this method impractical. The principle of the product integration method is similar to that of the collocation method. In this section, only the collo cation method and its extrapolation are introduced. Both the collocation and Galerkin methods can be brought under the framework of the projection method. The difference between them is that the former bases on the interpolation projection, and the latter on the orthogonal projection. In order to construct the interpolation projection, construct the first initial partition of the interval [0,1]: 0 = r 0 < r i < . - . < r r = 1.
(4.7.3)
r
Let I{ = [ri-i, rj], I = |J /,-, and refine the fixed interval I,- (i = 1, • • •, r) »=i
by the mesh width hi = (r,- — r t _i)/n,-. Let r,j = r,_i + jh{ (i — 1, • • •, r) and regard ft,- as an independent mesh parameter. The set {T,J} forms a partition on [0,1] : 0 = T0 < Tn < . . . < r l n i = n < < r r _ i < rr>1 < • • • < rr = 1.
(4.7.4)
h
Let S [0,1] be the set of the piecewise linear continuous functions under the partition (4.7.4), and denote by u1 = nhu the interpolation function of the continuous function u on S^O, 1], i.e., Hhu G S^O, 1], and rtufaj)
= ii(r t j ), i = 1 , . . . , r; j = 0, • • •, n,-.
(4.7.5)
Obviously, the linear operator is a projection operator and is called the interpolation projection. Applying nh, one can obtain a collocation solution uh(x) G S^O, 1] from the following operator equation: uh - nhKuh
= nV
(4.7.6)
The following lemma guarantees the existence of uh and the stability of the equation (4.7.6). L e m m a 7.1. If (I — K)'1
exists and is bounded, and ho = max A,l<*
is sufficiently small, then (I -
C[M].
rfK)'1
exists and is uniformly bounded in
7. SEM for ID weakly-singular integral equations
145
Proof. By Lemma 1.5, K is a compact operator. Obviously nh A 7.
(4.7.7)
It will be proved that (4.7.7) implies || nh K - # | | — 0, (ft0 — 0).
(4.7.8)
In fact, suppose that B is the unit sphere in C[0,1], and KB is a compact set, then for any e > 0, there exists a finite ^-system {vi, • • •, v,-}. By (4.7.7), there exist 6 > 0 and when ho < 6,
I K - n ^ H ^ , i = i,...,jv.
(4.7.9)
Take arbitrarily w £ B, then there exists v,- such that ||/^-t;t||<e.
(4.7.10)
Hence,
|| n" Kw - Kw\\ < ||(i - n > , | | +1|(7 - r\h)(Kw -
Vi)\\
< ( 2 + ||n"||)£.
(4.7.11)
According to the Banach-Steinhaus Theorem, (4.7.7) implies the uniform boundedness of {||n/l | | } . Since e can be arbitrarily small, then (4.7.8) holds. The conclusion of Lemma 7.1 follows from Theorem 1.1 and its corollary.
□ We now convert the operator equation (4.7.6) into a linear system. Let {^fj } be the basis functions of 5^[0,1]. That is,
{
1,
for i = /, j — m,
0,
otherwise,
i = l , - - - , r ; i = 0,...,n,-.
(4.7.12)
Set Mh
r
r»i
W = EEa*iW(0-
(4-7.13)
Substitute into (4.7.6) and obtain the linear system with respect to {a,j} : dim ~ ]T) ILS aii /
k Tlm T (
(
' ) Pij(T)dT
7=l,...,r; m=l,..,n,-.
= 9{T\m)> (4.7.14)
IV. SEM for integral
146
equations
After solving {a t J } from (4.7.14), one obtains the collocation-method so lution uh(t). Further by using (4.7.1) and uh(t), one can also obtain an iterative solution uh(t) = g(t) + I uh(r)k(t, r)dr. (4.7.15) Jo Obviously, (4.7.14) shows that at the collocation points, uh(t) = uh(rim). In the Chatelin's book!20', (4.7.15) is called the Sloan iteration. For the Galerkin method, Sloan and Thomee^ 122 ' has expounded in detail that the Sloan iteration can improve the accuracy, but for the collocation-method solution, it is not the case. What we are interested in is whether the extrap olation method can improve the accuracy. In the case of smooth kernels or the piecewise smooth kernels (the Green kernel), Q. Lin and J.Q. Liu gave a positive answer. Concerning weakly-singular kernels, F. Chatelint 20 } said: "If the kernel is not smooth, the method is no longer founded theo retically." The following lemma states the fact that the difference between Ehv, the orthogonal projection from L2(Q) to 5^(0,1), and the interpola tion projection \lhv has an asymptotic expansion. For details, see the work of P. HelfricrJ44!. L e m m a 7.2. Assume that v G # 4 ( 0 , 1 ) , then the asymptotic expansion E»v
-nhv = -J2 ^ f ( f t V ' ) + q(v) t=i
(4.7.16)
iZ
holds, where E^ : L2(I{) —► Sh(I{) is an orthogonal projection, rji is the characteristic function on /,-, and the remainder q(v) has the estimate ll«(«)IU»(o,i) < Cl4\\v\\Ht(0il).
(4.7.17)
Proof. By the Euler-Maclaurin expansion (1.2.12), the integral on the interval J,j = [Ti,j-iiTij] (* = 1> *, r ; j = 1, •••,»!,•) has the asymptotic expansion
/ w(i)dt = —'- f w"(t)dt +/i 4 /
wW{t)PA((t
- Ttj-i)/hi)dt,
Vw G # 4 ( 0 , 1 ) . (4.7.18)
Let w(t) = (v- vT(t))iP(t), where v G # 4 ( 0 , 1 ) , v1 is the interpolation of i>, and ip G Sh[0,1]. Substituting into (4.7.18), we have
/
Jin
(vW-v'itfimdt
7. SEM for ID weakly-singular integral equations
147 147
= -ii/./ww* = -fnJj.v"imt)dt
™-jf k w - ^ w w * +hf [t;<44>(<)W) + 3t; 3»<(s)8(<)^(*)] >(<)^(t)] +hf f/ [»< )(<)^(0 + •fry ■Pi((t 7ij-i)/hi)dt. •P4((* - nj-i)/hi)dt.
(4.7.19)
Noting that the second term on the right-hand side vanishes, and by (1.2.6a), PP4((t nj-i)/hi) 4((< - TiJ-j/hi)
= hiPi((t hiPL((t -
TiJ^/hi). rij-i)/hi).
Substituting into (4.7.19), and integrating by parts,
/f („(*)-,/(<))>(<)<«
•fry
== "-%l ^ / 12 12
Jin Jlij
v"(t)mdt + h} I[ v^\t)i>{t)dt v"(t)mdt+h} j*\twt)dt Jin Jlij
-h* -/if /[ 3v&(t)1>'(t)ft((t-nj-i)/hi)dt. 3«< >(iW(t)P5((t-7ij_i)/Ai)
(4.7.20) (4.7.20)
Hence,
/
Jo ./0
(v(t) -
v^t^^dt
= ("(O-^C))^*)* =E E EE/ / Wt)-At))*W t=l
7 j7== 1 l ^7»i
,=1
i2
-fr.
»=1
-- X > «? ? £E // !> «=1 »= 1
j =
JlJl
-fr*
4 4 7 21 3«< >(Otf'(t)A((*-n,i-i)A0
ll iJ'i
By the inverse estimate^**}, there exists a constant Co independent of /i,-, such that 1 k WIU»(i y ) < Coh^WnvVv), (4.7.22) II^HL»(J«) < Gofer II^IIL»(/«), VV VV- €€ s5*[0,1]. p>, i]. (4.7.22) It follows from (4.7.21) that for any i/> 6 Sh[0,1], 7
h
2
2 7 (v (O,I) = C ^ f - vw'. , V0L (O,I) (t; - vt>7,, V0L V>)i»(o,i) = (^ ^)i 2(o,i)
148 r
V h2
v
12
IV. SEM for integral , ^ ) £ a (A)
rr h? A? = - Yl TZ^id'y
i=l
iZ
equations
+ «(»)
^)L>(0,1)
+ «(f)
c l = = --Et^|f ^ ( ^( v^ ) ^") )L '» ^ ( ow , i ))+ «+ ( v?W. ).
4 7 23 ((4.7.23) -- )
where rji(t) are characteristic functions on I,-, i.e.,
{
1, l,
0,
Ti_i < < < Ti, T»-I << n,
otherwise.
(4.7.24)
Since
l«(«)l
(4) (4) (4) ft <<^Eii« ^ E n« iu»eoiMM'o+ah* E Ew(4) ii«ii^('«)ii^ii^«) iu»(i«)ii^iU'(/y) iU'('oii^iU'(io+^ oEEii *=i r
«=ij=i
r
< ^ ( I*=i > ( 4 ) I I L xi/nizunh ■(/«>)* tt==ii *=i
^hu ± f; n^)|ii,(/y)) * ( E E I iwiWo-)) wiWo-))** t = i ii == i t'=i
<
»=ij=i
(4.7.25)
C^II^H^O.DIIVIU^O.I),
where Ci and C are constants independent of h{ (i = 1, • •, r), then (4.7.25) and (4.7.23) imply
£*„ _ vi = £ _£*( V„'')^L Ehv -v'=J2 -^W ' ) ^ ++ ,(«), «(«), i=i
(4.7.26) (4.7.26)
^
and II«(«)HL»(O,I) II«(«)I|L»(O,I)
4 ^ Cfc CfeJHvllff^o,!). < IM|ir«(o,i)-
The proof of Lemma La 7.2 is complete.
(4.7.27)
D
Under the conditions that v = # 4 ( 0 , l ) n # o ( 0 , 1 ) and the mesh widths satisfy hi = ft2 = • • • = hr, Helfrich^ obtained h\ h A? h l , v = -—E v" Ehvh w-n - H g(v). f? w = - —1Zf 7 h w / / ++ «(»).
(4.7.28) (4.7.28)
7. SEM for ID weakly-singular integral equations
The argument here does not need the assumption v G
149 HQ(Q,1).
Since
r
2_] i]i(t) = 1, (4.7.28) is derived from (4.7.16) when the partition is isometi=i
ric. Lemma 7.3. Suppose that X is a reflexive Banach space, {Pn} is a sequence of projection operators, and Pn —► I for n —► oo, then the conjugate operator P* of Pn is also a projection operator of the conjugate Banach space X* f and P* -+ J,
for n -f oo.
(4.7.29)
Proof. Since Pn is a projection, then P% = P n , which leads to P*2 = P*, and P* is a projection operator of X*. In order to prove (4.7.29), we shall first prove that ~oo
U ft(Pn*) = X*.
(4.7.30)
n= l
where 5J(P,J) is the range of P*. Without loss of generality, suppose ^(P*) C &CPn+i), n = 1,2,--.. If (4.7.30) is not true, then there ex ists v, satisfying 0 / v G l = X**. For any tp G X* and any n, there is
(P^v)
=
(^Pnv)=0.
Let n —* oo, one obtains (V>, v) = 0, which implies i; = 0, this is contrary to the assumption v ^ 0. Second, by ||P*|| = ||P n ||, (4.7.29) implies that {||P*||} is uniformly oo
bounded. For every w G (J ^(P^*), there must be a sufficiently large integer i=i
m, such that tu G 3J(P^), hence, P , > = W, n>m.
(4.7.31)
oo
Applying the denseness of (J 5R(P^) in X* and the Banach-Steinhaus Theo»=i
, the proof is complete.
O
Corollary. Under the assumptions of Lemma 7.3, and ifK is a compact operator mapping X onto X, then \\K{I - Pn)\\x
- 0, f o r n ^ o o .
(4.7.32)
Proof. By P* — I (in X*) and p C ( I - P „ ) | | x = \\(I-PZ)K*\\X.,
(4.7.33)
IV. SEM for integral
150
equations
noting that K* is a compact operator of X*, using the argument that was used in the proof of Lemma 7.1, the proof can be completed. □ In the previous theorem and corollary, the reflexivity of X is an indis pensable condition. If Pn is the interpolation projection of C[0,1], (4.7.32) is invalid. In the following theorem, the norm || • || is taken in L 2 (0,1). Theorem 7.1. Suppose that u E # 4 ( 0 , 1 ) 25 a unique solution of (4.7.1), u is a unique solution of the collocation equation corresponding to the par tition (4.7.4), and u1 is an interpolation of u, then there exist functions Bi(u) (i = 1, • • •, r), which are independent of h — (/ii, • • •, hr) and depen dent on u, such that h
r
uh(t) - uT(t) = ^h?Bi(u)
+ 0(Afci),
(4.7.34)
»=i
with €l
= ||(/ - nh)K\\ + ||(I - Eh)K* || - 0, h -+ 0.
(4.7.35)
Proof. By (4.7.1) and (4.7.6), uh - u1 = nhK{uh h
- u) h
= n K(u - u1) + nhK(uJ - u), hence,
(J - nhK)(uh - u1) = nhK{u* - u) = tfKiu1 - Ehu) + nhK(Ehu
- u),
(4.7.36)
where Eh is an orthogonal projection mapping L2(0,1) onto Sh[0,1]. It follows from Lemma 7.1 that (I-^K)-1 exists and is uniformly bounded, hence, uh - u1 = (/ - n ^ ) " 1 n h Kiu1 - Ehu) -{I - nhK)-1
nh K(I - Eh)u
= Ji+J2-
(4.7.37)
First, by Lemma 7.1 and the error estimate of the interpolation of the linear finite element, we have
IWI < ve mf
||(7 - nVO" 1 n" K(i - Eh){u - v)\\
(4.7.38)
7. SEM for ID weakly-singular integral equations
151
Second, by Lemma 7.2, it follows that tfK)-1
Jx = (I -
Hh K{u* -
= (7 - K^Kiu1
-
1
Ehu)
Ehu)
+ [(/- n ^ ) - n* -(I - K)-1)^ = (7 - KYxK(v*
- Ehu)
- Ehu) - (7 - n * # ) - 1 ( J - nh)K(uT
- .Ehu)
- ( / - nh7s:)"1(7 - nh)7v:(7 - K)-1 nh Kiu1 - Ehu) = (7 - K)-1^1 - Ehu) + O(h20\\(I - nh)K\\) = - £ S ( J - Ky^KE^mu) + o(h20\\(i - nh)i<\\) t = l
1 Z
= - £ j | ( J - KY^iviu) + 0{hl\\K(I - Eh)\\) +0(hl\\(i-nh)K\\).
(4.7.39)
Let Bi(u) = — T^(I — K)"1K(rjiu),
i = 1, • • •, r, one obtains
r
(4.7.40) t= l
Corollary. For the nodal values of the piecewise linear collocationmethod solution uh(t), there is the following asymptotic expansion r
u\nj)
~ «(TJ>) = £
h?Bi(u(nj))
+
O(h2o£l).
(4.7.41)
»= 1
Further, the iterative solution uh defined by (4.7.15) can be extrapolated even at non-nodes. T h e o r e m 7.2. Under the assumptions of Theorem 7.1, there is r
uh(t) - u(t) = J2h2Bi(u(t))
+ O(Agei).
(4.7.42)
»= 1
Proof. By (4.7.15),
u-uh
=
K{u-uh)
= K[(I - KYl9 -{I- rfKY1 n" g] = K[(I - KY1 - (J - rtKY^g + K(i - n h 70 _1 ( J - nh)a = K(i - nhi
+ g) (4.7.43)
152
IV. SEM for integral equations
Substituting the identity K(I - nhK)-1
= (I -
Kn^K
into (4.7.43), u - uh = (I - KH^Kil
- n h )u
= (I - KH^KiE11 1
- nh)u + (7 - Knh)~lK(I
h
= (I - K)- K(E
+[(/ - # n*)
Eh)u
-
h
-f\ )u _1
- (7 - K)-x]K{Eh
-
nh)u
+(I-Kr\h)-1K(I-Ehu) = (I-K)-1K(Eh -nh)u -(I - Kn^Kil
- Uh)(I - K)-1K(Eh
fc
--n
)u
+(I-Knh)-1K(I-Eh)u = J1 + J2 + h-
(4.7.44)
By the identity
(7 - AXI*)-1 = I + K(i - nhK)-1nh, one obtains that {||(7— A'n A ) —1 ||} is uniformly bounded, hence there exists a constant M > 0, such that
l|/2||<M||(/-nh)^llll(^-n;'H| =
0(hl\\(i-nh)K\\)
|| Jsll < M\\K(I - Eh)u\\ < M\\K(I - Eh)(u -- « J ) i i <Mp:(j-.Efc)||||t«-t,'|| = 0(hl\\(I-Eh)K*\\). By Lemma 7.2, J
i = - E i|( z - *r1*£*(*•«)+O(AS)
= - E § i(7 - 'o-1* OH+(J - * r ** a-£*)(.»«)} ■ +0(ft$). Setting B,-(u) = - l ( I - J i r ) - 1 l f ( i K i i ) > t = ! , . . - ,r,
8. SEM for MD weakly-singular integral equations
153
and substituting into (4.7.44), it becomes r
«(*) - «*(*) = 5>?Bi(«) + O(fcfci). t= l
The proof is complete.
D
8. SEM for collocation-method solutions of multidimensional weakly-singular integral equations Consider the SEM for the weakly-singular integral equations on plane polygonal domains. Suppose that ft C M2 is a plane polygon. Consider the integral equation on ft u(x) = I k(x, y)u(y)dy + g(x),
(4.8.1)
where x = (a?i,x 2 ), y = (2/1,2/2), dy - dy^dy2, and k(x,y) is a weaklysingular kernel expressed by (4.1.3) or (4.1.4). For weakly-singular integral equations, the Nystrom method introduced in Section 4 cannot guarantee the convergence of the approximate solution, one has to use the modi fied quadrature formula method, the collocation method or the Galerkin method. Now consider the convergence and the splitting extrapolation of the collocation method. r
Let ft — (J ft, be the initial triangulation of ft. Each triangle ft, is t=i
divided into n,- = Ali smaller congruent triangles after /,- refinements, i.e., m ft, = (J Aij (i = 1, • • •, r). Let /i,- = 2~li (i — 1, • • •, r) be r independent i=i
mesh widths, h — (hi, • • •, /i r ), and ho — max hi. l<»
Denote by Shi(tt) the linear finite element space on ft,-, if v G Sfci(ftt-), then v G C(ft,), and v\ is a linear function. Again, let Sh(Q.) = {t)E Loo(ft) : v | n . G 5 h i (ft,), 1 < i < r } . Denote by AihBij and C^ (i = 1,. -,r; j = 1,- - , 4 n 0 the vertices of A,-,. n h : C(ft) -► S*(«) is an interpolation projection, i.e., Hhv G ^ ( f t ) , and nfct;(ZJ
A,B,C]
i = l , " - , r ; j = l,--,n,,
(4.8.2)
Thus construct the collocation equation ufc = HhKuh + n V (4.8.3) By means of the basis function on 5 h (ft), similar to (4.7.13) and (4.7.14), (4.8.3) can be converted into a linear system. The conclusion of Lemma
154
IV. SEM for integral
equations
7.1 remains valid, that is, if (I — K)~x exists and is bounded, and h is sufficiently small, then (I—rfK)'1 exists and is uniformly bounded. If uh has been found, one can then obtain the iterative solution uhh = Kuhh + g.
(4.8.4) h
h
The comparison of (4.8.3) and (4.8.4) shows that u and u take the same values at the nodes. Theorem 7.2 can be generalized as follows: Theorem 8.1. If I is not an eigenvalue of K and (4.8.1) has a unique solution u G H3(£l), then the collocation-method solution determined by (4.8.3) and the iterative collocation-method solution uh determined by (4.8.4) have the following asymptotic expansions: r
uh(x) {x) - «'(*) u\x) = Y,B Y,Bi{u{x))hl i(u(x))h?
+ 0{hlei), 0(hle1),
(4.8.5)
t= l
rr
uhh{x) ^2Bi(u(x))h] (x) - u(x) = Y /Bi(u(x))h?
+ O(fcfci), 0(hle1),
(4.8.6)
t= l
where functions Bi(u), i = 1, • •, r, are independent of h and depend on u, u1 = \lhu, h h e11 == \\(i-n \\(i-nhh)K\\ )K\\ + \\(i-E )K*\\, + \\(i-E )ir\\, h
and E
2
(4.8.7) h
is an orthogonal projection mapping L (Q) onto S (Q).
In order to prove Theorem 8.1, we shall first generalize Lemma 7.2 and then prove that Ehu — f]hu has an asymptotic expansion. Lemma 8.1. If u E H3(Q), then there exist functions Ui(u)) i = 1, • •, r, which are independent of h and depend on u, such that r
Ehu - Uhu = ^2hfE Y/h?EhUi{u) Ui(u) + + 0(hl). 0(hl).
(4.8.8)
»=i
Proof. Take any ip £ Sh(Q) and consider the inner product in L2(Q), (Ehu-nhu,xl>) r
=
ni
=
(u-uI,tl>)
-
«x)-uI(x))i>(x)dx
E E / /
t=ii=i- ^«i
= E E { / ("(*) - «'(*)MM0-)ds t= l j = l
+ / h + h, =h h,
JA
<3
(u(x) - «u71(x))(V>(x) (x)){4>{x) (u(x)
rP(Mij)dx\ i>{Mij)dx\ (4.8.9) (4.8.9)
8. SEM for MD weakly-singular integral equations
155
where M,j is the bary center of A, j . Using the argument in proving Theorem 4.1, it is known that there exist functions Ui(u),i = 1, • •, r, which are independent of h and depend on the second order derivative oft/, such that h = Y, hf ( Ui(u)xP(x)dx + qi(u, VO ,=i
J
n<
= E A.? / w(*)c'<(tt)^(*)d*+?i(«, v>) r
= 53fc?(^,^)+9i(ti,^).
(4.8.10)
»=i
Here, rji(x) are the characteristic functions of Q,-, and there exists a constant C, which is independent of ft, such that \qi(u,i>)\
/a = E E /
("(*)"w/(*)XV-(*)-i>(Mii))dx
=E E /
(«(*)-^("OXiK*) -v-(M,i))dx
,=1 i = l - ^ i i
t=l j =l J A O -
+ E E /
(«(*) - u/ (*))M*) - v>(M,v))«ix
= J 3 + 74.
(4-8.12)
/
Noting that (u(x) - u (x))(V'(x) - ip(M{j)) is a piecewise cubic function, which vanishes at three vertices and the barycenter, hence h = 0.
(4.8.13)
Obviously, by the inverse estimate, we have \h\ < C 1 /iS||u||„3 (n )||V'||jJHn) < C2fcSll«||*»(n)||^||. Substituting (4.8.10)-(4.8.14) into (4.8.9), we have
(4-8-14)
r
(Ehu - nhu, V>) = E hKmUi, VO + «(«, 1>) «=i
= j ^ h^Eh(ViUi), »= 1
i>) + q(u, V>), Vt/> € Sk(0).
(4.8.15)
156
IV. SEM for integral
equations
Letting U% = rjiUi and noting
l«(«,^)l
D
By Lemma 8.1, the proof of Theorem 8.1 is similar to that of Theorems 7.2 and 7.1, except in Lemma 7.2, for having a remainder with order 0 ( / I Q ) , v G # 4 ( 0 , 1 ) is required while for the extrapolation, only v G H3(0y 1) is required. Similar asymptotic expansions also hold for higher-dimensional weaklysingular integral equations. For example, let Q = [0, l] 5 be an s-dimensional cuboid, K be a weakly-singular integral operator, and 1 be not an eigenvalue of K. Consider the integral equation / k(xiy)u(y)dy
u(x)-
= g(x).
(4.8.16)
Jn Applying the mesh width h = (fci, • • •, /i 5 ), divide Q into N = (hi • "hg)""1 cuboid elements. We denote by Sh(Q) C C(Q) the corresponding piecewise s-linear finite element space. nh : C(Q) —► Sh(Cl) is an interpolation pro jection operator, Eh : L2(Q) —► Sh(Q.) is an orthogonal projection, uh is a collocation-method solution, satisfying uh - V\hKuh = Hhg,
(4.8.17)
and uh is an iterative solution, satisfying uh = Kuh +g. h
(4.8.18)
h
T h e o r e m 8.2. If u,u and u satisfy respectively (4.8.16), (4.8.17) and (4.8.18), and u G H3(Q), then there exist functions Bi(u) (i = 1, • • •, s), which are independent of h and depend on u, such that s
uh - u1 = £ft?£fcflf.(ti) + °(fto)>
(4-8-19)
»=i
and s
uh-u
= 53fc?S«(n) + 0{h3).
(4.8.20)
«=i
Proof. The proof is similar to those of Theorems 7.2 and 8.1. We need to prove that Ehu — F\hu possesses the following asymptotic expansion: s
Ehu - nhu = J2hiEhUi(u) «=i
+ O(h30),
(4.8.21)
8. SEM for MD weakly-singular integral equations
157 157
where Ui(u) are functions independent of h and depend on the second order derivatives of u. In order to prove (4.8.21), take \j) £ Sh(Q) and consider the inner product in £ 2 (ft), h h h h (E u u,,V>) ( « ---nnfcuti,A) {Ehuu --UU VO === (ti VO
= / (u(x) —- u11(x))t/;(x)dx (x))ip(x)dx Jci ./ft N
rt -u (z))^(z)
/e
» = 1 J*i N
f
= E/(«(*)-- « (x))di^(M.) /
»=1 » = 1 • ' «^'
+ E /(«(*)--u (z))(V'Oc)-V'(M;))
t=i
^
(4.8.22)
= Ii I1 + + IIa, 2,
where t/J = n'1!/, e,- is a cuboid element in Q, and Mt- is the barycenter of ti. Use the asymptotic expansion of the error of the trapezoidal rule JV N
Ii
» s
h2 f dd22uu j—2
= EE-12 4 t=i i=i
*t
= EEEE -12 | = t=i i = i »=i i = i
,vo
j
^(Aft-)
^
W
i
,
h f d a2«u At = EE- * * * * * JV N
5 s
•=lj=l t=ij=i
2
)
x
meas(e meas(ett)) -f + 52(^ * 2 (,V0 ^)
2
iZ ^e 12 yieCFX , j
+ 93(1* ,V0
5
12 7ft ax? ^
< + ^(u,V>)>
(4.8.23)
with l«s(«, VOI < Ci||«||j lfl3(«,^)|
(4.8.24)
We now consider the estimate of I^. The function (u(x) — u1(x))(ip(x) — \p(Mi)) vanishes at vertices and the bary center of e», which shows that if u is a piecewise quadratic polynomial, \\) is an s-linear function, then I2, as a, bilinear functional, has zero value. By the Bramble-Hilbert Lemma^^ or the technique used in the proof of Lemma 8.1, \h\ < C 2 ||u||ir.(n)||*||fc8. C2\\u\\m((1)U\\hl
(4.8.25)
IV. SEM for integral
158
equations
Substituting into (4.8.22), it follows
(Ehu-nhuJip) = £-i^?( i=i
B
; ^ ) + «(«^)> ^eSh(Q).
(4.8.26)
U
d2u Here, let B{ = — - , and |<j(u,VOI < CftolMljf^njIMIj therefore the proof
dxf
of (4.8.21) is complete.
D
R e m a r k 1. Only the proof of the SEM for the collocation method in the space of piecewise linear functions has been given. For the collocation method in the space of piecewise constant functions, one can also prove, by the asymptotic expansion of the orthogonal projection and the interpolation projection, that the accuracy is of 0(JIQ). However, the existence of a corresponding asymptotic expansion cannot be guaranteed. R e m a r k 2. The accuracy of the SEM depends on the estimate of = ||(/-n f c )iif|| + ||(/-^ fc )-K'*||, which bases on the analysis of properties of the kernel k(x,y). For the details, see I.H. Sloan and V. Thomeet 122 !.
Sl
Chapter V SEM FOR DIFFERENTIAL EQUATIONS
One of the main purposes of numerical computation is to obtain numeri cal solutions of partial differential equations by solving large scale algebraic systems yielded from appropriate discretizations of mathematical models. Different discrete methods lead to different research directions. The main discretization methods include the finite difference, the finite element, the collocation and the boundary element methods. The finite difference and the finite element methods remain to be the leading methods in solving partial differential equations. The finite difference method can be used to solve nonstationary (parabolic and hyperbolic) equations, but not for problems with complicated curved boundaries. The finite element method was initially applied to solid mechanics, and it was subsequently developed to be applicable in fluid mechanics. It can deal flexibly with complicated boundaries and there are many available softwares. The boundary element method can reduce the dimension of the discrete problem to be solved, and so it is suitable for the outer boundary value problems in particular. However, it can only be used for a class of typical problems and is thus un suitable for solving partial differential equations with variable coefficients. The collocation-method is easy to operate, and has wide applications in twopoint boundary value problems and boundary integral equations. But so far the research on the application of this method to partial differential equa tions is still inadequate. The Richardson extrapolation acceleration of these four methods are all successful. For details about the extrapolation of the difference method, see the monograph by Marchuk and Shaidurovt 90 ! and those of the finite element method, see the monograph by Q. Lin and Q.D. Zhuy']. However, the cost of global refinement required by the Richardson extrapolation is very high for higher-dimensional problems, it is the SEM that can fill this gap. Based on the multivariate asymptotic expansion, the SEM refines only one mesh parameter at each step and achieves the same order of error as that of the Richardson extrapolation, while the scale of the problem is greatly reduced. Obviously, the idea of reducing a big problem into several smaller problems is suitable for evaluating the problem in par allel on multiprocessor computers. It must be pointed out that the number 159
V. SEM for differential
160
equations
of independent mesh parameters does not necessarily equal to the dimen sion of the problem. One can construct several independent parameters even for a one-dimensional problem. In this Chapter, we discuss the SEM for the collocation-method solution of the quasilinear two-point boundary value problems in §1, then the SEM for the solution of the difference methods in §2, and finally, the SEM for the finite element approximation in §3. Only stationary problems are discussed in this Chapter. The SEM can also be applied to nonstationary problems effectively, because the time step can be regarded as an independent parameter. For details of the asymptotic expansion of nonstationary problems, see Marchuk and Shaidurovt 90 ]. 1. SEM for the collocation-met hod solutions of two-point bound ary value problems 1.1. SEM for the collocation-method linear two-point boundary value
solutions problems
of the
quasi-
Consider the boundary value problem of the following quasilinear ordi nary differential equation •
u" = f(t,u,u'), < [ aju(j) + bju'U) = dh
0<*<1,
(5.1.1a)
j = 0,1.
(5.1.1b)
The difference method, the finite element method and the method can all be used for the evaluation of the numerical (5.1.1), among them the collocation-method is the simplest and sively. In order to apply the SEM, divide the interval [0,1] into i.e., first choose the initial partition as 0 = TO < ri < • • • < r r = 1,
collocationsolution of used exten r segments, (5.1.2)
and then refine the interval /,- = [T,_I,T,] (i = l , - - , r ) by a new mesh width hi = (T,- - T t _i)/n t -. Let Tti = r,_i + jht (i = 1, • • •, r; j = 0, • • •, n t ) be the nodes, and the corresponding partition becomes Jh : 0 = T0 < <
< n | 2 < • • • < TI < T 2 ,I < Tr_i < • • • < r r = 1.
TM
(5.1.3)
Let Sp3(Jh) be the cubic spline function space corresponding to Jh, i.e., Sp3(Jh) = span{ 1,M 2 , (* - n>)+» * = 1, • • •, r; j = 0, • • •, n,-}. The cubic spline approximate solution u^ 6 Sp$(Jh) satisfies the following collocation
1. SEM for two-point boundary value problems
161
equations <{nj)
= f(rij, t i f o ) , < ( r t i ) )
<
« = 1> • • • i r; j = 0, • • •, H,-,
, fljUfcO") + *i*fc(i) = <*>> i = 0,1.
(5.1.4a) (5.1.4b)
As we know that if / has 3-rd order partial derivatives and the exact solution u* e C 4 [0,1] is an isolated solution of (5.1.1), then there exists a unique approximate solution Uh in a sufficiently small neighborhood of u*, and the following error estimate holds \\Bi(u* - uh)\\oo < C f t ' l p V H o o , j = 0,1,2,
(5.1.5)
d where D = —, and C is a positive constant independent of h. dt For the sake of simplicity, introduce the following notations: f(U) = /(*,«,«'),
f0(U) =
at/ f(U(t))
-f(U),
ou
= f(t, u(t),«'(«)),
/*(!/(<)) = ^ / ( ^ ( 0 ) - (5-1.6)
In the following, we shall prove, under certain conditions, that there is a multivariate asymptotic expansion with respect to h = (fci, • • •, /i r ). T h e o r e m 1.1.1°™ Suppose that u* € C 2 [0,1] is an isolated solution, Uh is the approximate solution of u* determined by (5.1.4), then there exist functions Wi(t) G C 2 [0,1], i = 1, • • •, r, swcA Ma/ r
„•(«) - uh(t) = £ > ' ( * ) * ? + O(*o),
(5.1.7)
*=1
wiM /in = max /i,-. l<j
Proof. Let r\h be the piecewise linear interpolation projection operator determined by the partition Jh. Since u'^ is a piecewise linear function, then (5.1.4) is equivalent to the system
' u» = nhf(t,uhlu'h)
(5.1.8)
< <*jUh(j) + l>ju'h(j)=dj,
i = 0,1.
162
V. SEM for differential
equations
Let eh = u* - uhy from (5.1.1), (5.1.5), (5.1.8) and Taylor's expansion, eh satisfies = (I - n*)/(l/*) + 0{h%)
el - h{U*)eh - f2{U')e'h ajehU)
+ bje'h(j) = 0,
j = 0,l.
(5.1.9)
Let H(r,t) be the Green function of the boundary value problem (5.1.9), and set g(t) = f(V) = /(*,«*,(«•)')• Noting that g(t) G C 4 [0,1], one obtains eh(r) = C H{r,i){I Jo
- nh)g(t)dt + 0(h*)
= J2 [ H(T,t)(I-r\h)g(t)dt
+ 0(h$).
(5.1.10)
.=i Jit Hi
Since I{ = (J Iih JtJ- = [r t > j -i, r t -j], for j = 1, • • •, n t , then
y
ff(r,i)(i-nfcMt)A
= E / #(r> ')(' - n *M0*.
(5.1.H)
By using the properties of the Green function, if r ^ 7 t j, then H(r^i) is a smooth function on 7,j. It follows, from the estimate of the remainder of Simpson rule, that H(r,t)(I-nh)g(t)dt
/ Jin
= ^(r,m,,0[,(m,,)-^^±^l]
+0(ft»)
1 = - - i f ( r , m 0 y ' ( m t i ) / » ? + O{h50) 12" = -j2h^l where mtJ- = n , j - i H
H(T,t)gH(t)dt
+ 0{hl),TtIti,
(5.1.12)
is the midpoint of 7,;-. If r € Iij, then the derivative
of H(r, t) is discontinuous at r = t. Considering the error expression of linear interpolation, for t € Iij, we have (/ - r\h)g(t) = ^ ' ( T S J - O C * - 7ij_!)(t - r 0 ) + O(ftg).
1. SEM for two-point boundary value problems
163
By using the mean value theorem, there exists t G iij, such that
H(r,t)(I-r\h)g(t)dt
I
= \f =
H(T,ty'(nJ-I)(*
\H(T,*)y" 2
= ~
j
(T»J_I)
- nj-iX* - m)dt + 0(hj)
f (tJin
H(r,t)g"(t)dt
nj.i)(t
+ 0(h$),
- Ttj)dt +
0(hf)
r € /,;•
(5.1.13)
Combining (5.1.12) and (5.1.13), we obtain
H{T,t)(I-nh)g(t)dt
J
= ™*? j Letting w;(r) =
H(T,t)g"(t)dt
f H{r,t)g"{T)dt,
+ 0(ht),
i = I, •••,!•.
(5.1.14)
it follows from (5.1.10) that
r
eh(r) = 5 > , ( r ) f c ? + 0(h*),
(5.1.15)
»= 1
this completes the proof of Theorem 1.1.
D
Further analysis can lead to a higher order expansion of the error, which is omitted here. In practice, it is most convenient to choose the cubic £-spline function as the basis function of Sps(Jh). For details of the construction of nonisometric 5-spline, see Schumakerl 114 !. 1.2. SEM for the collocation-method solutions Liouville type eigenvalue problems
of
Sturm-
Consider th e Sturm-Liouville type eigenvalue problem ( <
du d , ^ Lu = - ( P ( 0 — ) - «(*)<* = - M * K t/(0) =atr/(l) = at0,
/ p(t)u2(t)dt= i Jo
1.
(5.1.16)
V. SEM for differential
164
equations
Here, suppose that p(t),q(t) and p(t) are smooth functions, and p(t) > d > 0, p(t) > C2 > 0, Vt E [0,1].
(5.1.17)
All the finite difference, the finite element and the collocation methods can be used for evaluating the numerical solution of (5.1.16), among them the collocation method is comparatively simple and effective. Let Jh be the partition of the interval [0,1] defined by (5.1.3). To find the cubic spline collocation-method solution {A/^ti/j} of the eigenvalue problem (5.1.16) is equivalent to the problem: find Uh G Sp$(Jh) satisfying the collocation equation /
Luh(Tij)
-\hp(Tij)uh{Tij),
i = l , - - , r ; i = 0,--,n t -,
< \
=
(5.1.18)
tifc(0) = tifc(l) = 0, / p(t)u2h(t)dt = 1. Jo
It has been proved in references^!' ^ ^ that the approximate error satisfies |A - \h\ = O(fcg), \\D°(u - uOlloo = O(ftg), i = 0,l,2.
(5.1.19)
In order to apply the SEM to improve the accuracy, it is necessary to verify that the errors of the approximate eigenvalue and the corresponding eigenfunction have multivariate asymptotic expansions. For convenience, introduce the Hilbert space H = L2(p; 0,1), and define the inner product as
Jo Suppose that p(t) is continuously differentiate, then problem (5.1.16) can be rewritten as the following equivalent form: * Tu = u" - ai(t)u' - a0(t)u = \g(t)u, < ( «(0) = «(1) = 0, («,«) = !. And the corresponding discrete problem is to find uh e Sp^(Jh), 1 Tuh{Tij) = \
\hg{Tij)uh{Tij),
1 «fc(0) = u k (l) = 0,
(«k,«Jk) = l,
(5.1.20) satisfying (5.1.21)
1. SEM for two-point boundary value problems
where g(t) =
165
-p(t)/p(t).
T h e o r e m 1.2.1 4 "' //{A, u) is a simple eigen element o/(5.1..16), {A&, Uh] is the corresponding approximate eigenelement determined by (5.1.18), then the error of the approximate eigenvalue has the following multivariate ex pansion: X
" A* + ^ X > ? / rt*M*)«(4)(0* - O(*o). 12 . = 1 Ju
(5.L22)
and there exist functions Wi{i) (i = l , . , r ) , which is independent of h, such that r
u(t) - uh(t) = Y,wi(t)h? + O{h30).
(5.1.23)
Proof. Let Hh be a linear interpolation operator with base points {r,j : i = 1, • • •, r; j = 0, • • •, n,}. (5.1.21) is equivalent to finding Uh E Sps(Jh) and Ah satisfying f
<
u'l = Vr(a\v!h + aoi/^ + ^h9Uh), tifr(O) = Ufc(l) = 0,
(uhyuh)
(5.1.24)
= 1.
Let en = u — Uh. By using (5.1.20) and (5.1.21), it follows that (T - \g)eh = (A - \h)gu + (A - \h)geh +(I<
nh)uh,
(5.1.25)
e*(0) = e h (l) = 0, where Uh = a\ufh + ao^/i + ^h9^h- By direct evaluation it is easy to verify that -T is a self-conjugate operator in H. Divide both sides of (5.1.25) by a if g, and take the inner product with u, one obtains
A-A„ = -(A-A f c ) / /#)«(<)«*(')*+ / Jo
= f
Jo
p(tHt)(I-nh)uh(t)dt
P(t)u(t)(l-n
h
)uh(t)dt + 0(ht)
Jo = / p(t)u(t)(l-nh)u"(t)dt Jo
+ 0(ht).
(5.1.26)
166
V. SEM for differential
equations
Here u'^ = V\hv>h and the error estimate (5.1.19) are used. Evaluate (5.1.26) by the Simpson rule, and assume that u G C 6 [0,1], then A-
\h h
= [
P(t)u(t)(l-r\
)u"(t)dt + o(h$)
Jo
= E E r rt*Mwj - n")«"(()^+°(ht) i=ii=rT».M r
ni
r2
= ^2 J2 { vhiP(mii)U(mij)[U"(mij)
-,1"(r'j)+2""(r"")]+oW)}
+
oW)
= - E E J|P(™.. )"("■., )«(4)(™li) + 0(14) (5.1.27) *=I12
^
the proof of (5.1.22) is complete. In order to prove (5.1.23), define Hi = {
(5.1.28)
The operator 5(A) = ( - ( T - A p ) ) " ^ / - E) is meaningful^ . The corresponding Green function G(T, t) is piecewise continuously differentiate. Rewrite the identity (5.1.25) as - ( T - A^e^ 9 = (A-Afc)ii + (A-A fc )c fc
+ ! ( / _ nfc)ti" + V - nfc)(5fc - u").
(5.1.29) 9 9 Let Wh = Uh — u". Operate 5(A) on both sides of (5.1.29), it becomes (/ " E)eh
1. SEM for two-point boundary value problems
=
( A - A h ) / G(r,t)eh(t)dt+ Jo 1
167
f G(r, t)g-\t)(I Jo h
-
nh)u "(<)<*<
+ f G(r,0<7- (0a-n KW^ Jo
(5.1 .30)
= J1 + J2 + J3.
Using (5 .1.19), it is easy to obtain the estimate ||Ji||oo = O(ho). For J2, there is J2=
/ Gfafy-^tXl Jo
-nh)u"(t)dt G(T,t)g-\t)(I-nh)u"(t)dt.
= E E /
(5.1 .31)
If r g i t f , by using the Simpson rule, it is easy to prove that G(r,0«7-1(0a-nh)«"(0^
/ = ~
l
G{T,t)g-\t)uW{t)dt
+ 0{h
(5.1 .32)
If r € lij , by using the interpolation expansion (I - n V ( 0 = « ( 4 ) (nj-i)(< - 7i,i-i)(* - n,-)/2 + 0 ("•?). Vt G ly, and the :mean value theorem for integration, there exists £ G /<;, such that
L
G(r,^- 1 (0a-n A K(0^ :i /
G(r ) 0«7- 1 (0(<-n,y_i)(<-r i j )d<«( 4 )(r i , j _ 1 ) + O ( ^ )
= iG(r,o.r1(0«(4)K;-i) / (*-^-i)(*-ni)«ft + o(fc?) (5.1 .33) Substitute (5.1.32) and (5.1.33) into (5.1.31), it follows that J 2 = - - V > ? [ G(T,t)g-\t)uW(t)dt 12 fz? 7/;
+ 0(h$).
(5.1 .34)
168
V. SEM for differential equations
For the estimate of J3, obviously, there exists a constant C > 0, which is independent of fc, such that (51-35)
H/slloo < C M K I l o o , where w'h = — [ai(u'h - u') + a 0 (« h - u) + (XhUh - Au)]
= ^ M«k - u ')] + ^ h ( « * -«)] + A f c K - t i " ) + (Ak-A)«'.
(5.1.36)
By using (5.1.19) and (5.1.22), one obtains the estimate (5.1.37)
UKWoo = 0(hl). It follows from (5.1.30) that
( I - £ ) c f c = - - V > ? / G ^ O ^ ' W ^ W ^ + O^).(5.1.38) 12 . = 1 Jit Noting Eeh = (c fc ,ti)w= --{u-uh,u-uh)u
= 0(/i^),
taking iy,-(r) =
/ Gfatfg^lfiuWtydt, 12 hi tuting into (5.1.38), we obtain
(i = 1,
- , r ) , and substi-
r
eh(t) = u(t) - uh(t) = ^2hfWi(t)
+ 0(fcg),
t=i
and the proof of (5.1.23) is complete. 1.3. SEM
for singular
two-point
□ boundary
value
problems
Consider the quasilinear two-point boundary value problem d , , .du,
, .du
-*<**)*)+*)#
=*>")> *<*<>>
(5 . L39)
u(a) - u(b) = 0. Suppose that p(t) > 0, W G (a, 6), p(a) = 0, and
fw)dt<°°-
(5L40)
1. SEM for two-point boundary value problems
169
For this class of singular boundary value problems, Babuska^' discussed the error of a finite difference approximation in the case p(t) =ta, 0 < a < 1, and proved that the error at the isometric nodes is O(ho). In the following we shall prove that it is possible to eliminate the singularity by using a simple transformation. (5.1.39) is converted first into a regular two-point boundary value problem, and then obtain an asymptotic expansion of the mesh parameters obtained from an isometric partition or a piecewise iso metric partition. Under this transformation, the isometric partition in the new equation is equivalent to the gradual partition in the original equation, that is, the closer to the origin, the denser the grid points. The gradual partition depends on the property of the zeros of p(x). Consider the transformation r = r(t) = I p'itydt.
(5.1.41)
Ja
Obviously, r : [a, 6] —» [0,T], where T = r(6). By the assumption p(t) > 0, W £ [a, 6], r(t) is a monotonically increasing function, then the inverse function t = t(r) exists uniquely. Applying the derivative relation du
du
dr
dt
(5.1.39) can be converted into ( 5"(r) = q(r)u'(r) - p(r)f(r, ^ I u(0) = u(T) = 0,
«(r)) t
(5.1.42)
where « ( T ) = u(*(r)),p(r) = p(t(T)),q(r) = « ( * ( T ) ) , / ( T , « ( T ) ) = f(t(r), u(t(r))). (5.1.42) is a special case of (5.1.1), and so applying Theorem 1.1 one can obtain the asymptotic expansion of the collocation-method solution r
«(r) - « k (r) = 5 > , ( r ) A ? +
0(h*),
t= l
0 < r < T.
(5.1.43)
Substituting r by r(t), 0 < t < 1, in (5.1.43), we get that r
u(t)-uh(t)
= ^wi(t)h?
+ 0(h$),
0
(5.1.44)
i=i
This transformation can also be used in the eigenvalue problems described in (5.1.16).
170
V. SEM for differential
equations
The relation between the grid points on [0, T] and that on the original interval [a, 6] can be explained as follows: Suppose that p(t) = ta, 0 < a < 1, and [a, 6] = [0,1], then r(t) = j t'adr t(r) = (1 -
= *1-tt/(l - a),
a f - M / f H
If the new interval [0,T] is partitioned isometrically with grid points lj— }? , then, in the original interval [0,1], the grid points are (1 — ot)n *—° i * { (—) I +" } , which is a gradual partition.
1.4. SEM for two-point boundary value problems discontinuous coefficients
with
Consider the two-point boundary value problem -(pit')' + qu = / ,
a < t < b. (5.1.45)
u(a)
= WQ,
u(b) = ti 1 }
with p(t) > Ci > 0, q(t) > 0. If any one of the coefficients p(t),q(t), f(t) has a finite number of dis continuous points in the interval [0,1], then (5.1.45) is called the Dirichlet problem with discontinuous coefficients. For simplicity, suppose that there is only one discontinuous point £ G (0,1), then (5.1.45) can be regarded as the couple of two differential equations defined on (0,f) and (£, 1), and at t = f, it satisfies the following conditions of consistency:
«tt + 0) = 11(^-0), (5.1.46) Ptf + OXtf + 0) - ptf - 0)ti'(£ - 0) = g, where g is a fixed constant, u'(£ + 0) = limix'(£ + 6), and u'(£ - 0) = 6>0
limti'(£ + «). a—o We denote by Q* the set of the functions which possess piecewise con tinuous derivatives up to order k on [a, 6], and the first order derivative has discontinuity of the first kind at f.
1. SEM for two-point boundary value problems
171
Marchuk and S h a i d u W 9 0 ) proved that if p £ Q^ + 1 , / , ? £ <2£, r > 2, then (5.1.45) has a unique solution u £ <3£+2 H C[0,1]. They also proved that there is an asymptotic expansion of the difference approximation uh at grid points (cf. Theorem 3.2 in [90]). (5.1.45) can also be solved by the collocation method, provided that the discontinuous points of the coefficients are taken as the initial partition points. The procedure is as follows: convert (5.1.45), by the transformation (5.1.41), into a simpler form u"(r) = q(r)u(r)-p(T)f(r\ (5.1.47) u(0) = a, u(T) = b. If £ is a discontinuous point of the coefficients, then r(£) is a discontinuous point of p,q/f. Take the initial partition as 0 = r 0 < n < • • • < r r = T.
(5.1.48)
All the discontinuous points of coefficients r(£) £ { r , } . . Refine the subintervals respectively by mesh widths hi = (rt- — r,-_i)/nt- (i = 1, • • •, r), and let Tij = r,_i + jh{ (i = 1, • • •, r; j = 0, • • •, n,). Furthermore, let tih be the cubic spline approximate solution with respect to the nodes {rt-j}, u be the solution of the equation (5.1.47) with the approximation u^ of the collocation method, and ?& = u — Uh be the error. Repeating the derivation used in (5.1.9), it can be proved that e>>-qeh=(I-n»)(qu-pf)-(I-n»)(qeh),
(5.1.49)
eh(0) = e,(r) = 0. Let G(T, S) be the Green function of the problem (5.1.49). It follows from the discontinuity of q(r) that G(r, s) is possibly discontinuous at r,-. How ever, G{r> S) is smooth in (r,_i, Tj), then we still have
= / Jo = J2f
G(r,s)[(I-nh)(qu-pf)-(I-nh)(qeh)]ds G(r, s) [(I -nh)(qu-p
/)] ds
-J2[ G(T,s)[(I-nh)(qeh)}ds = E - ^ f t « ? / G(r,s)(qu-pf)"ds + 0(hi).
(5.1.50)
V. SEM for differential
172
equations
Here the smoothness of the integrand on /,- and the inequality
||(i-n f c )(« cOHoo,/, < Cft?||ff cOloo,* < ChtWD^uWoo are used. Consider now the original equation (5.1.45), the error of the approximate solution Uh of (5.1.45) has an expansion r
u(t) - uh(t) = ^2Wi(t)h? + O(ftJ),
(5.1.51)
*=i
where Wi(t) are independent of h = (/ii, • • •, hr) and depend on the coef ficients and their discontinuous points. SEM for the collocation-method solution of a quasilinear two-point boundary value problem with discontin uous coefficients can be considered similarly. 2. S E M for finite-difference a p p r o x i m a t i o n s 2.1. Difference ciple
equations
and the discrete
maximum
prin
Consider the Dirichlet problem of a linear elliptic equation Lu = > a » ( z ) - £ - 2 + l^hi(x)-^-
u{x) = g(x),
+ d(x)u
= f(x)
inficF, on dtl,
(5.2.1a) (5.2.1b)
where Q is a bounded open domain in Ms, a,-,6,,c/ and / £ C(Q),<7 £ C(dQ). Assume that a = min mina,(x) > 0, and the equation (5.2.1) has i<*<5 xen
a unique solution u. In order to construct a difference scheme of (5.2.1), define the grid lines I \ / and the grid Th on Ms as follows: First take the mesh width h = (/ii, • • -,/i,), and then let
i \ / = {x e Ms: XJ e m, x{ = mhif Hi e %>, i=l,-..,«, i ^ i } , Th = {xeMs : Xi = riihi, i = 1, • • •, *},
(5.2.2a) (5.2.2b)
where 2Z stands for the set of integers. Let e, be the unit vector in the i-th coordinate and let ft/* :={xe£l:x±hiei Qhti : = ( f l n r f c ) \ n f c ,
e f t , i = 1, • • •, s} H Th)
an fc :=flnnr fc| i, o M = nhunfc|l-udfifc.
(5.2.3a) (5.2.3b)
(5.2.3c)
2. SEM for finite-difference
approximations
173
Q/i is called the set of regular grid points, Clhti the set of irregular grid points, dClh the set of boundary grid points and Qh,t the union set of all three kinds of grid points. For a regular domain Qh,i = 0- Using the central divided difference, it is easy to construct the following difference equation of (5.2.1): s h h
Lu
= ]£
+ ^bi(x)[uh(x
+ hid) - uh(x -
hiei)]/(2hi)
t=i
+d(x)uh(x) = / ( * ) , uh(x) = g(x), Vx G dQh.
Vx€fifc,
(5.2.4a) (5.2.4b)
If ^lh,i ^ 0, and x G ftjif», then there exists at least a direction ej such that x -f hjej £ Q. There are many choices for setting the difference equation at the irregular point x G £lh,i- Suppose that x* = x + Sjhjej G d£lh, 0 < Sj < 1, is an intersection point of dCl with a segment between x and x -f hjej, we can use x* and x -f ^ e j , and replace the central difference in (5.2.4a) by a non-isometric difference. Another choice is called the simple trans port method, in which, let uh(x) :— uh(x*) and substitute into (5.2.4a). If a higher order scheme with an asymptotic expansion is required, then the Lagrange interpolation method should be employed. The detailed con struction will be introduced in the next section. Regardless of the methods of discretization, the difference equation can be described as
Vx€fi f c U «*,,-, where A(x, y) are coefficients which depend on x, y G fi/i U £lh,i and depends on a,, &», / , g and x.
(5.2.5) Fh(x)
Definition 2 . 1 . For any x G &h U Q^,,-, Me discrete set N(x) = {yenhU
Qh)i : A(x, y) ± 0},
is called a neighborhood of x. Definition 2.2. The set Qh U fi^,,- is said to be connected, if for any x,y £Qh U&h,i, there exists a finite sequence { x ( n ) } n = 1 , such that x^1' = x, x^m) = y, and x( n ) G ^ ( x ^ - 1 ) ) , n = 2,3, • • •, m.
(5.2.6)
174
V. SEM for differential
equations
By using Definition 2.1, the difference equation can be rewritten as
Lhuh(x) = £
A(z,y)u\y)=Fh(z), VsEflfcUfifc,,-.
(5.2.7)
The problems concerning stability and convergence of the difference equa tion (5.2.7) are of great importance. The discrete maximum principle, an analog of the maximum principle of differential equations, is the main tool for solving the problems. Before introducing the principle, the concept of operators of positive type should be defined. Definition 2.3. The difference operator Lh is said to be an operator of positive type, if a) b)
A(x, y) < 0, Y^
^(*>y)>0,
Vz G tth U Qfc|l-, y G M(x)y VxGftfcUQ*,,-.
x^y, (5.2.8a) (5.2.8b)
T h e o r e m 2 . l J 1 A ^ ( T h e discrete m a x i m u m principle) Suppose that Lh is an operator of positive type, and Q'h is a connected subset of Qh Utlh,i- If on the set
K = U ^(*),
(5.2.9)
uh(x) is not a constant, and Lhuh(x)
< 0, (or Lhuh > 0), Vx G
h
fi'fc,
then u (x) cannot reach the positive maximum (or the negative in Cl'h.
(5.2.10) minimum)
Proof. Suppose that Lhuh(x) < 0, Vx G fi^, and at the point x G Sl'hi u (x) reaches the positive maximum value h
uh(x) = maxu h (x) > 0, then Lhuh(x) = A(x,x)uh(x)
+
Y2 yeM'ix) y?x
A(x,y)uh(y)
(5.2.11)
2. SEM for finite-difference Approximations
=[ £
175
A(x,y)}u\x)
yeM'(x)
A(x,y)(uh(y)-uh(x)).
+ £
(5.2.12)
y£Jsf(x) y^x
It follows from the condition b) of Definition 2.3 that the first term on the right-hand side of (5.2.12) is non-negative, and from the condition a) and (5.2.9) that the second term is also non-negative, then Lhuh(x)
> 0.
Combining with the assumption Lhuh(x) Lhuh(x)
(5.2.13)
< 0, it follows that (5.2.14)
= 0.
According to (5.2.9), (5.2.12) holds only if £
(5.2.15)
A(xyy) = 0,
yeM'(x)
and uh(x) = uh(y), Vy E M(x).
(5.2.16)
By the assumption uh(x) ^ const., x E Qh, there exists a point x E &h, such that uh(x) < uh(lx). And by the connectivity of ftl, one can always find a sequence xW E fl'h (i = l , - - , m ) , such that x^1' E A / ^ ) , x^ E Af^-V), i = 2 . . - - , m l x € A/" (x< m )). By using (5.2.16), uh (x) = uh (x&). Without loss of generality, we may assume 0 < uh(x) = uh(xW)
= • • • = tx h (x( m )).
Otherwise, there exists x( m °), 1 < m 0 < m, then take x = We now evaluate L h ti h (x( m >),
Lfctifc(*<m)) = (
J2
(5.2.17) m +1
x^
° h
A(x(m\y))uh{xW)
y€JSf(x(m))
+ J2 ^(*(m),»)(«*(»)-«*(*(m))) yeJV(x<m>)
>
A(x^m\x)(uh(x)-uh(x^))
= i4(* ( m ) , *)(«*(*) - u*(3t)) > 0, which contradicts the assumption that Lhuh(x)
< 0, Vz € &'h-
(5.2.18)
176
V. SEM for differential
equations
Similarly, for Lhuh(x) > 0, replacing uh(x) by -uh(x) in the argument above, one can obtain that uh(x) cannot reach the negative minimum in Corollary 1. If in QhC\Clh ,-, Lhuh(x) < 0 (or Lhuh(x) exists at least one grid point x^°\ such that D(zW)=
> 0), and there
^(0)>2/)>0,
£ y€^(x(°))
x^enhunh)il h
(5.2.19)
h
then u (x) < 0 (or u (x) > 0) in Clh D Qhti. Proof. Take £l'h = fl/i O fij^-, if uh(x) ^const., then it follows from Theorem 2.1 that ^ ( x ) < 0 (or uh(x) > 0). If uh(x) = const., then
LV(*<°>) = D(*<°V(* (0) ) +
£
A(x(°\y)(uh(y)-uh(x0))
y€jV(x<°)) y**(0)
= D(x^)uh(x^)
< 0,
which leads to uh(x) = uh(x^) < 0, Vx G &h H fih,»- For the case Lhuh(x) > 0, a similar argument works. □ Corollary 2. i/* Me difference operator Lh satisfies the assumptions of Theorem 2.1 and Corollary 1, then the difference equation (5.2.5) has a unique solution. Proof. It is sufficient to prove that the homogeneous equation Lhuh(x) = 0, Vx G fi/i O Clhti has only a trivial solution. In fact, the condition Lhuh(x) = 0 of Corollary 1 guarantees that uh(x) < 0 and uh(x) > 0 hold simultaneously, which implies uh(x) = 0. □ Corollary 3. (Comparison Theorem) Suppose thatuh(x) is the solution of the difference equation (5.2.7) with the right-hand side F (x), and \Fh(x)\ < Fh(x),
Vx G Qh U n M >
(5.2.20)
then \uh(x)\ < uh(x)} V x G ^ U
fiM.
(5.2.21)
2. SEM for finite-difference
approximations
177
Proof. By using Corollary 1 and noting that -^h
Lh(uh ±uh) = F (x) ± Fh(x) > 0, one can derive uh(x) ± uh(x) > 0, therefore, ^ ( x ) ! < uh(x).
D
Theorem 2.2. Suppose that the coefficients of the difference equation
Lhuh(x) = Fh(x)y uh(x)
= gh(x),
xenhunh)i,
(5.2.22)
xetth
has the following properties: D{x) > 0, Vx G fi/» and D(x) > 0, Vx G Qh,i, then equation (5.2.22) has a unique solution, satisfying max\uh(x)\ < max\g h (x)\ + max\U h (x)\ + max\F h (x)/D(x)\,
(5.2.23)
where Uh(x) is a majorizing function, which is the solution of the following problem: -r;h,
Lhuh(x) = F"(X), Uh(x) T=T^/
where F'\x)
> \Fh(x)\,
xenhunh>i,
>0,
(5.2.24)
xedQhi , -rzh,
Vx G J2fc> an
Proof. The uniqueness of the solution (5.2.22) will be derived from Corollary 2. In order to prove (5.2.23), let ti fc (x) = tif(a:) + ti5(a:) + ti§(x) l
where uf(x), t = 1, 2, 3, satisfy respectively the equations Lhuh1{x) = 0, x€tlhUQhti;
h
Fh(x),
u$(x) = g(x), x e dQh, (5.2.25a) xeQh,
L u*(x) = { 0,
xEtth,i,
0,
a; e fih,
(5.2.25b) Ǥ(*) = 0, x G 5 f i A >
and
£*«£(*) = <
h
F (x),
(5.2.25c) x G fift,,-. «S(x) = 0, x E dQh.
V. SEM for differential
178
equations
By using Theorem 2.1 and its Corollaries, max|tij(*)| < max \g(x)\, flh,t
(5.2.26a)
s£dflh
and max\u$(x)\ < max\U(x)\.
(5.2.26b)
We shall now prove max \u$(x)\ < max \Fh(x)/D(x)\. nhunhti xerih,i
(5.2.26C)
For this purpose, replace Fh(x) in (5.2.25c) by F (x) = \Fh(x)\, the solution be 1X3(2;). By Corollary 3,
and let
h*3(*)l<*3(*)> Vx€SlfcUfi fc| «. Using Theorem 2.1, 1X3(2?) can attain its maximum value only at a point x(°) on £lh,i- However, at this point, \F(xW)\ = D(x°)uh3(x^)
A(x(°\y)(uh3(y)-uh3(xW))
£
+
t/CJV(*(°>) h
> D(x°)u 3(x(%
*<°> € n M >
which leads to <
max
!!*(*) = «*(*«»)
< |F(*<°>)/£>(s<0>)|. Thus the proof of (5.2.26c) is complete. The estimate (5.2.23) follows im mediately from (5.2.26). □ 2.2. Multivariate asymptotic expansion of the ence approximation on a domain with a smooth
finite-differ boundary
For simplicity, consider the Dirichlet problem of Poisson's equation -Aw = /, u = ,
inftCiR5,
(5.2.27a)
on <9ft.
(5.2.27b)
2. SEM for finite-difference approximations
179
Assume t h a t the boundary 9Q G C 4 + a . Take the mesh width h = (fti, • • •, hs), construct grid lines I \ / and grid points I \ , and denote, using (5.2.3), the set of regular grid points, irregular grid points and b o u n d a r y grid points by fth, Clh)i and dQh respectively. If x G Qh, then construct the difference equation 5
- ^ j V ( x ) = / ( x ) , Vx G J2fc>
(5.2.28)
7=1
where the central difference operator Sj is defined as Sjuh(x)
= [uh(x + hjej) - 2uh(x) + uh(x - hjej)]/h].
(5.2.29)
If x G £}&,*, and x + hjej £ Q, then (5.2.24) cannot be used to form (5.2.28). However, it is possible to stipulate first the meaning of uh(x + hjej) by the interpolation, and then to construct the difference equation by (5.2.29) in the following way: Suppose x -f hjej £ Q, b u t x — khjej G ^ / i U fi/i,i, k = 1, • • •, n, then the segment between x and x + hjej intersects dQ and let the intersection point be x -+- r\hjej, 0 < rj < 1. Construct a Lagrange interpolation polynomial p n ( 0 of degree ( n - f 1), such t h a t Pn(-khj)
= ^ ( x — khjej), fc = 0, • • •, n,
Pn(rjhj) = g(x + rjhjej).
(5.2.30)
Define Sffc(x + A i e i ) = p n ( f c i ) ,
(5.2.31)
and £ j V ( x ) = [ ^ ( x + fye,-) - 2ti / l (x) + u f c (z x G $}*,,-, x + fye,- £ Q.
hjej)]/h], (5.2.32)
Similarly, a difference equation for the case x G Slh,i but x — /ijej £ Q can be defined. T h u s , Vx G fi/i,», define a difference equation
- Yl *i"*(*)" X) ^ W = /W, VxGQ^, (5.2.33) with J x ( x ) = {; G { 1 , 2 , - - - , « } : x ± jhj (£ H, x G fifc|i}, J 2 (a?) = {1, 2, • • •, s} \ J i ( x ) . Combining (5.2.28) and (5.2.33), we obtain the differ ence equation of the problem (5.2.27). T h e convergence and stability of this equation depend on whether it satisfies the m a x i m u m principle. In other words, it is necessary to verify carefully whether (5.2.33) satisfies (5.2.8).
180
V. SEM SEM for for differential V.
equations
m+a ^(t) G G Cm+a tp(t) is is the the Lagrange L e m m a 2 . 1 . Suppose that \p(t) [—1,1], tp(t) of ip(t) with the the following set set of of interpolation base interpolation polynomial of points: {{^^jj,, 0, —/ij, •••,— nhj}yy where TJ G (0,1), then
p?m m
(o) == ^*iM _-r(ro) ) --^2(«Qo)) + ^»(-*,) i ) __ ^ ( 0 ) h
3
+,,
= Ep^)i* E(^^ (M+!, (»>+^. = (») + ^.
(5.2.34)
= [min{n — 1, m m — 2}]/2 loii/i wiiA remainder where rr = C55W, W, for for 7i ++ 22 << m , C
[
r C66/i™~ /if - 2 +2+o °, , [ C
(5.2.35)
l < m < n ++ l.
90 (cf. Marchuk and and Shaidurovt Shaidurovl90 By the the Lagrange Lagrange interpolaProof, (cf. ! .).) By tion formula, evaluate ip(h) ip(h)
n
(5.2.36)
k=0 fc=0
with aan,fc
"'*
aa
=
»." =
(-!)*+!(„ +1)!(1_,) (-l)*+i(„+ 1)!(1_,) (Jb+l)!(Jb + «,)(!»-*)!' ij)(n-t)!' (* + l)!(*+ " i +1 1 1 T~r~ 11
(5.2.37)
h be the the interpolation error. If If m m >> nn ++ 2, 2, then by Let ijRfc == ip(hj) — ip(hj) be the remainder estimating the
I**I
n +22)) ( 0 | / ( n + < (Ai fo - ^ •i ))Af ei (i 2( 2/ ih, -i ) .-.-.-((((nn ++ l ) hAi i))__™ ™x«i1| l^^»( + ( l / ( » + 2)! ( n\t)\/(n +2) h»+\l2 (l -- V,)) _max _maxii |V\4n+2 2)!. << A?+ (0l/(n ++ 2)!-
(5.2.38)
m< < nn + + 2, let let y>(i) be be an an arbitrarily polynomial of of degree m, obviously, If m y>(x) =~
V<x) = W * - ^ + s m + 2 -i H < 1 , r~T tt=0 =0
t! t!
m!
(5.2.39)
2. SEM for finite-difference
with
approximations
181
max \0X\ < ||V,||o^-i-<*[—1,13- Take (p(x) = 22x%—M—'
^ f°llows
t=0
that \Rh\
=
Mhj)-$(hi)\
+
\Hhi)-lp{hi)\
< Mfy) - f{hj)\ + ^an
-
Jb=0
■fan ,r)\Hr)hj)i m+a
<
m!
-
-||V>||c~+«[--i.iKi+Ei"" M*" fc — 1 k=i
(5. 2.40) +an ,nn" ,vm+i + a" )\ • (5.:
l+a
Using (5.2.37), lan,fc| < (n + 1)!, 77an)l, < 77^0 < n + 1. Substituting into (5.2.40), we obtain |JJ*| < C 4 /i™ +a ,
(5. 2.41)
for m < n + 2,
where CA = ^ J I H I C - H . [ - I , I ] { 1 + (" + 1)[(" + l)!»
m+
°
+ 2m]}
Using (5.2.36), we have £2V-(0) ^.)-2>(0) + V ( - M h
J
n
= 5 1 t>n,ki>(-khj) +
(5. 2.42)
bn^Tjhj),
Jt=0
with &n,0 = (O„,0 - 2 ) / / l | ,
6„,i = (On.1 +
l)/h],
K,T) = o.n,r,/h2j, bn>k = anik/hj,
k=
(5.2.43) 2,--,n.
Finally, by ?2V>(0) - V"(0) _ j,(hj) - 2V(0) + 4>(-hj) _
*J
+
fa)
- rP(hj)
*I
V. SEM for differential
182
equations
it follows from (5.2.38), (5.2.40) and Taylor's expansion that (5.2.34) and (5.2.35) hold. D L e m m a 2.2. For the coefficients 6n>jb, n = 1,2,3, k = 0,1, •••,?!, defined by (5.2.43), there is
( " *».o - J2 K*D/( 2 A; + E l6»>*0 > CT > 0, fc = l
(5.2.44)
Jb = l
where Ci is a constanty which is independent of hj and rj and is dependent on n. Proof. For n = 1, evaluate by (5.2.37), and the coefficients are 6l
"
2 7,(77 + 1 ) ^ '
=
2 -°= - ^ | ' ^
6l
2 (i+W
=
Hence, evaluate directly the left-hand side of (5.2.44), r 2
2
irjh]
1 r2 /
2 +
(l + f,)h]\ ih]
(l + fi)h]\
= 2 ^ 4 **,e{0,l). Thus, in the case n — 1, C7 = —. For n = 2, the coefficients satisfy 6
"•"- n{n + m
3-T,
+ 2)hy
h
-°-
4-2^
r,h]
)
1-7,
The left-hand side of (5.2.44) is 3-7,
\nh)
4-277
(l + t,)h]
i
r2
4-27,
4-2T?
6-
2r,2 + 7j3
^(13-T,2) > (6 - 5T? + 2 T I 2 ) / 1 3
> 3/13, 7,6(0,1).
/
l-m
r
,
4-2IJ
1
l
~* 1 (2 + r 7 ) f t ? J U | ( l T ^ | (2 + »7)ft?J
[3-«? 5TJ +
1-77
1-m
2. SEM for finite-difference approximations
183
3 Thus, in the case n = 2, C-j = —. For n = 3, the coefficients satisfy
24 , ( l + ,)(2 + ,)(3 + , ) / i f
h,rj = ■
7-577
32
3
2(2-,) ,fc? '
'°
-4(1-77)
k3=
^-(IT^J' * ' -"(2TW'
The left-hand side of (5.2.44) is
r2(2-,) L i,A?
7-5, (! + ,)/»?
4(1-,) (2 + ,)/.?
7-5,
4(1 - , )
■2 >
_
(^W
1-, i (3 + , ) / l ? J / 1- ,
I
(1 + ,)/.? + (2 + ,)/»? + (3 + ,)/>? J
+
ihj
1-T;
r2(2 - i|) _ 7 - 5 , _ 4(1 - , ) _ In
1
,
1 + 7? 2+ , 3-1' r 7 - 5 , 4(1-,) 1,
[2+ 1 + , +
2
+,
+
3J
> (24 - 3 2 , + 2 0 , 3 ) / ( 6 8 , + 12, 2 - 20, 3 ) > (6 - 8 , + 8 , 3 / 3 ) / ( 1 7 , + 3, 2 ) > ±, Thus, in the case n = 3, C7 = TJT. ou
, € (0,1).
D
L e m m a 2.3. //it*1 is a solution of the difference equations (5.2.28) and (5.2.33) satisfying the boundary condition uh(x) = 0 (x G <9ft/»), then there is an estimate max ^ ( x ) ! < £ max I/OOI + - i - A g max \f{x)\,
(5.2.45)
where b > 0 is a constant such that Q, C (—6,6)5, CV ts determined by (5.2.44), where n = 1,2,3, and h0 = max hj. Proof. Using Theorem 2.2, decompose uh = 1*2+1*3, where u* satisfies Lhu\{x)
= -J26]4(x)
= / ( * ) , Vx G Qh,
(5.2.46a)
1=1
1u5(*)
= 0, V x e f i M U 6 f i A ,
(5.2.46b)
V. SEM for differential
184
equations
and u% satisfies -J2t]u*(x)
= Q, Vx G Qfc,
(5.2.47a)
i=i
- J2 6H(z) - iZ SH(X) = /(*)■ V* € «M. (5.2.47b) ti§ = 0, V*€d«fc.
(5.2.47c)
Construct the following majorizing function: s
U(x) = (sb2 - Y
x2) max \f(x)\/2s.
(5.2.48)
»=i
Obviously, U(x) satisfies -AU(x)
= -y^6]U{x)
= max |/(x)|, Vx € n h , (5.2.49a)
;=1
cr(x)>o, Viefl^uatifc,
(5.2.49b)
which implies I^OOI < ^0*). V x G f t M -
(5.2.50)
On the other hand, by using (5.2.26c), it follows that max |*4(x)| < max \Fh{x)lD{x)\.
(5.2.51)
Therefore it is necessary to evaluate D(x), x G ftj»,i, by (5.2.47b). Since £ € ^M> ^ n e n ^i ^ $• Letting j G / i and using (5.2.42), we have 6Juh3(x) _ u%(x + hjej) - 2u%(x) + u%(x -
hje,)
n
= ^&n|*iifc(x-JbAiei),
l
(5.2.52)
Jb=0
where x G Slh.i, % + h ; e j G ft and ti h (x + rjhjej) = 0, the coefficients 6n)* are constants, which are dependent on x and j , and independent of hj. By
2. SEM for finite-difference Approximations
185
(5.2.44), the coefficients of (5.2.47b) satisfy
£>(*)> Mn.O-£>„,*|) fc=l
>CV(2//,? + X> n ,*|) *=1
> 2C 7 /IQ 2 , 1 < n < 3.
(5.2.53)
Since u%(x) = 0, (x G dft), then Fh(x) = f(x). By using (5.2.5), it follows that max|t*j|(*)| < max !/(*)/£>(*)I < -i-&»max|/(*)|,
(5.2.54)
and
i«kooi <\uh2(x)\ + \uh3(X)\
1 Vx G nh)t,
The proof is complete.
1 < n < 3.
□
T h e o r e m 2.3. If in the problem (5.2.27); <9Q G C 5+
+
0(h*+'),
i=i
x GJJfcUJJfc.i, where Wj(x) are independent of h.
(5.2.55)
X86
V. SEM for differential equations
Proof. Under the assumptions of Theorem 2.3, it implies u € C 5+<7 (fi) (cf. [52] ). Using (5.2.34), (5.2.28) and (5.2.33), one obtains
-LV*)-A«(x) = £ g ^
+ 0(/^).
xGfiftUftfc,,-. Let
I
d4u(x)
12 dxj problem
(5.2.56)
= Vj(x) G C 1+
- A W J ( X ) = VJ(X),
in Q,
u>j(x) = 0,
(5.2.57)
on <9fi, i = 1, •• - , s ,
with the solution Wj(x) G C 3+Qf . By using (5.2.5), we have -Lhwj(x)
- AWJ(X)
= 0(hl+a),
xetthV
fifc,,-.
Hence,
Lh(uh-u-J2h]™j) i=i
= - A w - Lfcu - ] £ A i w i " $ 3 ( A u ; i j=i
+
Lhw h
i) ]
i=i
= 0(/i^ +
(5.2.58a)
and u* - u - ^
^j WJ = 0, Vx G 9 0 .
(5.2.58b)
It follows from Theorem 2.1 and its Corollaries that (5.2.55) holds.
D
Theorem 2.3 shows that when n = 2, the boundary d£l and the func tions / and g are smooth enough, the extrapolation method can be applied once; when n = 3, the boundary and the functions are more smooth, the extrapolation method can be applied twice according to the following: T h e o r e m 2.4. If in the problem (5.2.27), the boundary dQ G C 7 +", the functions f G C 5 _ H T ( Q ) and g G C7+
2. SEM for finite-difference approximations
187
number of interpolation points n = 3, then the error of the finite difference solution uh(x) has a multivariate asymptotic expansion uh(x)-u(x)=
]T
hWwp(x) + 0(h\+*),
(5.2.59)
1
where wp(x) are independent of h. Proof. Under the assumptions of Theorem 2.4, it implies u G C 7+<7 (Q). By using (5.2.34), we obtain -Lhu-
Ahu
- l V / , 2J ^ 12f-f j=i
dxjJ
1 Af,t . 3 6 W
+ 36oE )^6+°(/lo+<7). i=i i Let v* = J
innkunh,,-.
(5.2.60)
1 d6"
, v,- G C 1 + a (Q) and let u>7- be the solution of the following 360 0s? '
equation: —AWJ = Sj,
in Q,
Wj — 0,
(5.2.61)
on 5Q, j = 1, • • •, s.
Obviously, Wj G C 3+<7 , and = 0(/iJ+a), j = l,..-,«.
-Lhwj-Awj
1 #4" „ , Vj G C 3+<7 (Q) and let Wj satisfy
Furthermore, let v; = 3
(5.2.62)
ndxj
-AWJ
J
= 5y,
{?j = 0,
in Q,
(5.2.63)
on <9ft, j = 1, • •, s.
Obviously, Wj G C 5 + a ( Q ) . Let iu^ be the finite difference approximation of Wj) satisfying LhS^ = VJ, J
wf = 0,
mQhunh}i, ond£lh.
(5.2.64)
188
V. SEM for differential
equations
By using Theorem 2.3, there exist functions Wjtk, which are independent of ft, such that the following multivariate asymptotic expansion holds 5
3 + 2.65) u nhM,,-. .(5.: wf(x) -- W WJ{X) xx 6€ ft/, Slh U (5.2.65) j ( x ) == Y, kl&jA*) 0 °), w* (*) " ,fc(x) + + O(h 0(A2+*).
fc=i
Using (5.2.61)-(5.2.64), we have 53
*J) E™J■*J)
Lh(uh -- u i=i
i=i
5
5
j=l
j=l
= -A« - Lhu - Y^^h] - ^vjhj =
+ O(h50+a °), O(^o )>
in
+ 0(hl+") (5.2.66a)
mQUJlfc,,-, «fc hunh,i,
and 5 S
h
Q h
u -u-Y^tftf-H2 3 t j=i ;=i
S3
= °>
i j=i =i
on5n on dfifc. *-
5 2 66b ((5.2.66b) -- )
By using Theorem 2.1 and its Corollaries, it follows that
2
^E^ = E^(
uh/l - - IX u — —
14
i3=1 =i
5s
i 3=1 =i 3 3
Sj
j=l
fc=l Jt=i
^l ++ o0(0 (^) 53
:hj .0 + JE"i ^ =l
in fi Q,hA.UQfc,,-. Ui]/, ( i. inQfcUQft,,-. in
;'=i
+f O (0(h?'), A^), (5.2.67)
In (5.2.67), grouping the same terms and letting wp(x) be the coefficient functions of h2f3 (1 < |/?| < 2), we obtain (5.2.59). D R e m a r k 1. When n > 4, Lemma 2.2 is invalid. However, Marchuk proved that the inequality (5.2.44) still holds if a larger ft = ph is used, where p is an appropriate positive integer. Obviously, the corresponding difference scheme will be more complex. R e m a r k 2. Theorems 2.3 and 2.4 are also valid for the elliptic differential equations of type (5.2.1). The crux of the proof is to verify the maximum principle. For details, see B o n W 1 0 ] .
2. SEM for finite-difference
approximations
2.3. Multivariate asymptotic difference approximation
189
expansion of the on a rectangular
finiteparallelepiped
Consider the quasilinear elliptic partial differential equation
Ati = /(*,ti),
inft = IKOAOi
(5.2.68a)
»=i
u = 0,
on dft.
(5.2.68b)
In order to guarantee that (5.2.68) has a unique solution, assume that / ( x , u) is a smooth function on ft x JR, and
(5.2.69)
fi(u):=K(x,u)>0.
Take the mesh width h = (h\, • • •, /i 5 ), where hi = 6,/iV,-, i = 1,•• •, s, and iV,- are positive integers. Let ft/i = {x = (xi, •••,*,) : Xi = jhi} 0<j<
dnh = Qhn 5ft, nh = nh\ 3ft.
Niy 1 < i < s},
Construct a difference operator s
Ahu(x)
= ] T [u(x + hkek) - 2u(x) + u(x -
hkek)]/h2ki
*=i
VxGft*,
(5.2.70)
and the corresponding difference equation Ahuh(x)
= / ( x , tifc(x)),
i t ^ x ) = 0,
in n fc , onfifift.
(5.2.71a) (5.2.71b)
Here, the scheme is simple since there is no irregular point. However, ft is a nonsmooth domain, it is hard to guarantee the smoothness of the solution of the auxiliary problem. The following lemma points out the conditions for the smoothness of the solution, and the relevant proof will be found in [90]. L e m m a 2.4. Consider the equation - A n + du = f,
u = 0,
in ft = (0,b x ) x (0,6 2 ),
(5.2.72)
on 3ft.
If the coefficients d and f G C 1 + a (ft), wh ere 0 < cr < 1, and / ( x ) = 0 at the corner points o/ft, then the solution u G C 3+
190
V. SEM for differential
equations
L e m m a 2.5. If the coefficients of the equation (5.2.72) d and f G C 3+<7 (Q) and satisfy the coherence condition at the corner points of Q d2f d2f = 0, 0, -— (Wx ) - - W r(*) = = 0, 0, //(x) (s) =
(5.2.73) (5.2.73)
then uGC 5 + < 7 (ft). It is easy to see that in Lemmas 2.4 and 2.5, the coherence condition at the corner points are not only sufficient but also necessary. In fact, if u G C 3+
+
8d2u(x) — = /(x). dx\
°=^r ^r= / w -
(5(5.2.74) 4)
"
In addition, if u £ C5+" at the corner points we also have the following: iz(x) d4u(x)
00 = =
= (
dx\ 5d2
2 dx dx\ d2f
3d44u(x) u(x) 5d2
dx\
dxl dx\ d2f
)Au{x) )Au(x)
== dxf — o(x)o —dx\ 0 y(x). 0.
(5.2.75) (5.2.75) K }
2
dx dx\ Lemmas 2.4 and 2.5 can also be generalized to multidimensional cases. Lemma 2.6. For the equation (
s3 6 in ft = n (°> j)> 5s >> 3>3, (0, &j), i=i on 3Q, on 3Q,
-Au + du = / ,
< II
uu = = 0, 0,
(5.2.76) (5.2.76)
if the coefficients d and_J G C 1+<7 (Q), then the necessary and sufficient condition for u G C 3+ *(fl) is /f(x) ( x ) = 0, Vx G G M.
(5.2.77)
3+or
Furthermore, if d and f G C (ft), then the necessary and sufficient conditions for u G C 5 +"(Q) are (5.2.77) and 02f
d2f
3xt3xf
<9x^ <9xj
w/iere M 15 the set of all k (< s — 2)-dimensional boundaries of£l.
(5.2.78)
2. SEM for finite-difference approximations
191
Proof. First, consider the case 5 = 3. It is easy to prove the necessity of the conditions. Consider an arbitrary point on one of the edges, e.g., A = (ai,0,0), 0 < ai < 6i, if ii G C 3+Qr (ft), then the vectors with A as a vertex, and rays parallel to any one of the three coordinate axes, are all on 9Q. Hence by the boundary condition, the following equality holds, 2
rd
d2u
u
d2u-i
LsrsrsjU"'™-
0 = 5+C
<5 9)
"
If u G C *(Q), then two inward normal vectors at point A lie on 80, and are parallel to 0 x 2 and Ox^ respectively. Hence, dAu
0
dAu =—(A)-—(A) ox\
ox\ K
dx\ 2
,d = ( K
dx2
d2f
dx\'^dx\ 2
d
dx\> '*=*
d d2 32N ) ( — + — + — In 2
w
dx\)K2 dx2
dx2j 1-=^
dx\
df
= — - (A) - —(A). dx\ dx\ Here the condition
92u
(5.2.80)
(A) = 0 is used. Therefore, the coherence conditions
dxi
on the vertices and edges for s = 3 are necessary. In order to prove that the conditions are sufficient, take an arbitrary point A = (ai, a 2 ,03) G ft. Let fii = ftf){x : xi = a i } , which is a cross sec tion perpendicular to the xi-axis and passing the point A. By tii(x 2 , X3) = u(ai,X2,X3), obviously, u\ is the solution of the two-dimensional problem
d2u\ ~
=
d2u\ r +
du
i
dx\ d2ux
dx\ d2ux d2Ul
dx\
dx\
dx2
= /(ai,x2,x3), in fii, i/! = 0, onffii,
+ dux (5.2.81)
and satisfies the coherence condition of Lemma 2.4 (or Lemma 2.5 respec tively), which implies that the 3-rd order (or 5-th order respectively) deriva tives of u(ai, X2, X3) with respect to x 2 and X3 are Holder continuous. Since
192
V. SEM for differential
equations
di may be any value within (0, fei), then for a fixed xi, u(xi, x 2 , X3) belongs to C 3+<7 (Qi) (or C 5+
+
d2v1 ^ +
dVl
= f
*>(ai>X2>X3)>
vi = 0, with fXl(x1)x2ix3)
inQl
>
(5.2.82)
on 3fii,
d
If / G C 1+ "(Q), then fXl G C"(0)
= —f(xuz2,x3). axi
and t;x G C 2+<7 (Qi). Since vi(x 2 ,x 3 ) = -r—(ai,x 2 ,x 3 ), thus - — — , o*\ dxidx^ d3u 33ti 3 2+a dx\dx\ and dxidx dx3 G C'(Q). If / G C +"(ft), then / P l G C (fi). 2
It is easy to prove that / r i ( a i , x 2 , X3) at four corner points of fii satisfies: - - = 0, which leads to v1 G C 4+Qf (Qi), or dx\ dx\ Dau G C^Q), |a| = 4, c*i = 1. Similarly, construct a cross section passing the point A and perpendicular to the x 2 -axis, one can also prove that the conclusions are valid for other mixed derivatives of u(x). For the case n > 3, the conclusion of Theorem 2.4 can be proved by induction, which is omitted here. D f(ai,x2}x3)
= 0,
Theorem 2.5. Let u be the solution of the problem (5.2.68). If u G C 5+
h
+ ^Tft'u/,- = 0(&o+")>
u -u
in Qh
>
(5.2.83)
t=i
where W{ G C 3+(7 (ft) (i = 1, • • •, s) are independent ofh. Proof. Since u G C5+"(Q), then v ^
A*II-AII = £ i=l
Let Vi
1 04« = ——e I2dxj
C1+°(n),
/i2
&
u
^ _
:0r?
+ o(fcj|+*)f
i = 1, • • •, s. Noting that
uh = u + O(h20),
innh,
infl*.
(5.2.84)
2. SEM for finite-difference approximations
193
there is (Ah - Mu))(uh
- u)
h
= f(x, u ) - f(x, u) - A h u + Au - /i(w)(« h - «) 8
= - X > , ? f . + 0(ft|!+<7),
infi„.
(5.2.85)
«= 1
Construct an auxiliary problem (A - f\(u))wi
= t;,-,
in Q,
<
(5.2.86) w;t- = 0,
on dfi, t = 1, • • •, s.
We shall now prove w{ G C 3 + a (ft). Since fx(u) and v{ G C 1 + *(fi), by using Lemma 2.6, it is sufficient to prove Vi(x)
= 0, Vx G M,
(5.2.87)
where M is the set of all k (< s — 2)-dimensional boundary points. In fact, take any x G Af, and draw a sufficient small segment /,- starting from the point x and parallel to the x,-axis, obviously, /,- C d£l. It follows from the boundary condition (5.2.68b) that ^ = vi(x) = 0, Vx e M, 12 5x?
(5.2.88)
which leads to Wi £ C3+
(5.2.89)
Substituting (5.2.86) and (5.2.89) into (5.2.85), one can derive
(Ah-f1(u))(uh-u
+
J2^wi) »= 1
*=i
t=i
= £>?(**-A)«*+0(0 »=i
= O(h30+a),
infi^,
(5.2.90a)
and the boundary condition uh
-
u
s
+ ] C A i ?u;i = °'
o n dQh
'
(5.2.90b)
V. SEM for differential
194
(5.2.83) follows from the discrete maximum principle.
equations
□
If u G C7+
^2
h w
^ ^
= °( fc o + *)>
in
"»>
(5.2.91)
1
where /? = (ft, ••-,&) and wp G C 7 - 2 M+"(n) (1 < |/?| < 2) are indepen dent ofh. Proof. Since u G C 7+
Ahu-Au=
^2
h2pvp+0(hl+°),
(5.2.92)
1<|/?|<2
where vp G C5+<7~2'^(f2) are independent of h. Using Theorem 2.5 and (5.2.92), one can derive = / ( x , uh) - / ( x , u) - /i(t*)(tifc - ti) + Ati =
X)
^ 2 % + 0(^+
Ahu
inO fc>
(5.2.93)
1<WI<2
where wp G C5+<7~2I^I(Q) are independent of h. Construct an auxiliary equation ( A n - fi(u))wp = wp,
in Q,
<
(5.2.94) wp = 0,
on dft,
and the corresponding approximate equation r
( A * - / i («))«£ = £ , , <
innft, (5.2.95)
u>/j = 0,
on dfi/,.
Now we shall prove wp G C7+a-2M(Ti), 1 < |/?| < 2. First, consider the case \f}\ = 2. In order to prove u^ G C 3+ "(ft), by using Lemma 2.6, it is sufficient to prove wp(x) = 0, Vx G M, where M is the set of all k (< 5 - 2 ) dimensional boundaries. Comparing (5.2.93) with the identity f(x , « A ) - f(x, u) - h(u)(uh
- u) = 0, Vx € M,
(5.2.96)
2. SEM for finite-difference approximations
195
one obtains vp(z) = wp{x), V 1 6 M , 1 < \/3\ < 2.
(5.2.97)
But vp is a 6-th order partial derivative of u, hence, it follows from u\dn = 0 that vp(x) = 0, V* e M, and then wp € C 3+
= 0, V* €
= ^-(x)
5+
M, i = 1, • • .,*. By Lemma 2.6, wp G C (fi). Applying Theorem 2.5, the approximate solution w^ (\/3\ = 1) of wp has an asymptotic expansion h2aw0a
E
Wp - Wp =
+ 0(h30+°),
in Qh,
(5.2.98)
|o|=l
where wpa are independent of h. By (5.2.93) and (5.2.95), we have
E **S)
( A * - A («))(«*-«-
1
= ( A * - / ! ( « ) ) (tl* •-u-
E
i<\P\<2
^ % + £ V'frfi-WfiJ) i^l/^l^s
2
- « - E * % + £ E/i2(/J+0f)^
= (A* - / ! ( « ) ) ( « * ■
1
|/9|=l|o|=l
+<W)) = 0{hl+"),
(5.2.99)
infih
Grouping the terms with the same power of h in wp and u;^ 0 , from (5.2.99) one can derive ( A * - A ( u ) ) ( « 'k - « -
ft2%)=0(/«^).
E
infih,
1
(5.2.100a) and uh-
u-
£
h2/3wl) = 0, on<9Qh.
(5.2.100b)
1
(5.2.91) follows from the discrete maximum principle.
□
R e m a r k . Marchuk and Shaidurov a s s e r t e d ^ that the auxiliary prob lem may not satisfy the coherence condition, and then the expansion to
V. SEM for differential equations
196
higher order may not hold even for the case / G C* + a (Q), k > 4. However, the proof of Theorem 2.6 is under the assumption u E C 7 + a ( Q ) , hence, the expansion to higher order can also be proved provided that u is sufficiently smooth.
2.4. Numerical
examples
E x a m p l e 2 . 1 . Consider the two-dimensional Poisson equation - A n = /, u = 0,
in Q = (0, l ) 2 ,
(5.2.101)
on dfi,
with the exact solution u(x, y) = x(l—x)y(l—y) cos(7rx/2) cos(iry/2). Using the five-point finite difference scheme and solving the linear equation by the conjugate gradient method, the results of the Richardson extrapolation and Types 1, 2 splitting extrapolation methods are given in the following tables: Table 2.1. Step size hx = hy = J number of
Richardson extrapolation
extrapolations max error
CPU(sec)
Type 1 SEM
Type 2 SEM
max error CPU(sec) max error CPU(sec)
0
2.236E-3
0.01
2.236E-3
0.01
2.236E-3
0.01
1
1.230E-5
0.17
6.564E-5
0.12
6.564E-5
0.14
2
4.113E-8
1.59
6.159E-7
0.71
5.977E-7
0.52
3
7.008E-11
14.60
1.406E-8
3.80
1.754E-8
1.80
4
9.6E-14
125.13
3.215E-10
17.69
8.865E-9
4.35
8.739E-12
79.32
5.266E-9
9.84
5
Table 2.2. Step size hx = hy = ^ number of
Richardson extrapolation
extrapolations max error
CPU(sec)
Type 1 SEM
Type 2 SEM
max error CPU(sec) max error CPU(sec)
0
6.111E-4
0.14
6.111E-4
0.13
6.111E-4
0.13
1
8.825E-7
1.65
4.724E-6
1.43
4.724E-6
1.45
2
6.786E-10
15.10
1.168E-8
8.16
8.086E-9
5.74
3
1.729E-10
40.02
5.635E-9
17.60
4
3.803E-12
179.00
9.000E-9
42.83
3.0514E-9
93.20
5
The solution of this example is smooth, and so three kinds of extrapola tions are all effective. The Richardson extrapolation has the best stability but the amount of work and storage increase rapidly with the number of
2. SEM for finite-difference
approximations
197
extrapolations, while it increases much slower for the splitting extrapola tion, especially for the Type 2 methods. However, due to the poor stability, the accuracy will be limited when the number of splits is large. E x a m p l e 2.2. Consider the two-dimensional Poisson equation (5.2.101). Its exact solution u{xy y) = sin(16x -f 16y)x(l — x)y(l — y) is an oscillatory function. The numerical results are listed in Tables 2.3 and 2.4. Table 2.3. Step size hx = hy = J number of Richardson extrapolation Type 1 SEM Type 2 SEM extrapolations max error CPU(sec) max error CPU(sec) max error CPU(sec) 0 2.704E-1 0.02 2.704E-1 0.02 2.704E-1 0.02 1
6.110E-2
0.19
2.823E-1
0.14
2.823E-1
0.15
2
2.730E-2
1.95
9.970E-2
0.75
1.019E-1
0.56
3
1.733E-3
18.04
4.23
4
4.895E-4
152.11
1.591E-2 1.122E-3
19.92
1.927E-2 2.227E-3
4.69
4.064E-5
89.47
1.83
1.751E-4
10.69
6
9.860E-6
20.76
7
2.696E-6
37.83
5
Table 2.4. Step size hx = hy = ^ Type 1 SEM number of Richardson extrapolation Type 2 SEM extrapolations max error max error CPU(sec) max error CPU(sec) CPU(sec) 0.17 2.673E-2 0.16 0.16 2.673E-2 0 2.673E-2 1
1.485E-3
1.86
4.992E-3
1.79
4.992E-3
1.64
2
6.040E-3
17.28
3.538E-4
9.70
3.518E-4
6.53
3
1.551E-5
47.58
1.630E-5
20.25
4
3.538E-7
213.64
6.464E-7
48.49
5.578E-8
105.91
5
In this example, the accuracy of the first few approximations obtained by SEM is not improved significantly since the relatively large initial mesh widths hx and hy cannot reflect the oscillation of u(xy y). The accuracy in creases rapidly with the number of extrapolations. In addition, for the Type 2 methods, the CPU and storage only slightly increase with the number of extrapolations. By contrast, the accuracy of the approximations obtained by the Richardson method decreases with the number of extrapolations, because the global refinement yields a large linear system. E x a m p l e 2.3. Consider the 3-dimensional Poisson equation
f Au = f, u = 0,
inn = (0,l)3, on d£l,
V. SEM for differential equations
198 3
TTXi
t=i
2
with the exact solution u(xi, x2) x3) = ]\ [*«(! — x%) c o s (
)] • The com-
parison between the numerical results and storage of the Richardson ex trapolation and the splitting extrapolation are given respectively in Tables 2.5 and 2.6. And the comparison of CPU for solving subproblems with dif ferent degree of parallelism on a multiprocessor computer is given in Table 2.7. =fco= i Table 2.5. Step size h\ — = hh~ 2 = ^3 = J Type 2 SEM
Type 1 SEM
Richardson extrapolation
number of
extrapolations max error
max error CPU(sec) max error CPU(sec)
CPU(sec)
0
3.91E-4
0.04
3.91E-4
0.04
3.91E-4
0.04
1
2.44E-6
0.71
6.79E-6
0.52
6.79E-6
0.49
2
7.61E-9
13.6
1.90E-7
2.53
1.90E-7
2.09
3
3.23E-11
285.67
5.06E-9
12.98
5.32E-9
7.41
4
1.41E-10
21.94
5
7.89E-12
57.11
Table 2.6. Comparison of storages for three extrapolation methods
(hx = h2 = h3 = J.) number of extrapolations Richardson extrapolation Type 1 SEM Type 2 SEM 0
162
162
162
1
1769
339
339
2
16956
945
945
3
149063
2135
2135
4
1250370
4410
3430
5
2048543
9051
5411
Table 2.7. Comparison of degree of parallelism for three extrapolation methods number of
Richardson extrapolation Type 1 SEM Type 2 SEM
extrapolations N
max CPU
JV max CPU N max CPU
1
2
0.33
4
0.13
4
0.13
2
3
12.94
10
0.38
10
0.38
3
4
272.01
20
1.42
20
0.67
4
5
35
35
1.44
5
6
55
55
2.54
Here, N denotes the number of subproblems and max CPU denotes the CPU needed by the largest subproblem. Table 2.7 shows that the degree of parallelism of the SEM is high.
2. SEM for finite-difference
approximations
199
E x a m p l e 2.4. Consider the 4-dimensional Poisson equation AM
= /,
U = 0,
in Q = (0, l ) 4 ,
on 3ft, WXi
with the exact solution u(xi, x 2 , a:3, z 4 ) = fj[ [x,(l - a?,-) cos ( ) ] . For in*=i 2 formation on the accuracy, CPU time, storage and the degree of parallelism, see respectively the following tables: Table 2.8. Comparison of accuracy for three extrapolation methods (*! = h2 = h3 = fc4 = \) number of Richardson extrapolation Type 1 SEM Type 2 SEM extrapolations max error CPU(sec) max error CPU(sec) max error CPU(sec) 0 6.8E-5 0.18 6.8E-5 0.14 6.8E-5 0.14 4.76E-7 1 8.23 3.24E-6 1.85 3.24E-6 1.91 2.10E-9 2 407.30 8.21E-9 14.48 8.26E-8 12.73 2.00E-9
3
60.86
Table 2.9. Comparison of storage for three extrapolation methods (hX = h2 = fc3 = *4 = \) number of extrapolations Richardson extrapolation Type 1 SEM Type 2 SEM 0
486
486
486
1
12197
1350
1350
2
253368
1656
1656
3
4617929
7980
7980
Table 2.10. Comparison of degree of parallelism for three extrapolation methods (hi = h2 = h3 = /14 = - ) number of Richardson extrapolation Type 1 SEM N max CPU extrapolations N max CPU 5 0.46 8.04 2 1 1.47 15 399.05 3 2 3
31
Type 2 SEM N max CPU
5
0.46
15
1.32
31
3.52
This example shows that the higher the dimension, the more effective the SEM, and the higher the degree of parallelism. E x a m p l e 2.5. Consider the two-dimensional inhomogeneous boundary value problem of the divergence type d du d <9u -(e-5*—) + — (e-5*— ) = / , inQ = (0,l)2, dx on dQ, u = 9,
200
V. SEM for differential
equations
with the exact solution u = (1 — e 5 ^"" 1 )) sin(7ry). The numerical results are given in Table 2.11. T a b l e 2 . 1 1 . (hx = hy = number of
Richardson extrapolation
extrapolations max error
max CPU
\)
Type 1 SEM
Type 2 SEM
max error max CPU max error max C P U
0
2.22E-2
0.09
2.22E-2
0.02
2.22E-2
0.05
1
3.68E-4
0.23
1.48E-3
0.15
1.48E-3
0.20
2
1.73E-6
1.96
2.74E-5
1.22
2.72E-5
0.60
3
3.93E-9
105.13
3.96E-7
11.03
5.33E-7
2.08
4
2.68E-12
591.71
3.75E-9
47.42
8.57E-9
7.88
5
3.0E-15
1282.95
1.76E-11
274.8
1.14E-10
22.78
6
1.20E-12
52.85
7
3.00E-14
155.45
3. S E M for finite-element approximations The extrapolation of the finite element method is a new research field evolved in the last decade, it develops rapidly and has a wide scope of application. Recent works show that the extrapolation and asymptotic ex pansion of the finite element approximation have become the bases tone for studying the theory of finite element superconvergence, algorithms with a high order of accuracy and a posteriori error estimate. For details, see the monograph by Q. Lin and Q.D. Zhu^'^ and the summary by R. Rannacher^ 10 ^. The extrapolation of the finite element method has a great advantage over that of the finite difference method. First, the latter has its limita tion: for smooth domains, the boundary interpolation technique can help to derive asymptotic expansions, but at the same time, it also destroys the uniformity of the scheme and makes the coefficient matrix nonsymmetric. Second, the finite difference method has theoretical defects, e.g., in order to extrapolate once, it requires the exact solution u € C 5+
3. SEM for
finite-element
approximations
201
equal to the dimension of the problem, while in the finite element method, the independent mesh parameters can be set according to the shape and size of the problem and the characteristics of the computer. And a large prob lem can be converted into several smaller subproblems and then solved by SEM on a multiprocessor computer. It is concluded that the finite element splitting extrapolation is more suitable than the Richardson extrapolation in solving large scale problems on multiprocessor computers. 3.1. Finite-element equations
approximations
of second
order
elliptic
In the following we shall introduce briefly the three main parts of the finite element method, namely, the weak form of a problem, the subdivision of a domain and the Ritz-Galerkin method. Consider a second order elliptic equation: s
(
- ^2 DjiaijDiu)
+ du = / ,
in fi C JR5,
(5.3.1a)
on Ti C dQ,
(5.3.1b)
onr0 = an\ri,
(5.3.ic)
M=l I
\
5
^2 aijDiUUj = g, »,i=i
( ti = o,
where v = (i/i, • • •, i/s) is a unit outward normal derivative, (5.3.1b) is called the natural boundary condition, and (5.3.1c) the coercive boundary condi tion. In order to derive the weak form of (5.3.1), construct the following function space: V={vGH1(Q):v\rQ
(5.3.2)
= 0}.
Obviously, V is a Hilbert space, satisfying the inclusion relation Hi(Q) CVCH
= H*CV+
C
H'\il)y
where H = L2(Q,). Construct a bilinear form on V a(u,v) = I (Y\
aijDiuDjV + duv)dx.
(5.3.3)
a ( , •) is said to be V-elliptic, if there exists a constant y > 0, such that a f t i . t O ^ T l M f t , Vti€ V.
(5.3.4)
Obviously, if the matrix [a^-]. = 1 is symmetric and uniformly positive def inite on Q, and d G L°°(ft), d > 0, then (5.3.4) holds.
V. SEM for differential
202
Definition 3.1. ueV
equations
is called a weak solution of (5.3.1), if it satisfies a(u,v) = f(v),
VvG7,
(5.3.5)
where the right-hand side of (5.3.5) f(v) = / fvdx + /
Jo,
Jr1
gvdx
is a continuous linear functional on V. By using the Lax-Milgram theorem^231, the ^-ellipticity guarantees the existence and uniqueness of the weak solution. The Ritz-Galerkin method is to find a class of finite dimensional subspaces Vh! C Vh2 C V for hi > h2, and the corresponding approximate solution uh EVhy satisfying a(ti\i;) = /(«), V w € % .
(5.3.6)
Furthermore, by using the Lax-Milgram theorem, the approximate solution satisfies uh = Rhu,
(5.3.7)
where Rh : V —► Vh is a Ritz projection, i.e., the orthogonal projection takes a(-, •), defined in (5.3.3), as an inner product. In other words, uh G Vh is an optimal approximation to u in the sense of inner product a(-, •). Thus, the density of \J Vh in V implies the convergence of the Ritz-Galerkin method. /i>0
The finite element method is a special case of the Ritz-Galerkin method. The difference between them is that the finite element space Vh is a set of piecewise polynomials, and the classical Ritz-Galerkin method often regards Vh as the set of smooth functions. Hence, the partition Jh — {e} of a domain Q is the first step of the finite element method, where e stands for a partition element. However, the following restriction is imposed: suppose that t\ and e2 are two different elements, then either t\ D e2 is an empty set, or they have a common vertex, or a common side. The most important elements are the triangular and rectangular in two-dimensional problems and the cuboid in three-dimensional problems. If Jh is a triangulation of fi, one can then define linear finite element subspace Sh(Q) = {ve C(Tl) : v\e € Fi(e), Ve €
Jh}}
where P\(e) denotes the set of linear functions on e. If Jh is a rectangular partition of fi, then one can define a bilinear finite element subspace S h (fi) = {v e C(ty : v\e € Qi(e), Ve €
Jh),
3. SEM for
finite-element
203
approximations
where Q\{e) denotes the set of bilinear functions on e. For the threedimensional cuboid elements, the trilinear finite element subspace can be established similarly. Now take Vh = Sh(Q) C\ V> the corresponding finite element approximation uh G Vh satisfies a(uh,v) = f(v),
Vw€ % .
(5.3.8)
Applying the basis functions (pj, (5.3.8) can be expressed as an equivalent algebraic equation. There is an one-to-one correspondence between a basis function and a node. The j-ih basis function
= 6ij
=
0,
i±h
hj = l , - - - , m ,
where N{ are nodes, m is the number of nodes excluding those on To, and it is the dimension of Vh. Let m
"*(*) = 5>;WOO'
(5.3.9)
where rjj = uh(Nj), j = 1, • • •, m, are unknowns. Substituting (5.3.9) into (5.3.8), one obtains a linear system of 77 = (771, • • •, rjm)T Arj = 6,
(5.3.10)
where A = [a,j] is an m x m matrix with elements atJ- = a(
204
V. SEM for differential
3.2. Basic
expansion
of bilinear
rectangular
equations
elements
For simplicity, consider the Dirichlet problem of Poisson equation - A n = /,
in ft, (5.3.11)
u = 0,
on dft,
where ft C M2. Suppose that Jh is a rectangular partition of ft, and So (ft) C ^ o ( ^ ) ls ^ n e corresponding space of bilinear elements, the finite element approximation uh G SQ satisfies a(uh,v)
:= / (DiuD\v + D2uD2v)dxdy
= / fvdxdy,
Vv G S#(ft),
(5.3.12)
d d with D\ = — and Di = — . Define the interpolation operator Ih : C(ft) —* 5g(ft), where the in terpolation basis points are the nodes of Jh. Let e G J ^ be an element defined by e = (xe - h e , X e + / l e ) X (T/C - * e , 2/e + * e ) ,
(5.3.13)
where (x e ,y e ) is the center of e, while /i e and ke are the halves of two edge-lengths of e respectively. Let hi = max{/i e },/i2 = max{ibe} and eeJk eeJh h0 = max{/ii, A 2 }. We shall prove the following integral identity: Lemma 3.1.1''' For any rectangular element e G Jh, the following integral identity holds: / Di(u — Ihu)Divdx.dy Vi; G S*(ft),
= I B{y)Dl{Dl{u^Ihu)Dlv)dxdy,
(5.3.14)
where (5.3.15)
B{y)^\[{y-yef^kl\. Proof. Since B"(y) = 1, IDi(u-
Ihu)Divdxdy
= I B"(y)Dl{u
-
Ihu)Divdxdy.
3. SEM for finite-element approximations
205
On the right-hand side, integrating by parts with respect to y twice, one obtains (5.3.14). a L e m m a 3.2.( 7 7 1 If u G % 3 ( f i ) (2 < p < oo), then there 15 a superconvergent estimate a(u - Ihu, v) < Ch20\\u\\3)P\\v\\iy,
Vt, G S£(fi),
(5.3.16)
where p' satisfies - + — = 1. p p P r o o f . Note t h a t F(y) = \[B{y)}2.
B{y) = F"(y) - V o
(5.3.17)
0
Substituting into the right-hand side of (5.3.14), integrating by p a r t s and applying the inverse estimate, one obtains
J B(y)D2(D1(u -
h^D^dxdy
= j Biy^DlDxuDiV
+ 2DiD2(u
= J B(y)D2D1uD1vdxdy =
+ 2DlD2v
I B{y)D\D1uDxvdxdy-2
< C*fc>| 3 ,p, e |t;|i, p ',e +
< Ck2e\u\3,pMhP>,',
-
Ihu)DlD2v]dxdy J B(y)D iZ?2(« — Ihv)dxdy
j F'\y)D1D2v
■
DiDJudxdy
Ck3e\u\3,Pte\vh,P',e
Vt;€50h(n),
(5.3.18)
where, and hereafter, C is a constant independent of the partition and the notation | • \m,p,e denotes a semi-norm of Wpm(e). Substitute (5.3.18) into (5.3.14), it becomes / Di(u - Ihu)Divdxdy
< Ch20\u\3)P>e\v\1)P>te.
(5.3.19a)
j D2(u - Ihu)D2vdxdy
< Chl\u\3,ptc\v\ilP;e.
(5.3.19b)
Similarly, there is
Finally, by the Holder's inequality, we obtain a(u-
IhU,v)
206
V. SEM tor differential
= £
r /
< Ch20 ]T)
equations
2
(.52Di(u-Iku)Div)dzdy \u\3,p,e\v\iy,e
e€Jh
< chK Y: \
e£jh
< Cfc2||n||slPln||t;||ilp'ln, Vu G 5j(0). T h e proof of (5.3.17) is thus complete.
O
It is known t h a t for a bilinear finite element space, t h e interpolation error satisfies \\u-Ihu\\1)P
(5.3.20)
For a linear finite element space in a uniform triangulation or in a piecewise uniform triangulation, a similar super convergent estimate as (5.3.16) is still valid, b u t t h e proof is n o t as simple as t h a t for t h e case of rectangular elements. In t h e following argument, t h e crux is t o apply an auxiliary function B(y) which was suggested by Q. Lin in [70]. By this technique, we have t h e following theorem: T h e o r e m 3.1.'''J Suppose that Jh is a regular rectangular partition, n SQ(Q) is a bilinear finite element subspace. If u G ( f l ^q(e)) #o(^)> eeJh (1 < q < oo), then forVv G SQ(£1), there is an asymptotic expansion a{u — u1 ,v) =
D
+
-2F'(y)D22D1uD1D2v]dxdy
{ / [B{y)D\DluDlv
I [A{X)D\D2UD2V
-
2E'{x)DlD2uD1D2v\dxdyy
(5.3.21a) ifu G ( n
< ( e ) ) Hi/oHQ), (1 < q < oo), then
e€Jh
a{u — u1 > v)
3. SEM for
finite-element
207
approximations
2
= 2
h { " J [%DlD2uD2vdxdy-
e£Jk
-1
e
j^DlDmD^dxdy e
+ / [ - D32D1uD2{F{y)Dlv)
+ 4F(y)D^D1uD1D2v]
+ j [- DfD2uD1(E(x)D2v)
+
dxdy
AE{x)D\D2uD1D2v]dxdyV (5.3.21b)
and ifu € ( n
W*(e)) n Wf(Q) n #<J(fi),
a(w — u1, v)
= £ { ? [(hl + k*)DlDl«-vdzdy + f [E(x)(D^DiuDiv +F(y)(D\DiuD2v
-
ADlDluDiv)
- AJ^2DluD2v)]dxdy^,
(5.3.21c)
where u1 = I^u, and E(x) = W ) ] 2 , o
A(x) = l-[(x - xef - h*\, 2
F(y) = W ) ] 2 ,
B(y) = \[(y - yef - k%
0
Ac = he/hi,
(5.3.22)
I
He =
ke/h2.
Proof. By using the integral identity (5.3.14), it follows that / D\(u — uI)D\vdxdy = j
B(y)Dl(D1(u-uI)Div)dxdy
= j B(y)DlD1uD1vdxdy+
I 2B(y)D1D2(u
-
uI)D1D2vdxdy
= Vi + J 2 , VvES£(fi).
(5.3.23)
By using (5.3.17) and integrating by parts, we have J2 = 2 jF'\y)DlD2{u
-
u^DxD^dxdy
208
V. SEM for differential
2 f —k 2 e / o
equations
DiD2(u-uI)D1D2vdxdy
Je
= - 2 /F'{y)DlD 1 uD l D 2 vdxdy.
(5.3.24)
Substitute into (5.3.23), it becomes / Di(u — uI)D\vdxdy =
IB{y)DlDiuDivdxdy - 2 f F\y)DlD1uD1D2vdxdy.
(5.3.25a)
Similarly, / D2(u — uI)D2vdxdy -
j A(x)D\D2uD2vdxdy
- 2 I
E\x)D\D2uD1D2vdxdy.
Vt; E Sj(fi).
(5.3.25b)
Summing up the integrals for all e G Jh, the proof of (5.3.21a) is complete. For u G ( I ] W?(«)) n # £ ( 0 ) , substituting (5.3.17) into (5.3.23), and integrating F"(y) by parts twice, we obtain / Di(u — uI)D\vdxdy --k2e
= --JbJ I DlDmDivdxdy + j
D2D1(u-uI)D1D2vdxdy
F"(y)DlD1uD1vdxdy
+2 j F(y)Dl[D2D1(u
- tiJ)Z}iJD2t;]
= ~-k2efDlD1uD1vdxdy+ +ADlDiuDiD2v]dxdy = --k2e
I
JDlDxuDivdxdy
fFiy^DmDxv
3. SEM for finite-element approximations
209
+ [[-D$D1uD2(F(y)D1v)
+
AF{y)D\DluDlD2v]dxdy. (5.3.26a)
Similarly, / D2(u — uI)D2vdxdy = — h \ I D\D2uD2vdxdy o Je I[-D\D2uD1{E(x)D2v) +■ Je ->3 +4E(x)DfD 2uD1D2v]dxdy.
(5.3.26b)
Summing up the integrals for all e G Jh, the proof of (5.3.21b) is complete. For u G W*(Q) C\ ( U W*(e)) H H%(Q), noting that for the rectaneeJk gular partition, when integrating by parts in the y-direct ion, Ae remains unchanged, then ^2
I ^lD\D2uD2vdxdy
= - ^
I
\\D\Dluvdxdy.
Similarly, when integrating by parts in the ^-direction, fie remains un changed, and ^2
I nlDlDxuDxvdxdy
= - ^2
I&V\V\wdxdy.
Therefore, integrating (5.3.21b) by parts, (5.3.21c) is obtained. Corollary 1. If u G ( II
w
?(e))nHo(fy>
(? =
2
D
or oo), then there
are the following superconvergent estimates:
where ||u||' m>| = ( £
\\uh - u% < Chl\\u\\'3,
(5.3.27a)
\Wh ~ u'Hi.00 < Ch\\ lnfto|||«||^oo.
(5.3.27b)
IMIX,,,,.) 17 '
and
Nlm.oo = ™ « I H k o o . . .
210
V. SEM for differential
equations
Proof. Set v = uh — u1 in (5.3.21a). By using the inverse estimate, it follows that
ii« k -« 7 n?
< Ca(u-u\uh
-u1)
= chl\\u\\3\\uh-u%. Eliminating the common factor on both sides, the proof of (5.3.27a) is complete. In order to prove (5.3.27b), the estimate of the discrete Green function G\ in W\(Si) norm is needed. The discrete Green function Gj is defined to be the Ritz projection of the Green function GZi where z = (x,y) is a fixed point in Q, so Ghz satisfies a{Ghz,v) = v(z), Vv G 5j(Q).
(5.3.28)
Denote by dzGhz the partial differential of Ghz with respect to x or y. There is the following estimate*36)' I 132 !; Il«-G*|| ltl = 0(|lnAo|).
(5.3.29)
By using (5.3.28) and (5.3.27a), we have |0,(«*-«')(z)| =
\a{u-uI,diGhz)\
< CASHiillUP-GjlU,!
D
Corollary 2. Under the assumptions of Theorem 3.1, if u £ ( J~[ eeJh W*(e)) n i / £ ( Q ) , (g = 2,oo), then there exist wi,w2 € #o(ft), such that \\uh - u1 - h\Rhwl
- h\Rhw2\\i
< CAg||ii||i,
(5.3.30a)
and \\uh -u1
- h\Rhwl
- h22Rhw2\\liO0
< Chl\ lnfcolNIi, (5.3.30b)
3. SEM for finite-element approximations
211
where J?j, stands for the Ritz projection operator. Proof. Define linear functionals on Hl{Q) as follows: F v
^ ) = -» £ [%DlD2uD2vdxdy, 6
Je
eeJ»
and F
*(v)
= - : E
flilrftDiuDivdxdy.
1 Obviously, |F,(v)| < — ll^llsllvj|i» > = 1,2. By using the Lax-Milgram o theorem, there exist two functions
ti^w^njnflftn), • = 1,2, such that a(wiy v) = Fi(v),
Vi; G ffJ(fi), i = 1,2.
(5.3.31)
Substituting into (5.3.21b), we have a(u — u , v) = ft?a(u;i,i;) + /ila(w 2 ,t;) + 0(ftg)||ti||i||t;||i, Vw€5j(fi). Note that a(u;t-,t;) = a(i?/ju;t-,i;), Vv G 5 Q ( ^ ) . h\RhWi — h\RhW2, from (5.3.32), we get \\uh _
M/
(5.3.32) Setting i; = uh — u1 —
_ fc2/JfcU;i - &liifcu;2||i < C&g||ti||i.
By the definitions of w\ and W2, from (5.3.21b), it follows that a(y
_ _ / ^ _ ^
w)
< c/igllixll^Ml!,!.
Let v = 9 2 G£. By the estimate (5.3.29), the proof of (5.3.30b) is complete. D
Theorem 3.2.1*'] Under the assumptions of Theorem 3.1, if Q, is a convex angular domain, and u G W*(Q) C\ ( J ] W*(e)) O H$(Q), then eeJh
there exist functions w\ and W2 defined by (5.3.31) such that \\uh _ ui _ h*wi _ h\w%
< Cht\\u\\'^
(5.3.33a)
212
V. SEM for differential
equations
and ||« h - u1 - h\w[ - fcX||o,oo < C^|ln/io||l«ll4,oo> (5.3.33b) where w[ and w2 stand for the interpolation functions of w\ and w2 respec tively. Proof. Using the Nitsche technique^], we shall prove I),!* _ , / _ hlRhW! - hlRhw2\\0
< CTiolMi;,
(5.3.34a)
and (In* - „ ' - hlRhw1 - hlRhw2\\0,oo < Cht\ lnfco||Mi;f00. (5.3.34b) Let
Vv e J5To(0).
Obviously, the following a priori estimate is valid 1Mb < C\\uh -u1-
h\Rhw1 - h\Rhw2\\«.
(5.3.35)
Hence, by using (5.3.30), it follows that \\uh^uI^h\Rhw1^h\Rhw2\\l = a{uh — u1 — h\RhWi — h2RhW2icp) = a(uh -u1
- hlRhivx - h\Rhw2)
+a(uh - u1 - hlRhWx - h\Rhw2, (p1)
- hlRhWx - h22Rhw2i(pI)i
where - + — = 1, q = 2,00. Applying (5.3.21b), c^u* - u J - h\Rhw1 - h\Rhw2y (p1) [DlD1uD2(F(y)D1
= Z) " + J2
I h
eeJ '
^F(y)D^D1uD1D2(pIdxdy
(5.3.36)
3. SEM for finite-element approximations
+ ^2 eej»
213
- IDlD2uD2(E(x)D2
+ ^2 I 4E(x)DlD2uD1D2tpIdxdy (5.3.37)
= Ji + J2 + Ja + JAEstimate J\ by integrating by parts, obtaining
Ji = - ^2 f DtDiuF{y)D1
Again, applying integration by parts to the first term of the right-hand side, it follows that
Ji = - Yl [DtuF(y)Dl(fI -
/
DtuF(y)D1(
-
Jde
eeJ"
(D$uF{y)D\
e€J»Je
*
^ |«k«,e(|v :, |2,«',e + / » o 1 | V / - V | l , 9 ' , e ) k ee^
< CAj|ti|i i f ||v|| 2 , f ..
(5.3.38)
Here, de2 stands for the two sides of the element e parallel to the t/-axis, and the trace theorem is applied in the derivation above. An estimate for J3 can be obtained similarly. The estimates of the right-hand side of (5.3.38) for J2 and J4 are obvious. Substitute the estimates of J i , J2J J3 and J4 into (5.3.37), it follows that h
a(u
- u1 - h\RhWl
- h22Rhw2iiPI)
< Cht\\u\&tq\\
Applying the a priori estimate (5.3.35), and using (5.3.39), the proof of (5.3.34a) is complete. In order to prove (5.3.34b), set q = 00 in (5.3.39), and take (p = g2 G 2 HQ(£1) H i / (ft), where gz is the regular Green function defined by FresheRannacher^"] it becomes |( M * _ _ h\Rhwx
-
h\Rhw2){z)\
214
V. SEM for differential
= a(uh -u1
- h\Rhwi
- h\Rhw2,
equations
G*)
< C&S||n|r4f00||^ - GJIU.1 + CfcJ||ti||l4f00||^||2|i
Hence, by the interpolation estimate of Sobolev space, there are \\RhWi - w{\\o < Chi
i = 1,2,
(5.3.40a)
and \\RhWi - "flkoo < Chm'™W"-£\
(5.3.40b)
where e > 0 is any positive number. Replacing Rh^i and RhW2 in (5.3.34) by w[ and w^ they become \\uh - u1 - h\w[ - hlwT2\\o < Ch40\\u\\'4i
(5.3.41a)
and \\uh - u i - h\w[
< Chmm^^^2-£h
_ h\w%iOQ
(5.3.41b)
Judging from the fact that a convex domain in which rectangular partitions can be implemented must be a rectangle, (5.3.41b) can be improved to (5.3.33b). D T h e o r e m 3.3.1*'] j
addition to the assumptions of Theorem 3.2, if uew?(n)n( n wf5(e)) ni^n), g = 2oroo, eeJh n
then \\uh -u1-
h\RhW! - hlRhw2\\i
||u* - u1 - h\RhWl
||„* _ ui - h\w[
< Chi\\u\\'5,
- h\Rhw2\\hoo
_ h l ^
(5.3.42a)
< Cfc$|lnfeo||Ml5,co> (5.3.42b)
< Chl\\u\\'s,
(5.3.43a)
3. SEM for finite-element approximations
215
and \\uh - u1 - h\w[ - ftXlli.00 < Ch%\ln/i0|||t/|r5. Proof. First, let v = uh - u1 — h\Rhwi
— h\Rhw2
(5.3.43b)
and use (5.3.21c),
IMI? = a(uh - u1 - h\Rhwi =
- h\Rhw2,
Z ) { /E(*)[DiIhuDiv ~ e€J"
v) ^DluDlV]dxdy
Je
+ I F{y)[D\DiuD2v
-
ADlD\uD2v]dxdy\
< C^llwlUlluH'j.
(5.3.44)
Eliminating the common factor on both sides, (5.3.42a) is obtained. Second, set v = dzGhz, we have dz (uh - u1 - hlRhwx = a(uh -u1
-
- hlRhwx
h22Rhw2){z) - hlRhw2i
G))
< C^Hiilli.ooll^Gjlli.i
(5.3.45)
and hence (5.3.42b) is proved. Finally, replace RhW\ and RhW2 in (5.3.42) by w[ and w\, and apply the finite element estimate, the proof of (5.3.43) is complete. □ R e m a r k 1. If the rectangular partition is uniform, then Ac and /i e are constants, these means w\ and w2 are functions independent of h. Hence (5.3.33a) shows that the mean square error of the splitting extrapolation value is 0 ( / I Q ) - (5.3.33b) shows that the maximum error of the splitting extrapolation at the nodes is O(fto|lnfto|). (5.3.43) reflects that the SEM is also effective when it is applied to derivatives of u. R e m a r k 2. It is possible to apply the splitting extrapolation twice, this depends on the smoothness of w\ and w2. If Q is a rectangle, one can judge the degree of smoothness of w\ and w2 by the construction of functionals Fi (i = 1,2) and Lemmas 2.5 and 2.6. R e m a r k 3. Suppose that Q is a smooth domain or a polygon, then there is no rectangular partition of Q. But if Do CC D C ^ , D is covered by a uniform rectangular partition, and D includes no corner point of Q, then by using the theory of interior estimate, one can prove ||,4* _ u1 - h\w[ - ^11^111,00,1)0 ^ C/^|ln/io|,
(5.3.46a)
216
V. SEM for differential
equations
and \\uh - ti J - h\w[ - h22wl\\ltDo < Ch3Q.
(5.3.46b)
Furthermore, if u G W*(D) H ( U W*(e)) n # £ ( 0 ) , then H ^ - _ fc^f _ / i l ^ H x ^ o < Chi
(5.3.47a)
||U* _ u1 - h{w[ - /iMlk°°,Do < Ch$\lnh0\.
(5.3.47b)
and
These facts imply that for some complicated domains where a uniform rect angular partition cannot be implemented, one can first apply the uniform partition to the interior of the domain, and then accelerate the finite ele ment approximation by SEM. The extrapolation method remains effective when it is applied on concave domains, but the convergence may be slow. R e m a r k 4. The bilinear form of a general second order elliptic equation is .
2
2
£ ( u , t ; ) = / (^^aijDiuDjV J*1 t=i j = i
2
+ yY^bjDjv
+
duv)dxdy)($.ZA$)
j=i
and the corresponding Dirichlet problem is: Find u G #o(fi),
sucn
B(u, v) = (/, v), Vr; G Hi(Q).
that (5.3.49)
However, finite element expansions under a uniform rectangular partition are still included in the discussion of the asymptotic expansions of B(u — ii 7 ,t;), Vv G S#(ft). By detailed derivation*771, one can prove that if u G W%{Q)C\Hl(ti)C\ ( [ ] Wj(e)), there exist fourth order differential operators e
L\ and Li independent offt,such that
B(u — u7, v) = hl
$ ^ ( i l W , v ) L 2 ( c ) + ft2 ^ ( L 2 W , v ) L 2 ( e )
+0(Aj)IH| / 5 i f ||t;||i f f ' l
V«6 5 j ( n ) , « = 2,oo.
(5.3.50)
3. SEM for finite-element approximations
3.3. Expansion
217
of three-dimensional
problems
In the following, we shall generalize the above results to three dimen sional cases. Consider the three-dimensional Poisson equation d2u dx2
d2u d2u , + + = / , in Q C M3, dy2 dz2 " u = 0, on afi.
(5.3.51a) (5.3.51b)
Suppose that there exists a cuboid partition Jh on Q. Let Sh = {v e C(Q) : t;|e € Oi(e), Ve € J h } , where SQ = Sh C\ satisfying
HQ(£1),
a(uh,v)
uh is a finite element approximation of (5.3.51)
= f VuhVv Jci
= f /w, Vi; G 5j(fi), Jet
(5.3.52)
where V = ( — , — , — ) stands for a gradient operator, and the integral dx dy dz element dxdydz is omitted. In order to derive an asymptotic expansion of the finite element ap proximation, it is necessary to prove the relevant integral identity similar to that of the two-dimensional cases. First, if e E Jh, let (xeiyeize) be the center of e, and 2hej2ke, 2re be the edge-lengths of e, hence, e = (xe-/ie,xe
+ he)x(ye-keiye
+ ke)x(ze-re,ze
+ r e ). (5.3.53)
Construct the following auxiliary functions:
E{x) =
\[{x-xey-h%
F(y) =
l[(y-ye?-kl],
and G{z)=l-[{z-zef-r%
(5.3.54)
which have the following properties: F'(y) = y-ye, (**)<»> = 6 ( » - y . ) , (F3)<4) = 4 5 ( y - i / e ) 2 - 9 f c e 2 .
(5.3.55)
V. SEM for differential
218
We denote by u1 the interpolation of u on Sh(ti).
equations
Let w = (u - u% =
1
by parts it is easy to prove the following —(u — u ). Using ntegration i dx integral identity:
=
fa fa
F'(y)G'(z)wyM), (5.3.56a)
= J (F(y)wxx + G(z)w2Z ye)w = f (-(F\y))'wyy Je 6
-klG>(z)w9,), (5.3.56b)
3
ir-
■ ze)w = / (-(G2(z))'w„ Je 6
-r2eF'(y)wyz), (5.3.56c)
3
and
J (V-Ve)(z--
ze)w = f (-{F\y))'G\z)wyy Je 6
Lkl{G2)"wyz).
18
(5.3.56d) In the following, let = max/ie,
/i2 = maxfce)
= maxr c ,
h0 = max{/ii, /i 2 , ^3},
(5.3.57) and Ae,l = he/hi, Theorem 3.4.^
Ae,2 = fce/^2, Ae,3 =
If ueW^(Q)C\H^(Q)n(
rt/h^.
(5.3.58)
U Wf5(c)), 2 < q < oo,
Men Mere eris/ fourth order differential operators L\,Li
and L3, such that
a(u — t/,t;) = X > ? £ [Linv + e$J*Je » =1 =
0(ht)\\u\\'tJv\\lit,,
\/vesS(p). h
(5.3.59)
Especially, if J is a uniform partition, then the differential operators are independent ofh. Proof. First, we shall prove the following expansion:
f1
\u — u )—v
dx
3. SEM for finite-element approximations =
219
o 1 3 / (*eW**yy+ ^ * * « ) v + 0(&o)IIHk«||«||i l j', 3
fc 3 c€J^^ /rk ^e
VveSj(Q).
(5.3.60)
Note that v € 5j(f2). Applying Taylor's expansion on the element e, one obtains vx(x, y, z) = vx(x, y c , z c ) + (y - i/e)tf*y (*, y, * c ) +(z - *«)«**(*, yc *) + (y - Ve)(z - z c )v r y z , vx(x,ye>ze) = vx(x,y,z) - (y - y e )v x y (x,y,z) - ( z - ze)vxz(x, y, z) + (y - y c )(z - ze)vxyZi
(5.3.61b)
Vxy(x, y, Ze) = ^ ( x , y, z ) - ( z - Ze)Vxyz,
(5.3.61c)
vXz(x, ye,z) = vxz(x, y, z) - (y - ye)vxyz.
(5.3.61d)
(5.3.61a)
and
Applying (5.3.61a) and letting w = -Q—(u — w7), it follows that
WV-E
+ ( z - Z e ) t ; ^ ( x , yc, *) + ( y - Ve)(z - * e K y * ] •
Since i; x (x,y c ,z e ), v x y (x,y,z c ), vxz(x,ye,z) using (5.3.56) and (5.3.61), we derive
and v xyz are constants, by
/ wvx = / { [ ^ f o K v v + G(z)ti M 1 - F'(y)G'fc)u; y ,] **(*, y c , ze) 1 1 + [~(F 2 (y))Vyy 6 3 1 1 + [-(G\z))'uxzz 6 3 + [-(F2(y))'G'(z)uxyy 6
-hlG'{z)wyz\vxy{x,y,ze) -r2eF'(y)wyz]vX2(x,ye,z) - -kl{G2(z))"wyz}vxyz}. 18
(5.3.62)
V. SEM for differential
220
equations
Substituting
F(y) =
6
-(F\y))"--kl 3
and
G(z) = l-(G\z))" -
^
into (5.3.62) and integrating by parts, it becomes / wvx rf 1 = / { [~F2(y)u*yyyy Je K 6 1 —r\uxzz -
1 1 - - M * y y + -G2(z)uXZ222 3 6 F(y)G(z)uxyy2Z]vx(x,ye,ze)
o
1 -[-F2(y)UXyyy 6 1 -[-G2{z)uX222 6 1 + [-F2(y)G(z)uxyyy2 6 1 r
■ - / (k2uxxyy 3 */c
— { /
f r
3 1
^G(z)U X y «] VXy (x , V, Ze)
3
1
k2eG2(z}Uxy 18
zz ^ Vxy z
. | dxdydz }
)v(x,y€ize)dxdydz
}(fr2u*yy + r*ux^)t;(:r, t/c, ze)dydz
J S?
l
l
+ / {["^(yWyyyy + «/e ^ 6 1 1 + [-J,2(y)1l**yyy ~ 6 3 1 1 + [-G2(z)uXX22Z 6 3 1 2 + [--F (y)G(«)tiiCJflfJ
-\J
rlF(y)uXyy2]vX2(x,ye,z)
+ r^ti xxzz
- /
J Si
o
1
- G 2 ( z ) t i ^ ^ --F , (y)Cr(z)n a . yy „]v J .(x,y e ,^e) 6 'k2eG(z)uxxy22]vy(x,y,ze) r2eF(y)uXXyy2)v2{x,ye,z) 1
^ k2eG2(z)uxy2ZZ]vxyzjdxdydz
~ y }[-^2(y)M*yyy«y(*»y»«e)
3. SEM for finite-element
221
approximations
1 1 —k 2 e G{z)u xyzzz v y {x, y, ze) + -G2(z)uxzzz 3 6 12 —r e F{y)u xyyz v z {x y y ey z)]dydz.
vz(x, ye, z) (5.3 .63)
Here 5i and 52 stand for two boundary surfaces of e which are perpendicu lar to the x-axis. Summing up the integrals for all e G Jh, and noting that the surface integrals are cancelled in the summation procedure, we get / (u - u*)xvx Jet
= -3 X ) Je/ { [k*u**w + rlu****\ *ej»
\v -{y-
ye)vy - (z - ze)vz + (y - ye)(z -
ze)vyz]}dxdydz
+0(hi)\\u\&Jv\\1j = - X ) / {[fceU**yy + r2eUXX2Z]v 3
eeJh
Je
+ klF(y)UXXyyyVy 2
+r eG{z)uxxzzzvz
+ r\F'(V^ggy
g ZVy + k\G{z)UXXyy
-
k eF{y)G'{z)uxxyyyvyz
-r;F'(y)G(z)uxxzzzvyz
}dxdydz +
= ~ X ^ / [klu**yy +
ZVZ
2
0(ht)\\u\\f,jv\\liq>
r2euxxzz]vdxdydz
+0(^)ll«ll'5,,ll»l|l.,'.
(5.3 .64)
Thus the proof of (5.3.60) is complete. t
t
«// o il
•// o it
Similar expansions can be obtained for / wyvy and / ming all of them, we obtain (5.3.59).
wzvz. By sum-
□
h
Corollary 1. If J is a uniform partition, then there exist functions Wi,W2 and w% 6 HQ(Q,) independent ofh, such that 3
uh - u1 = J2^RhWi
+ O(fc*|lnfto| 2/3 ),
1= 1
where RhW% is the Ritz projection of W{. Proof. Since Jh is a uniform partition, Theorem 3.4 shows
a{uh - „ ' , « ) = W
/ ^«» + 0(*$)IM|'B|M|i,
(5.3 .65)
222
V. SEM for differential
Vv G S j ( 0 ) .
equations
(5.3.66)
Let Wi G #o(ft), (i = 1,2,3) satisfy a(wi,v) = [ Uuv,
Vv G # o ( ^ ) -
(5.3.67)
Substituting into (5.3.66), it follows that a(uh - u1 - £ t f i f c u ; , , „) = O ^ I M I s l M l i . 1= 1
Vv G Sj(t2).
(5.3.68)
3
Let v = uh — u1 — Y^ h?RhWi, then »'=i
||„» _ u / _ ^hfRnWiWi
= 0(ht).
(5.3.69)
Vt> € S%(Q),
(5-3-70)
Applying the inverse estimate, IMIo.oo < C\lnh0\V3\\v\\lt which leads to 3
\\uh -u1
-J2hlRhWi\\0)oo
(5.3.71)
t=i
Corollary 2. //u;,- G W*(fl) H H^O),
(i = 1,2,3), Men
3
tifc - ti 7 = ^2h?wi
+ O(&S|ln& 0 | 2/3 ).
(5.3.72)
»=i
Proof. If wi G W^(fi), by using the finite element estimate! 23 1, I K - RhWi\\0ioo < Chl\\wi\\2too. Replacing RhWi in (5.3.71) by wi9 (5.3.72) is obtained.
(5.3.73) D
Corollary 3. If w{ G W*+e(Q) 0 ffj(n), (i = 1,2,3), anrf 1 > £ > 0, Men 3
« * - « ' = 2 A , V + 0(&2 + *|lnfto| 2 / 3 ). 1= 1
(5.3.74)
3. SEM for
finite-element
223
approximations
P r o o f . By the inverse estimate, \\RhWi - titfHo.00 < C|Infc 0 | 2/3 ||iJfcti;,- 2 3
w{\\lt3
f
D
R e m a r k 1. Corollary 3 is only a global estimate. In fact, applying the theory of interior estimate, one can show t h a t at the interior points there exists an asymptotic expansion of the form (5.3.72). R e m a r k 2 . T h e results in this section can be generalized t o the mul tidimensional cases (cf. the work by A. Zhou et a/.^**^).
3.4. Piecewise parameter
uniform asymptotic
rectangular partition expansion
and
multi-
T h e uniform rectangular (cuboid) partition is the foundation of asymp totic expansions in the two previous sections, where the number of indepen dent mesh parameters is equal to the dimension of a domain. But for large scale problems, one can apply the piecewise uniform rectangular (cuboid) partition to increase the number of independent mesh parameters, which not only facilitates parallel evaluation, but also reduces the amount of work and storage. For example, the domain Q = ( 0 , 2 ) x ( 0 , l ) i s partitioned with three independent mesh widths /ii,/i2 and hs (Fig. 3.1). Fig. 3.1.
hs
hi
/i2 m
W i t h o u t loss of generality, suppose t h a t 0, = [J Q t , jf1
are uniform
»=i
rectangular partitions of Q,-, and the global partition Jh
m
= (J jj1
sat-
t=i
isfies the regular partition condition. Let 2/i t) i and 2/i t| 2 be the edgelengths of the rectangular element of j / 1 . Among the 2m mesh parameters
224
V. SEM for differential
equations
{hi)iihi2}^l=li there are only / independent mesh parameters, denoted by /ii,'.-,A/ ( 2 ~ < / < 2 m ) . Theorem 3.5. If Jh is a piecewise uniform rectangular partition of fi C M2, and suppose that Ai, ••-,/»/ are I independent mesh parame ters, and u G ( ]1 wq(e)) H #£(fi), (q = 2,oo), then there exist Wi G HQ(£1)
(i = 1, • • • ,p), such that hfRnwiW! < Ch30\\u\\'4i (5.3.75)
\\uh _ „/ _ h\Rhw1 and ||« h - u1 - hlRhWl
hfRhw,\\ii00
< C^|ln/io|||u||^
(5.3.76)
m
Proof. Since Jh = [J j / 1 , and each j / 1 is a uniform partition of fi,, *=i
then the identity (5.3.21b) can be rewritten as a(u — u1, v) = ^£{-9^,1 /
D\D2uD2vdxdy--hl2
+ J2 { f [~
/
DlD1uD2vdxdy\
DlD1uD2(F(y)D1v)^4F(y)DlD1uD1D2v]dxdy
+ j [ - D\D2uD1{E{x)D2v)
+
4E(x)DfD2uD1D2v]dxdyV
Vt;G5j(Q).
(5.3.77)
Let witj G #
= -~
I D\D2uD2vdxdy,
Vt; G # £ ( 0 ) , (5.3.78a)
a(wit2iv)
= - - I DlDmDtvdxdy,
Vv G ftf(Q). (5.3.78b)
and
The Lax-Milgram theorem guarantees the existence and uniqueness of the solution Wij. Substituting (5.3.78) into (5.3.77), we have m
fl
i
,
(« -« -E(*'i^.i i=l
+ A 'jW-.j).«) < <7*oll«llilMli(5.3.79)
3. SEM for finite-element approximations
225
Noting that among the 2m mesh parameters {ft,,i, fc,-,2, * = l , - - - , m } , there are only / independent ones. Packing them, one obtains /
\\uh-uI-'£ih*Rhwi\\1
h
Similarly, let v = dzG
and substitute into (5.3.79), we have
z
/
11"* ~
uI
h RhWi
-Y, *
Wi>«> ^ CfegllnftollHli-
The proof of the theorem is complete. Corollary 1. If w^ G W^fa) expansion
D
H HQ(Q),
then there is an asymptotic
l
uh = u1 + Y,h?Wi + O(fcg|lnfc 0 |),
(5.3.80)
where w^ £ 11^(0,), (i = 1, • • • , / ) , are functions independent of the mesh parameters. Proof. By the inverse estimate and (5.3.75), / h u W" ~ * ~ J^tiRhWi\\0loo < C|ln/i 0 | 1 / 2 /i?.
(5.3.81)
•=i
In addition, if W{ £ W^(Q) C\ estimate:
HQ(Q),
noting the following finite element
\\RhWi - Wi\\ot00 < Ch0\lnhol and replacing Rh^i by it;,-, the proof of (5.3.80) is obtained.
(5.3.82) Q
Applying the Nitsche technique, similar to Theorem 3.2, it is possible to expand the remainder and improve the estimate to 0(/io|lnfto|), provided that u £ W*(n)C)H%(£l)n( [ ] W*(e)). All these results can be generalized to three-dimensional cases.
e£jh
Theorem 3.6. If Jh is a piecewise uniform cuboid partition ofCl C M3, and suppose that fei, ••*,/*/ are I independent mesh parameters, and u £ Wf(Q)r\Hl(n)C\( n W*(e)), then there exist functions w{ E H^Q), (i = e£jh
1, • • •, /), such that
l
\\uh - u1 - ^hlRHWiWi
= 0(AJ),
(5.3.83a)
226
V. SEM for differential
equations
and I
/3 \\uhh -ii -u1J--^fc?/Jfcti;,-||o X > ^ , | | o f ,oo~ = O(fcJ|lnfc O ( ^ | l n / i 0 0| |22/3 ). ||ti ).
(5.3.83b)
»=i m
Let Jh
Proof.
= |J j / 1 , and jj / /1 1 be uniform cuboid partitions of t=i
fit, i = l , - - , m . Assume that 2Af->i,2At-j2 and 2/it>3 are the three edgelengths of elements in J-1, then (5.3.60) can be expressed as f
d
dv dxdydz
T
{u u)
LTx - Yx =
5Z
Q
,-=i
d
(hl2u**yy + hl3uxxzz)vdxdydz
/
+
O^IMI'SJMIM'I
^
Vi;eS£(ft), VveSJfm,
(5.3.84)
and a(w — i / , t>) m
-I
1
»
r
= ££ *W/ {*iV>?(^ k i ^ M ++ DX»g)« Dl)u = l> ++^^, , 2a^(D? ^ ( ^ ++Dl)u ,=i •/«* ,=1
33
l
•/«* l
+hl3Dl{Dl +hl D»)«}t«tedy
(5.3.85)
Let u;,-j 1.2,3),satisfy satisfy t^ij G € #o(fi), #o(^)> (*' ( l = 1> l i -*' *• i*mi!"*!i =i =1) 2,3), D
u vdxd dz
«(^,i<») ) u '• vdxdydz, «(£.•,;-») == ** // £>J(53 £>J(S^ *) v >
(fi), VVww Ge ^^(fi),
hence m m
33
h R a(u a(«h --u u1 - 53 53-^2^2hijRkWij,r) i,i h™i,j.") h
1
ft = V«€S$(n). = 0(ht)\\u\\' 0(Ao)ll«ll5,5fJv\\ ll»lli,.', 1>ql, V»GS 0 (Q).
(5.3.86)
But out of these 3m mesh parameters {/i»j, i = 1, • • •, m; j = 1, 2, 3}, only / parameters are independent. In (5.3.86), packing hij and setting g = 2, one obtains
||«ft - «*J' - Y,tfiRk*i\V o(h*). IK E*?^^lli = 0(h$). «t=i= 1
(5.3.87)
3. SEM for finite-element
227
approximations
Using the inverse estimate for
O
Corollary 1. If w( G W£(fi) n ff£(0), (i = 1, • , / ) , then i
uh = ui
+ J^h'?wi + O(h*\lnh0\2/3).
(5.3.88)
t=i
Corollary 2. If wt G W^1"1" n # £ ( 0 ) , (i = 1, • • •,/) and 1 > e > Q, then l h
1
u - u = ^h?w!
+ O(fc^|lnfc0|2/3).
(5.3.89)
»=i
These corollaries can be proved directly by the inverse estimate of the finite element theory. The results show that the accuracy of splitting ex trapolation depends on the smoothness of the solution Wij of the auxiliary problem, the smoothness of the solution u and the shape of the domain Q. However, there is always \\wj — RhWi\\ot00 = O(h€0)) hence, in many cases the SEM can be used to improve the accuracy of an approximation.
3.5. Piecewise
strongly-regular
partition
For general domains, instead of the rectangular (cuboid) partition, which cannot be globally implemented, one can apply the piecewise stronglyregular partition. A strongly-regular partition is obtained from joining the equally-spaced points on the opposite sides of a convex quadrilateral. If a domain is first decomposed into convex quadrilaterals, and each of them is subdivided into a strongly-regular partition, then the union is called a piecewise strongly-regular partition. For example, a plane polygon can be dem
composed into m convex quadrilaterals ft = (J ft,-, and the equally-spaced »=i
points on two opposite sides of each Qt- are joined to form quadrilateral ele ments. In Fig. 3.2, a pentagon is decomposed into 2 convex quadrilaterals, on which a piecewise strongly-regular partition can be constructed. Fig. 3.2.
228
V. SEM for differential
equations
Let $, : [0, l ] 2 —► £li be a bilinear mapping: f xxtf, f)) = a,-i(l- 0 ( 1 " *?) + « « * ( 1 - i?) + 0,-sf i| + a i 4 ( l - 0*?> ^ (5.3.90) ( *2«, 17) = M l - 0 ( 1 " l) + M ( l - 17) + M*7 + M l " 0*7, which maps four vertices of [0, l ] 2 onto four vertices of ft, with the coor dinates (a«j,k»j), (j = 1,2,3,4). Obviously, under the mapping $,, the square [0, l ] 2 is mapped onto the quadrilateral Qt. For a function v defined on fii, there is a corresponding function v, defined on [0, l] 2 , such that v = vo$i.
(5.3.91)
2
Similarly, for a function t; on [0, l] , one can also define the corresponding function v = v 0 S r 1 onfi,. The space of test function is SQ
(fi) = {v e
HQ(SI)
n C(fi) : v 0 $, are the piecewise
bilinear functions on [0, l] 2 , i = 1, • • •, m } .
(5.3.92)
Let IhU G S'o(fi) ^ e an interpolation function, IhU = IhU 0 $,- be the corresponding bilinear interpolation of 2 on [0, l ] 2 . Note that each £2,-, (1 = 1, • • •, m), has two mesh parameters /i t) i and hij, but only / (< 2m) are independent (e.g., only three independent mesh parameters in Fig. 3.2), which are denoted by / i i , - - , / i / . A piecewise strongly-regular quadrilateral partition works more flexibly than a piecewise rectangular partition, and there is an asymptotic expansion with respect to mesh-parameters h\, • • •, hi. Since f d dv / —(u — Ihu)—dxdy Jcti dx dx f
J[o,iP
ft
r L
ft
d£
dr,
ft
dr,
ft
8i
J
+P11—(« - Jh«)—v +P22—(w - ift«)—v }>d£dn, d£ d£ dr, dr, V» € Sj(n), where Pll = ( — X2(Z,V))2/J, Of] P22 = ( —
X2((iT]))2/Ji
or) ft ft
P12 = or,
-—X2{Lr,)—x2{i,ri)/j, d£
(5.3.93)
3. SEM for finite-element approximations
229
and \d d d d I J = — xxiCn)— x2(Cr))-— xi(£,i?) — x2(t,l) I *d£ orf or] o£ ' (5.3.93) converts the integral on £2, into that on [0, l ] 2 . In addition, hu is a bilinear interpolation on the uniform rectangular partition on [0, l ] 2 . Hence, by (5.3.50), there exist fourth order differential operators L{ti and Lt>2, such that /
V(u — Ihv)Vvdxdy
= T,hh / ^ j w ^ y + o(fcS)||nir5fnJHIi.ni. Vv G Sj(O).
(5.3.94)
Hence, a(u — u , v) m
= EE
2 f t
t=ii=i
.
'i/
i.-Jtinrfxdy+O^HiillilHl!,
7n
»
VvGS£(ft).
(5.3.95) m
Determine Wij G //o(ft), (* = 1> * > 5 3 a(wij,v) = /
=
Lijuvdxdy,
1> 2), such that Vv G #d(ft),
i = l , . , m , i = 1,2.
(5.3.96)
By using (5.3.95), it follows that m
2
II"* " « ' - E E ^ i ^ i l l = ° ( f t o ) .
(5.3.97)
Since only / mesh parameters are independent, packing them, one can ob tain /
ll«* - «' - Y,tiRkV>i^
= °(fto)-
(5-3.98)
1= 1
If the domain is convex, then Wij G H2(Q.) H HQ(£1). Using the inverse estimate, \\IhWij -RhWij\\0tOO
< C\lnh0\1,2\\IhWij
1/2
,
-RhWij\\i (5.3.99)
V. SEM for differential
230
equations
then, uh
=
/ i -^jThlhwi u
= 0(^|lnAo|1/2)
(5.3.100)
t=i
holds. For concave angular domains, the error has a different asymptotic expansion, but the accuracy can also be improved by the application of SEM. These results can be extended to three-dimensional cases. Suppose that m
ft C -R3, and there exists an initial partition : ft = (J ft,, where each ft, is »=i
a convex hexahedron, then there exists a trilinear mapping $,- : [0, l ] 3 —► ft,-. A uniform cuboid partition on [0, l ] 3 will be mapped by $, onto a stronglyregular partition on ft,-. Similar to the previous discussion, one can obtain an asymptotic expansion of uh — u1 with respect to the independent mesh parameters. For a domain with curved boundary, ft,- may be a quadrilateral with curved boundary. It is possible to implement similarly a piecewise stronglyregular partition, and there is also an asymptotic expansion of uh — u1 with respect to the independent mesh parameters. m
Cross points must be avoided in the initial partition ft = (J ft,, because *=i
at cross points, the error in finite element solutions cannot be accelerated by extrapolation methods.
3.6. Numerical
examples
E x a m p l e 3.1.t 131 l Solve the following equation by the SEM with multimesh-parameters, Atx = / ,
u = 0,
in ft = (0,2) x (0,1),
on 3ft.
(5.3.101)
The exact solution is u = 1 0 x j / ( l - x / 2 ) ( l - y ) e x + y . Decompose ft = ftiUft2, where ft2 = [0,1] x [0,1] and ft2 = [1,2] x [0,1] (Fig. 3.1). Partition ft with three independent mesh widths h1,h2yh3l and take the initial mesh widths as hi = /i 2 = h3 = —. Let t/0 denotes the finite element solution with the 16 partition (hi, /i 2 , /13), uuu2 and u 3 denote respectively the finite element solutions with the partition (/ii/2,h 2 } A3), (huh2/2,h3) and (hu ft2, ft3/2).
3. SEM for finite-element
231
approximations
The result of SEM is
uC
4
3
15
Ui
=~T,»=i
(5.3.102)
UQ.
3
6
Table 3.1. SEM with multi-mesh-parameters solution
max-error
root mean square error
1.380634E-2 1.246342E-2
6.585123E-3 6.426419E-3
«3
1.080608E-2 5.959988E-3
5.156504E-3 2.759510E-3
uc
3.051758E-5
9.569507E-6
"0 "1 «2
E x a m p l e 3 . 2 . ^ ^ Consider the following Poisson equation: f Ati = / , < I u = 0,
infi = (0,l) 2 , on dCl.
The exact solution is u = sin 7rx(cos 2iry — 1). The comparison between the results of SEM and Richardson extrapolation is shown in Table 3.2, where the initial partition is h\ = /12 = - . 4 Table 3.2. Comparison of SEM and Richardson extrapolation Point ( 1 , 1 ) Point ( 1 , 1) Point ( 1 , 1 )
Point (J, J)
Richardson extrapolation
4.9E-4
7.2E-4
6.9E-4
1.0E-3
Splitting extrapolation
2.5E-4
8.6E-4
3.5E-4
1.2E-3
These results show that the accuracy in both cases are of the same order, but the amount of work of the Richardson extrapolation is three times more than that of SEM. Without applying the extrapolation methods, the order of accuracy will be much lower, e.g., the maximum error oiuhlih2 is 1.1E-1, and the maximum error of uhl/2>h2/2 is 2.9E-2.
Chapter VI COMBINATION METHODS FOR ACCELERATING THE CONVERGENCE
Extrapolation and combination methods are two kinds of methods for accelerating the convergence. They both combine several independent ap proximate solutions to one with higher order of accuracy, but they are dif ferent in essence. The extrapolation method combines several approximate solutions obtained by applying the same algorithm on different grids, and hence it is not suitable to be evaluated in parallel. On the contrary, the com bination method combines approximate solutions obtained synchronously by applying different algorithms to grids with similar scales, and then ob tain an approximate solution with higher order of accuracy. In this way, the approximate equations of similar scales are suitable to be evaluated in parallel. The principle of combination methods was proposed by Q. Lin and T. Lilt 64 ). For the application in the solution of boundary integral equations, see T. Lii and C.Z. Mat 84 !. In this Chapter, we introduce first the combination principle, and then its applications to differential equations, integral equations and boundary integral equations. Zenger's combination techniques for the solution of sparse grid problems will be discussed in the next Chapter. 1. C o m b i n a t i o n m e t h o d s
1.1. Combination
principle
Consider a linear operator equation in =7,
(6.1.1)
where L maps a subspace of a Banach space U into the Banach space F. Assume that (6.1.1) has a unique solution u. It is usually impossible to solve (6.1.1) exactly. However, one can construct m approximate equations dependent on the mesh parameter h: L*u$ = f*,
f = l,...,m, 232
(6.1.2)
1. Combination methods
233
where V> : U-1 —► Fj1. U-1 and FJ1 are finite dimensional Banach spaces depending on mesh parameters, and u^ and //* are net functions defined on the grid ft^ C ft. In a special case, L* is also a linear operator mapping U into F, and so the right-hand side of (6.1.2) can be taken as //* = / . Assume that the approximate operators L* (i = l , - - , m ) satisfy the following conditions: A i ) Operators V> (i = 1, •, m) are invertible, and there exist constants d > 0 independent of ft, such that
\m)-l\\
(i=l,...,m).
(6.1.3)
A2) For the solution u of (6.1.1), there exist /t-(ti) G F such that = hP(h(u) + r?) -> 0,
L*u -Lu
as ft -» 0,
(6.1.4)
where p > 0, and r^ —► 0 (as /i —► 0). A3) The following problem Xt/;,- =/,-(ti)
(6.1.5)
has a unique solution iut-, and the corresponding approximate problem Llw* = U(u)
(6.1.6)
also has a unique solution w^, with ef = u£ - Wi — 0 (ft -> 0).
(6.1.7)
L e m m a 1 . 1 . ^ ' ( C o m b i n a t i o n principle) Under the assumptions A\), Ai) and A3), if there exist combination coefficients a,- (z = 1, • • -,771), such that
X > = i,
(6.1.8a)
»=i m
5>fc(«) = o,
(6.1.8b)
t=i
then
u-j2^i=hpJ2a'(ei+(Li)~lri)*=i
(6-L9)
»=i
Proof. By using A2), there is
L$(u-u$-hpWi) h.. - LuT„. PpLL-w h hh = L^u t = ft*(/i(ix)-L>t + rf), i = l , . . . , m .
(6.1.10)
234
VI. Combmation methods for accelerating the convergence
It follows from A3) that u — 11% — hpW{
= &"(e? + ( X ? ) " 1 ^ ) , i=l,.--,m.
(6.1.11)
Using (6.1.8), we have m
m
] T a,.«;,. = L " 1 ^ and (6.1.9) follows.
a,-Zf-(u) = 0,
(6.1.12)
D
For difference equation, the combination error estimate (6.1.9) is valid m
only at grid points x £ Q Q^. t=i
R e m a r k 1. There is a different property between the coefficients of the combination method and the extrapolation method. The combination coefficients usually satisfy at- > 0 (i = 1, • • •, m), while the signs of extrap olation coefficients are alternate. 1.2. Combination
methods
for integral
equations
Consider an integral equation f h{x,y)u{y)dy = f{x). (6.1.13) Jo Assume that the kernel k(x,y) and f(x) are sufficiently smooth, and 1 is not an eigenvalue of the integral operator u(x)-
I k(x,y)u(y)dy. (6.1.14) Jo Using the rectangular cubature rule, construct an approximate operator oiK: Ku=
n-1
K£u = h^*(*, i=o
1
j
(«' + «)fc)ti((t + -)&), z
(6.1.15)
z
meanwhile, using the trapezoidal cubature rule, construct another approx imate operator
K$u = - J2 [*(*> ift)w(ift) + *(«, (« + l)h)u((i + l)ft)], (6.1.16) 2 »=o
1. Combination methods
235
1 where h = —. n Again, construct two Nystrom's approximate equations ti* - Kful
= / , i = 1, 2.
(6.1.17)
Let L? = I - K?, (t = 1,2), and L = I - K. We shall now verify the conditions Ai), A2) and A3). By the theory of collective compactness: \\(K$ - K)K*\\ - 0, (h - 0), i = 1,2.
(6.1.18)
It follows that Ai) holds. A 2 ) and A 3 ) are derived from Taylor's expansions Kfu - Ku = h2li(u) + 0(h4),
i = 1,2,
(6.1.19)
with /
1 ,1 d2 i( ) = - — / —-Hx,y)u(y)dy, u
1 ,1 a 2 ^(«) = — / — 12 Jo dv
(6120)
k(x,y)u(y)dy.
Thus, there is 1 -h(u)
2 + -l2(u) = 0.
(6.1.21)
O
o
By Lemma 1.1, we obtain u - - ( i i j + 2w§) = 0 ( / i 4 ) . (6.1.22) o Note that u\ satisfies a linear system with n unknowns while u^ satisfies a linear system with n + 1 unknowns. It needs almost the same amount of work to solve these two systems. However, in order to reach the same order of accuracy by using extrapolation methods, solutions of linear systems with n + 1 and with 2n unknowns are needed. R e m a r k 2. For a weakly-singular kernel, (6.1.13) is equivalent to the following form (l-J
k{x,y)dy)u{x)~
J
k(x,y)(u(y)-u(x))dy
= f(x). (6.1.23)
This form has no singularity, and the Nystrom approximate solutions u\ and u\ can be obtained, hence, the accuracy of the combination approxi2 1 mation -u\-\—itf} can still be improved.
236
VI. Combination methods for accelerating the convergence
R e m a r k 3. The Simpson rule and the Gauss-Legendre two-point rule can also be combined to obtain approximations with higher order accuracy. It is explained as follows. By using the Simpson rule, one can construct a Nystrom approximate operator: 1 K%u = h"%2 [*(*> ih)u(ih) + 4fc(s, (i + -)h)u((i 2
t=o
1 + -)/i) 2
+ i ( x l ( t + l)ft)u((i + l)&)]/6.
(6.1.24)
And by using the Gauss-Legendre two-point rule, one can construct another Nystrom approximate operator:
KU =h-£ L [*(«, *±A*+±h)u(2l±lh + ±h) 2 ^
2
^
2
,/$
+*(*, ?i±±h - -U)„(?!±ifc _ _LA)1. z v3
*
V3
(6.1.25)
Since /^tx-^ii=
/i4
-/i4
K%u-Ku=
<94
+ 0(/i6),
fc(x,y) d4
135 dt/ 4
i(*,y) + 0 ( / i 6 ) ,
(6.1.26a) (6.1.26b)
which imply that if u?1 satisfy the following Nystrom approximate equations t i * - ^ t i ? = /,
z = 3,4,
(6.1.27)
then 4
3
7
7
and the unknowns of systems (6.1.27) are respectively 2n -f 1 and 2n. 1.3. Combination
methods
for difference
equations
Consider a three-dimensional Poisson equation 3
d2u
Lu = Y* = /, ttdxl u = (7,
in fi = ( - 1 , l ) 3 , K >' on dQ.
(6.1.28)
1. Combination
methods
237
For a given mesh width ft, one can construct two difference operators. The first one is a seven-point difference operator
= —2 { u(x1 + ft, x 2 , x 3 ) + ii(xi - ft, a?2, «3) ft ^ +ti(xi, x 2 + ft, x 3 ) + ti(xi, x 2 - ft, x 3 ) + ti(xi, x 2 , x 3 + ft) +ti(xi, x 2 , x 3 - ft) - 6ix(xi, x 2 , x 3 ) | ,
in 0 fc ,
(6.1.29)
and the second one is a nine-point difference operator LgU =
4ft 2
\ ti(xi + ft, x 2 -f ft, x 3 + ft) L
4-u(xi - ft, x 2 4- ft, x 3 4- ft) -f u(xi 4- ft, x 2 —ft,x 3 -f ft) +ti(xi - ft, x 2 - ft, x 3 4- ft) -f u(xi 4- ft, x 2 4- ft, x 3 - ft) 4-u(xi - ft, x 2 4- ft, x 3 - ft) 4- u(xi 4- ft, x 2 - ft, x 3 — ft) +u(xi - ft, x 2 - ft, x 3 - ft) - 8tc(xi, x 2 , x 3 ) | ,
in Qh. (6.1.30)
It is easy to derive that the truncation error is Llu-Lu = h2U{u) + 0(hA), 1 = 7,9, with *d4u
1
i7(U) =
-y/—, 4
« ^i 9 uU
1 ^
12 £J £?&*
^_^
34u ),
(6.1.31)
iZl^dxj
hence,
-/7(«) + -/„(«) = —^2« = L(—f).
3 3 12 Now construct two difference equations
12
inQA)
L*u* = f,
(6.1.33)
2
u? = g
(6.1.32)
h
/,
on dQh, * = 7,9.
12 It can be proved, by using the combination principle, that if u € C 6 (Q), then 1 h? u--(2u!} + 4) f = 0(hA). (6.1.34)
238
VI. Combination methods for accelerating the convergence
In fact, let W{ (i = 7,9) be the solution of an auxiliary problem Lwi = h(u),
in Q, (6.1.35)
Wi = — / , 12
on dQ, i = 7,9,
then L^u-u^ -h2Wi) = L* u — Lu — h2 LiW{ = h2U{u) - h2Lwi + h2(Lwi - L^Wi) + 0(hA) = 0(/i4),
inftfc, 2
u - u^ - /i wt- = 0,
(6.1.36a) on dftfc.
(6.1.36b)
It follows from the discrete maximum principle that u-u1>-h2wi
= 0(h4),
inftfc, 2 = 7,9,
hence, 1 u - -(2ti£ + ti§) o
h2 o
(2w7 + w9) = 0(h*)y
in f2h.
(6.1.37)
i Note that w = —(2wr + wa) satisfies 3 2
1 / 1 x Lw = -/7(u) + -/„(«) = L(—f),
3 1 w = —/, 12
3
in «,
12 on Oil, (6.1.38)
i.e., w = — / . The proof of (6.1.34) is complete. Remark 4. Bramble^ " constructed a 19-point difference operator Xf9 and a difference equation ii9«i9 = / + — A / , «19 = 9,
infi„,
(6.1.39a)
on 5Q,
(6.1.39b)
1. Combination methods
239
and obtained
u-u\z
= 0(hA),
inQ h .
As compared to the Bramble method, the combination method is sim pler as the evaluation of A / in Q^ is omitted. R e m a r k 5. The combination methods can be applied to eigenproblems, nonlinear elliptic equations and problems with curved boundaries. E x a m p l e 1.1. Consider a weakly-singular integral equation (cf. Baker and Miller^) u(x) - I \n\xJo
y\u{y)dy = g(x)}
(6.1.40)
with 0.5{x 2 In x + (1 - x 2 ) ln(l - x) - (x + 0.5)}.
g(x) = x-
In order to discretize (6.1.40) by using cubature rules, (6.1.40) can be expressed in an equivalent form u(x)(l -
- / l n | x - y\dy) In \x - y\ \u(y) - u(x)jdy
= g(x))
(6.1.41)
and the singularity is cancelled. By using the mid-rectangular cubature rule, the nodes are taken as 2 i +1 . r-m-i Xi = , i = 0, • • •, n — 1. The approximate solutions {^t/ l=0 satisfy 2n the equations f1 Hi (l - / In \xi -y\dy) Jo = 9(*i),
1
n
] T In |x,- - XJ \ (UJ - tl,-)dx
nJ=0
t = 0,.-.,n-l.
(6.1.42)
After obtaining {u{}, the Nystrom approximate solution is given as follows: 2f + 1 Ui, u(x) = {
for x = jn-i
, i = 0, • •, n — 1,
^(x)+~/Zmlx~xilwi'
*^{«o,---,*n-i}.
(6.1.43a)
(6.1.43b)
VI. Combination
240
methods for accelerating
the
convergence
By using the trapezoidal cubature rule, the nodes are taken as x,- — —, i = 0, • • •, n. The approximate solution {£, }" = 0 satisfies the equation ,1
=
!
» "
ui (1 - / In \xi - y\dy) - - ^ In |x,- - xs \ (UJ - u{)dx Jo ni=0 (6.1.44) 9(*i)i * = 0, ■
where n-l
£
U(i) =
«(0)/2 + «(n)/2 + 2 «(.'). »=1
i=0
After obtaining {tit}, the Nystrom approximate solution is: it,-,
for x = —, i = 0, n
u(x) = <
(6.1.45a)
>™,
i n "
9(*) + " ! " > I* - x i l " i - * ^ { - : i = 0, • • •, n}. (6.1.45b) And the combination approximate solution is 2_ 1_ uc{x) = -u(x) + -u(x).
(6.1.46)
Table 1.1 lists the errors of tZ(xt-), 2(x,-), and uc(x{). Since the nodal values obtained directly from (6.1.43a) and (6.1.45a) have a better accuracy than that of the nonnodal values obtained from (6.1.43b) and (6.1.45b), Table 1.1 lists only the nonnodal values. Table 1.1. h = 1/4 Xi
tx(xi) - u(*i) u(xi)-u(xi)
u(xi)-
uc(xi)
0.0625
1.03E-2
-1.31E-2
2.52E-3
0.1875
-2.04E-2
1.21E-2
-9.59E-3
0.3125
-1.32E-2
3.24E-3
-7.75E-3
0.4375
-4.22E-2
3.39E-2
-1.68E-2
0.5625
-3.40E-2
3.37E-2
-1.14E-2
0.6875
-6.16E-2
7.84E-2
-1.49E-2
0.8125
-5.02E-2
1.04E-1
1.07E-3
0.9375
-6.44E-2
2.07E-1
2.62E-2
E x a m p l e 1.2. Solve the equation (cf. Baker and Miller®, p. 358) 1 Z"1 1 «(*) " - / ; rr«(y)<*y = 1irJ-il + (x- y)2
(6-1-47)
2. Combination methods for collocation-method
solutions
241
The kernel of (6.1.47) is smooth. For the results with n = 8 and n = 16, see Table 1.2. The efficiency of the combination method in this case is obvious. Table 1.2. u(xi) Xi n = 8 ±1 -1.67E-3 ± 3 / 4 -2.54E-3 ± 1 / 2 -3.02E-3 ± 1 / 4 -3.17E-3 0 -3.21E-3
uj(xi)
ti(*i)-«J(*0
n = 16 -4.11E-4 -6.33E-4 -7.55E-4
n = 8 3.31E-3 5.00E-3 5.97E-3 -7.86E-4 6.30E-3 -8.00E-4 6.35E-3
n = 16 8.33E-4 1.25E-3 1.49E-3 1.58E-3 1.59E-3
e «w-« w n = 16 n = 8 -1.25E-5 -2.73E-5 -2.71E-5 -1.56E-5 -2.16E-5
3.72E-6 -6.09E-6 -6.17E-6 3.57E-6 -3.21E-6
E x a m p l e 1.3. Consider an eigenvalue problem (cf. Baker and Miller® p. 177) / e*yf(y)dy = \f(x). (6.1.48) Jo By using the trapezoidal cubature rule with h = 1/80, the maximal approx imate eigenvalue is 1.35306. Using a more accurate method, the maximal approximate eigenvalue is 1.35303. The results of the combination method, see Table 1.3. Table 1.3. n
A*
1.34756 1.36445 8 1.35164 1.35583 16 1.35268 1.35372 4
Ac = j-Ajvf + j AT 1.353187 1.35303 1.35303
These examples show that using the trapezoidal cubature rule and the mid-rectangular cubature rule, the errors of u\, u\ and Ajvf , XT have op posite signs. Using this property, we obtain the following a posteriori esti mations min{itf,w5} < u < max { u\, v% },
(6.1.49)
and min{Ajvf, XT} < A < max {AM, A T } .
(6.1.50)
2. Combination methods for collocation-method solutions of qua dratic and cubic splines Consider a nonlinear two-point boundary value problem ( u" = f(t,u), 0<*<1, { I u(0) = ti(l) = 0.
(6-2.1)
VI. Combination methods for accelerating the convergence
242
Assume that / is a smooth function on [0,1] x M, and the problem (6.2.1) has a unique solution. Let {t,}o be equally spaced nodes, ti = ih (i = 0 , - - - , n ) , h = —, n and rrii — ~(£,-i + *»), (i = 1, • • •, n), be the midpoints of It- = [£,_i, t,-]. We denote respectively by i?2(0> ^ 3 ( 0 the quadratic and cubic i? spline functions^ 4 1, and by 1*2 (0> u$(i) the quadratic and cubic spline collocationmethod solutions, i.e., "2(0 = ]C
C
«52(T--0'
n+1
u3(t)=
(6.2.2a)
t
^diB3(--i),
(6.2.2b)
t=-i
where the undetermined coefficients {c,-}^ and {d,}"!" 1 are respectively determined by the collocation equations u'2'{mi) = /(m t -,ii 2 (m t )), i = l, — ,n,
(6.2.3a)
ti2(0) = ti 2 (l) = 0, and
"aft) = /fa, «3(*.-)), i = l , - - - , n ,
(6.2.3b)
ti 3 (0) = n 3 (l) = 0. By (5.1.5) u2 and u 3 have the following error estimates respectively " < P r ( « - « 2 ) | | 0 0 = 0(A 2 ), i = 0 , l ,
(6.2.4a)
| £ ( U - U 3 ) L = 0 ( / > 2 ) , j = 0,l,2.
(6.2.4b)
T h e o r e m 2 . 1 . The combination approximation u(t) = 2u2(t) - u3(t) has the following error estimate at the nodes o m«|«(tO-«(i,-)|
= 0(A 4 ),
(6.2.5)
and the global estimate satisfies \\u-u\\oo=0(h3).
(6.2.6)
2. Combination methods for collocation-method
solutions
243
Proof. Let II2 be a piecewise constant interpolation operator with basis points {m t }J; and let II3 be a piecewise linear interpolation 0]aerator with basis points {£,}o • Then, the collocation equation (6.2.3) is equivalent to
< k
«2=na/(*,«2),
(6.2.7a)
«a(0) = «a(l) = 0,
•
< «S = ns/(*,t»3), k
(6.2.7b)
u3(0) = « 3 (i) = 0.
Let ek(t) = u(t) - uk(t) (k = 2,3). Subtracting (6.2.7) from (6.2.1), one obtains
< k
e* = /(*,«) - /(*, «*) + ( I - n*)/(*, «Jt),
(6.2.8)
e*(0) = e fc (l) = 0, i = 2,3.
Set wk = /(*, uk) - /(*, ti), (*) = / ( * , ti(t)).
By using the estimate (6.2.4), it follows that
||^t* a || o o = O(fc 2 ),i = 0,l>
(6.2.9a)
| | ^ t « , L = o(* 2 ).i = o,i,2.
(6.2.9b)
We have /(t,ti) - /(t,ti*) = /i(t,ti)c fc + 0 ( / i 4 ) , Ar = 2,3, 9 where /^ = —f(t,u). Substituting into (6.2.8), we get du [ e'i - fi(t, u)ek = (/ - nfc)<7 + (/ " nk)wk + 0 ( / i 4 ) ,
[ efc (0) = c t ( l )
= 0 , 4 = 2,3.
(6.2.10)
VI. Combination methods for accelerating the convergence
244
Let 11(8,1) be the Green function of the boundary problem (6.2.10). It is well known that H(s, i) is continuous and piecewise smooth but its derivative is discontinuous at s = t, then «*(*) nk)g(t)dt ^ I = / H{8,t)(i-n .,*)(!--k)g(t)dt+ = ['His Jo
Jo 4 + 0(A 0(h4),), JtJt = 2,3.
= Ji(s) + JJ22(«) (s) = Ms) +
4 ff(, S)* t)(/-n + o(/i fcH(tyt(*)<*<+ °C» 4)) ) ( ' - - n*)™*
(6.2.11)
For the case k = 2, i.e.,
MU) = Y, / A(*0
H(ti,t)(I-n 2)g{t)dt 2)g{t)dt ,<)('-- n
~~ ^ 2 4 d<2
( f f ( I - -n 2 )ff)(m 'i) +
0(* 5 )]
= E [ ^2 ( ^ - n a ) i r ) ( m i ) + o(fc«)] " h n
^2
dH ,
1 ~bT
= A,yhr t
2
+ Hg"](m*) + 0(h*)
dH
9 'dt +
r dH ,
h2 f 1 —2 I Hg"dt]+0(h' ) 24 h. h r
+ Hg"dt + 0{h 1+ ), (/l) 9 dt + — / 2iX/^ = E^X-ar^ ° ~ 12 Jo -el 24 ./o 8H
/l 2
/•!
/l 2
/•! 0 #
4
y
ft2
rl
where H = H(ti,t). Integrating by +parts using the condition, = — / g'dt — and / Hg"dt + boundary 0(h4), one obtains 12 Jo dt 24 Jo where 7/ = H(ti,t). Integrating by parts and using the boundary condition, h2 /■! one obtains — / H(ti,t)g"dt + 0(h4). [6.2 .12) 2 MU) = h24 Jo/-i 4 MU) = / H(ti,t)g"dt + 0(h ). (6.2.12) Jo and s E /,-, since For the case s ^ti, (1 = 1, • •24 •, n),
lH{S
For the case s ^ ti,- (i = 1, • • •, n), and s G /,-, since Mi- n2)g(t)dt
y
ff(«,0(J-n3-H(s, M0
,0-
m , ) ) ( / - - H2)gdt + H(s,
mi) (i
L~
- n2)gdt
= (#(s, t) - H(s, rm))(I - H2)gdt + H(s, rm) J (I - U2)gdt = jf0(h*), we get we get
= oV). M») = J (s) = 1
h22 /•! h //•! H(s,t)g"(t)dt / H(s,t)g"(t)dt 24 Jo
+ 0(h3). + 0(h3).
(6.5>.13) (6.2.13)
2. Comb/nation methods for collocation-method
solutions
245
By using the estimate (6.2.9a), one obtains in a similar way h2 fldH h2 /•! J2(U)= — / w'2dt+— / HwUt + 12 Jo dt 24 Jo 2 h /-i 8H = — w'ndt 24 Jo dt = 0(h*), 3
J2(s) = 0(h ),
8±tit
0(h4)
(6.2.14) (6.2.15)
»=l,.-.,n.
Substituting (6.2.12)-(6.2.15) into (6.2.11), one obtains asymptotic expan sions h2 e2(ti) =
/-i / H(ti,t)g"(t)dt + 24 Jo t = l,---,n,
0(h*), (6.2.16a)
L2
e2(s) =
/ H(8,t)g"(t)dt + 24 Jo 8&{tu-',in}>
0(h3), (6.2.16b)
For the case k = 3, i.e., •M«) = Z ) /
(6.2.17)
H(s,t)(I-n3)g(t)dt.
i=iJl>
For ,;' / i, by using Simpson's rule and the estimate to the remainder, it follows that J
H(s,t)(I-n3)g(t)dt 4 g(tj) + 9(ti-i) = -MTfrmjlfrm,)-*" ]+0(/>5) 6 2 /i 3
= =
12
H(s,,mj)g"(mj)+0(h5)
/ H(s,t)g"(t)dt 12 Jij
+
(6.2.18)
0(h5).
For j = i, by using the following interpolation expansion (/ - n3)<7 = g"(U-i)(t - <,_!)(< - U)/2 + 0(h3),
t €U)
246
VI. Combination methods for accelerating the convergence
and the integral mean value theorem, one obtains f
H(8,t){I-V\3)g(t)dt
= ]-H(Sl09"(U-i) I (t - U-i)(t - U)dt + 0(/i4) Jii
2
2
h
r / H(s,t)g"(t)dt + 0{h*). 12 Jii Summing up with respect to j , there is
(6.2.19)
=
+ 0(h4).
Ji(s) = ~J*H(s,t)g"(t)dt
It follows immediately from the estimate (6.2.9b) that J2(s) =
0(h4),
and so one obtains the asymptotic expansion e3(s) =
/ H(s,t)g"(t)dt 12 Jo
+ 0(h*).
Combining (6.2.16) with (6.2.20), the proof of Theorem is complete.
(6.2.20) D
E x a m p l e 2 . 1 . Consider a two-point boundary-value problem u" — Au = 4cosh(l), i/(0) = ti(l) = 0, with an exact solution u(x) = cosh(2x —1)—cosh(l). The comparison among numerical results obtained from the quadratic and cubic spline collocationmethods and the combination method at the nodes with h = — are listed 10 in Table 2.1. Table 2 . 1 . Xi = ih ti(xi) xx *2 *3 *4 *5
u2(xi) u(xi) -
2.58E-4 4.36E-4 5.52E-4 6.17E-4 6.38E-4
u3(xi) u(xi) -
u(xi)
5.16E-4 8.72E-4
3.72E-7 6.68E-7
1.10E-3 1.23E-3
8.82E-7 1.01E-6 1.05E-6
1.27E-3
3. Combination methods for the Nystrom
247
solution
Table 2.1 shows that the combination methods are very effective, and that the quadratic spline collocation-method is better than the cubic spline collocation-method. 3. Combination methods for the Nystrom solution of boundary integral equations of the second kind 3.1. Boundary
integral
equations
of the second
kind
Partial differential equations can be converted usually to boundary in tegral equations by the potential theory. As an example, the Dirichlet problem of the plane Laplace equation Au = 0,
in ft, (6.3.1)
u = g,
on T = dft,
can be converted to the boundary integral equations of the second kind by using the double layer potential theory, i
a(z)$(z)
where x = (xux2),
-\
d f / $(x) ln]x - z\ds(x) = g(z), 2w Jr dnx z e T,
z = (zuz2),
(6.3.2)
\x - z\2 = (xx - zx)2 + (x2 - z2)2.
d
-— dnx denotes an outward normal derivative. Let a(z) = - , if z is a smooth point on T, otherwise, a(z) —
6(z)
, where 6(z) is an exterior angle. Once $(z)
2-K
is obtained, the following formula 1
f d /
VI. Combination methods for accelerating the convergence
248
standard methods such as Galerkin's method and the collocation-method will consume a great deal of CPU in evaluating integrals; the discrete ma trix is full and nonsymmetric, and its order increases rapidly in successive refinements; for a nonsmooth boundary, the integral kernel is singular at the corners. In the following we shall discuss how to overcome these difficulties. One possible way is to replace these standard methods by the Nystrom method to save the evaluation of integrals (cf. R.A. Kress*50!). Another way is to replace the solution of a large system with mn unknowns by having m systems, each with n unknowns, then simply average their results, so as to reach the same order of accuracy as that obtained from the large system. Such a combination method for boundary integral equations is proposed by T. Lii and C.Z. Ma*84!. 3.2. Combination
methods
for the Nystrom
solutions
Assume that a boundary curve is expressed by a parameter s as
r = {*(*) = (*i(*),*2M): o < s < i}. If r is a closed curve, then x(s) is a periodic function with period 1. In particular, if T is a smooth curve, then x(s) is a smooth periodic function, and x(s) : [0,1] —► T is a one-to-one mapping. Hence, there exists a constant C > 0, such that (6.3.4)
|*'(*)| > c , v * € [o,i].
By means of the transformation, the equation (6.3.2) is expressed as the following integral equation on [0,1]
I k(t, s)
and k(t,s)
1 s'aMxifo) - »i(t)) - x'1(s)(x2(s) x
(Xl(s) -
2
Xl(t))
+ (x2(s) -
k(t, s) = <
(6.3.5)
has the following - ga(0)
x2(t))2 (6.3.6)
2
*
(*i(«)) +(*,(«))' = t.
3. Combination methods for the Nystrom solution
249
It can be seen that if x(s) G C / + 2 [0,1], then jfc(s,*) G C'([0, l]x[0,1]), where C'[0,1] denotes the set of continuously differentiable periodic functions of order /. For convenience, rewrite (6.3.5) in the following operator form (I-K)
(6.3.7)
= g.
Since the trapezoidal rule of periodic functions is highly accurate, it can be used to construct the Nystrom approximate operator *»=-!>('• ">(-)> n t=1 n n
(6.3.8)
and the approximate equation (6.3.9)
(I-Kn)
R.A. Kress^50! gave an error estimate of
J^'V = - ! > ( * >
M
n i=1 mn and m approximate equations
), 0 < j < m - l ,
(6.3.10)
mn
(/ - K&)
(6.3.11)
Solve in parallel
m —1
?mn = -Yl m
*&•
(6'312)
i=o The following theorem shows that (pmn and (pmn have almost the same order of error. T h e o r e m 3.1. Ifk(t,s) an error estimate
G C'([0,1] x [0,1]), y G C'[0,1], then there is
lk-£m„||oo = 0((mn)-' + n-2').
(6.3.13)
VI. Combination methods for accelerating the convergence
250
Proof. By (6.3.8) and (6.3.10), one obtains 1 m— 1
Kmn
= -Yj m
Jtf>.
(6.3.14)
i=o
Using the error expression of the trapezoidal rule of periodic function
r1
«
(KP - K)
d
'
)) — mn ds
(k(t,s)
where Pi(s) is a Bernoulli's periodic function with Fourier's series expansion ( Pi(s) = (-1)V-W
°° 2sin(27TJ5) £ 1 — 1 , for / odd, ~
fl(«) = (-l)P-2)/2 V i=i
2COS(2TTJS)
-, for / even.
(b.d.lbj
(2^i)'
Without loss of generality, assume that / is odd. Let c\ v)l\t)
f1 = (-l)('-3)/2 / Jo
dl 2sin
[27rjn(s - i/(mn))] — [*(*, s)
Substituting into (6.3.15), one obtains oo
( M ° - K)
(6.3.18)
Using the identity m-l
^ sin (27rjn(s - i/(mn))) •=o
= 0, for m Jtf,
it follows that m-l w 0
') = 2$ > jj ° (W = 0, 0, for mjlj. mj[j. »=o
(6.3.19)
Construct an auxiliary function wy satisfying the integral equation (I - K)wf
= vf\
0 < i < m - 1.
(6.3.20)
3. Combination methods for the Nystrom solution
Its approximate solution un
251
satisfies the equation = vf\ sf = ■*°. 0 < 1i < m --- 11..
(I - - K^)wf Jtf >)■ (I-
(6.3.21) 5.21) (6.:
l<
It follows from (6.3.7), (6.3.11) and (6.3.18) that
)
oo OO
,
(I-KP)bp(i - itf))(p _-^vm-n_ n-< ^ r ' 5si,f')))==0. o, D2J'i)"' 7=1
0 < i < m - 1.
(6.3.22) (6.: J.22)
Since Knl —► K, (n —► oo), and (6.3.7) has a unique solution, the inverse operators ( l — Kn ) , 0 < i < m — 1, exist and are uniformly bounded, hence, CO oo
- M -- n
<«f
i=i OO oo
= V -- V®
oo
+ ""
■ ' « , «
i=i
= 0,
- n
-
i=i
'(-f- -«S°)
0 < i < m - 1.
(6.3.23) (6.: J.23)
By using k(t,s) G C'([0,1] x [0,1]) and the error estimate of Nystrom's approximate solution (see Chapter 4), it is easy to prove that there exists a constant C independent of n,j and i such that
M° "
1 Ww^ - - wf\\ . iu^ll < CnCn~'.
5.24) (6.3.24) (6.:
Substituting into (6.3.23), we have oo oo
J . (0
(6.:(6.3.25) 5.25)
i=i
Noting the identity m- l m
£
w?\t)
= 0,
for mjlj, for mj[j,
(6.: 5.26) (6.3.26)
t=0
and summing up with respect to i in (6.3.25) and taking average, the proof of (6.3.13) is complete. □ In order to discuss the error estimate of the combination method for the solution of analytic periodic function, we denote by B„ the set of analytic functions with period 1 in a strip region = {zeC:\I : \Imz\ < 5S<7 , = {* G Cm:z\<
*}.
(6.3.27) 3.27) (6.:
252
VI. Combination methods for accelerating the convergence
where C is a complex plane. If f(s) is a restriction of f(z) E Ba on the real axis, then using the trapezoidal rule on the interval [0,1], the error has the following expansion^] ETn(f)
= £ e x P ( " 2™\k*\)ak(f), l*l>o
0 < |
(6.3.28)
where the sign of W £ (—
(6.3.29)
(6.3.30)
the solution
(6.3.31)
In the following we shall prove that the combination method can improve the order of error quadratically. T h e o r e m 3.2. Under the condition (6.3.30), we have \\
(6-3.32)
Proof. Using (6.3.28) and Fourier's expansion of (p, we have (K - KP)
(6.3.33)
where
obviously,
f1 vW(t) = / k(t, s)
1
{
))ds,
(6.3.34)
2n
= 0.
(6.3.35)
1
Let u/ ) and itA ) satisfy respectively the equations (7 - K)w& = «(«'), i = o, 1,
(6.3.36a)
(I - ffW)£(0 = t;W, i = 0,1.
(6.3.36b)
3. Combination methods for the Nystrom solution
253
By (6.3.31), ||™(° - £ W ||oo = 0(exp(-27m|<7|)), i = 0,1,
(6.3.37)
hence, = 0 ( e x p ( - 4 7 r n | ^ | ) ) , i = 0,1. (6.3.38)
(l-KP)(
Since (J — Kn ) _ 1 are uniformly bounded, it gives
(6.3.39)
By using (6.3.35), it follows that ^(i)
w(o) +
=
o.
Therefore,
= 0(exp(-47rn|F|)). □
(6.3.40)
In general, the combination of more than two approximate solutions cannot raise the order of accuracy. However, it can cancel more remainders in the error expansion and hence can still improve the accuracy. In addition, as the computation is done in parallel, the combination of more approximate solutions will not need much CPU.
3.3. The nonsmooth
cases
If T is a nonsmooth boundary, e.g., a polygon or a polygon with curved boundary, then the kernel of the boundary integral equation (6.3.2) is dis continuous or even singular. In this case, the combination methods men tioned above do not work. However, they can still be applied if the Mori double exponential formula is used. Consider an integral 1=
f f(s)ds. Jo
(6.3.41)
Using a transformation s = \j)(i) — — [tanh (-7rsinh(tf)) 4- l ] , we have oo
/
f(j>{t))xl>'{t)dt. -oo
If there exist positive integers ao, ai, a^ such that l / W W W I ~ flo exp ( - ai exp(a 2 |*|)), t — ±oo,
(6.3.42)
254
VI. Combination methods for accelerating the convergence
then taking an appropriate mesh width fc, the error of the truncated trape zoidal rule
/<"> = /, JT f{rl>{ih))j>'{ih)
(6.3.43)
i=-N
is I - Z ^ = O (exp ( - CN/ IniV) J , where C > 0 is a constant. Here the convergence rate is of exponential order (cf. [94]). In order to avoid the 4 overflow, the mesh width is usually chosen as h = —. N (6.3.43) can be regarded as the cubature rule on the interval (—hN, hN). Since the integrand and its derivatives of various order decrease to zero double exponentially as \t\ —» oo, it is possible to regard this integrand approximately as a smooth periodic function on (—hN, hN). Construct rn cubature rules
)¥'(h
4? = ^ E fW
)> 0 < t < m - 1,
m
J=-N
m (6.3.44)
and i
m —1
C°o — £«Pm
(6.3.45)
,=o If T is a nonsmooth boundary, e.g., a polygon with curved boundaries, r r , r = 1, •••,/, then the kernel of equation (6.3.2) is piecewise smooth, and the corner points may be the singular points of the kernel. However, the integral on T is still the summation of the integrals on T r . T r can be expressed as Tr = {xr(s) = (xri(s), xr2(s)) : 0 < s < l } . Set <pr(s) = 9(xr(s))
= $(z), z e T r ,
(6.3.46)
then 1 r d — / $(x) \n\x 2TT Jrj dnx =
/ Jo
z\ds(x)
kjr(t,s)(pr(s)ds,
zeTr)
j , r = l,...,/.
Define the integral operators Kjr
kjr(t,s)(p(s)ds,
j,r=l,...,/.
(6.3.47)
3. Combination methods for the Nystrom solution
255
The boundary integral equation (6.3.2) can be expressed by the integral system
"j(*)
j = l, ••,/,
(6.3.48)
r=l
where 1 _ I g'
for t G (0,1),
aj(t) =
0(*)/(2TT),
for t = 0 or t = 1,
and 6(0) and 0(1) are the exterior angles at two end points of Tj respectively. Let (p = (
(6.3.49)
Using the cubature rule (6.3.44), construct Nystrom's approximation T)[,of T and m equations
(A - T ^ V i ? = 9,0
(6.3.50)
After obtaining m approximate solutions independently, take an average i
f
m— 1
= -I>1?,
(6-3.51)
which is just the combination approximation. Similarly, when JV, h are appropriately chosen, the contributions of the integral (6.3.42) on (—oo, —Nh) and (Nh, oo) can be omitted. The error of the cubature rule (6.3.43) is 0 ( e ~ c / / l ) , and the error of the cubature rule (6.3.45) is 0 ( e ~ c m / l ) , where c is a positive constant independent of h. The value Ip evaluated from (6.3.50) and (6.3.51) is more accurate than each tp^h) (i = 0, • • •, m — 1), moreover, its error will decrease quadratically. For details, see the following numerical examples.
3.4. Neumann
boundary value
problems
Consider the following Neumann boundary value problem of Laplace equation Au = 0,
in fi,
8u — = 9, on
on r ,
r f g(x)ds(x) = 0.
( 6 - 3 - 52 )
VI. Combination methods for accelerating the convergence
256
By using the single layer potential theory, (6.3.52) can be converted to the boundary integral equation of the second kind i ±a(y)a(y)
f
/ a(x)
2TT JT
d
In \x - y\ds(x) = g(y),
(6.3.53)
dny
where the sign of the first term on the left-hand side depends on that (6.3.52) is an interior problem or an exterior problem. Certainly, for the exterior problem, in order to guarantee the uniqueness of the solution, the values of the solution at infinity have to be restricted. For the interior problem, the trivial solution is a constant, hence the solutions of (6.3.52) are not unique. In order to guarantee the uniqueness, usually
L
a(x)ds(x) = 0
(6.3.54)
is taken as a constraint condition. Once
u(y) =
Vy G
ft.
(6.3.55)
Similar to the case of Dirichlet problem (6.3.2), if T is a smooth bound ary, then the kernel of the integral operator is also smooth; if T is a polygon, then the kernel is piecewise smooth. For smooth kernels, one can obtain approximation by using Nystrom's method, and improve the accuracy by the combination methods; for piecewise smooth kernels, before these pro cedures, one has to apply the double exponential formulae. All these algo rithms have the same efficiency as those for DirichletJs problems. In the following we shall consider the solution of the discrete equation of the Neumann's problem. Assume that, by using the quadrature rule, (6.3.53) and (6.3.54) are converted to the following discrete equation Ax = 6,
(6.3.56a)
n
lTx = Y,Uxi = 0,
(6.3.56b)
»=i
which is an inconsistent system. By the least square method, we can derive a normal system BTBx T
T
T
= BTb, T
(6.3.57)
where B = [A : /], b = [b : 0] . However, to solve (6.3.57) directly is expensive: considerable amount of work is needed for evaluating BTB, moreover, the condition number of BTB is the square of that of B.
3. Combination
methods for the Nystrom
solution
257
In order to avoid solving equation (6.3.57), it is possible to regard (6.3.56) as finding the Galerkin approximation in the subspace H = {x £ Mn : lTx = 0}, i.e., find x G H satisfying yTAx = yTbi
Vy G H,
(6.3.58)
which is equivalent to the following bordered system \ (6.3.59) Obviously, the solutions of two systems (6.3.57) and (6.3.59) are almost identical, while solving (6.3.59) is easier than solving (6.3.57).
3.5. Numerical
examples
E x a m p l e 3 . 1 . Consider the Dirichlet problem of Laplace's equation An = 0, u = g, x
2
in ft,
(6.3.60)
on T,
y 2
where H is an ellipse: (—) + (—) < 1. The exact solution u = In [(x — 5 4 10)2 -f y 2 ], and so both T and g are smooth. Table 3.1 lists the errors of the relevant Nystrom approximation and the combination approximation at interior points, where m denotes the combination number. Table 3 . 1 . n = 4
n = 8
m = 1
m = 2
m = 3
m = 1 m = 2 m = 3
(0,0)
1.4E-1
9.1E-4
3.8E-4
1.6E-3
(i.o)
7.1E-2 -2.5E-3
(0,1)
2.7E-1
(*i>yO
(2,0)
l.OE-2
0
9.3E-5 -1.3E-3 4.7E-7 l.OE-3
-9.8E-2 -7.0E-4 -3.1E-4
0 0
1.1E-2 2.3E-5 9.5E-7 5.6E-4
0
0
The results show that the combination method is very efficient. For example, the result of n = 4, m = 2 is almost as good as that of n = 8, m = 1, which coincides with Theorem 3.1. E x a m p l e 3.2. Consider again the Dirichlet problem (6.3.60), here the boundary T is expressed in parametric form: T = { (xi(t) = —— sin - , #2(0 v3 2 = — sint) : 0 < t < 2TT}. It is easy to verify that (0,0) is a corner point
258
VI. Combination methods for accelerating the convergence
2
3 3 2
of T with an interior angle -w. The exact solution is u = r / cos (-0). Construct the Nystrom approximate equation by using double exponential formula ip(t) = - [tanh(sinh(*) + 1)] and the cubature rule (6.3.44). For the errors of Nystrom's approximate solution and combination solutions, see Table 3.2. Table 3.2. m = 1 4.9E-3 (0.2,0) 3.6E-2 (0.6,0) (0.2,0.2) -1.8E-2 (0.5,-0.1) 1.4E-2 (*i>yi)
n = 20 m = 2 2.0E-5 4.2E-4 -2.6E-2 6.3E-5
m = 3 1.6E-6 1.8E-6 -4.2E-3 -1.9E-6
n = m = 1 -7.0E-6 8.5E-4
40 m = 2 2.2E-8 2.9E-7
-3.0E-2 -5.2E-4 5.2E-5 6.0E-8
The solution u in this example is a nonsmooth function, but it can be seen from Table 3.2 that the combination method based on the double exponential formula is very efficient. E x a m p l e 3.3. Assume that the domain ft = (0,1) x (0,1) of the problem (6.3.52) is a square. The exact solution is u = y. For four sides of this square, construct a cubature rule by using the double exponential formula. For the errors of Nystrom's approximate solution and combination solutions, see Table 3.3. Table 3.3. (xi,Vi)
n = 4X 6 n = 4 x 12 m = 1 m = 2 m = 3 m = 1 m = 2
(0.25,0.25) 4.8E-3 -3.4E-5 -2.6E-5 1.9E-5 -2.2E-7 (0.75,0.75) 4.9E-3 -1.1E-5 -1.0E-5 -2.0E-5 -4.0E-7 (0.5,0.5) 9.7E-3 -1.9E-5 -3.7E-5 9.5E-7 -3.0E-8
E x a m p l e 3.4. Consider the Neumann problem (6.3.52) with a domain ft which is the ellipse in Example 3.1. The exact solution is u = In [(x — 10)2 + y2]. One can treat this problem in two different ways: solving the bordered equation (6.3.59), or solving the normal equation (6.3.57). Both methods have almost identical results, but (6.3.59) needs much less amount of work. Table 3.4 shows the errors of Nystrom's approximation and the combination approximation.
3. Combination methods for the Nystrom solution
259
Table 3.4. n = 8
n = 16
m = l
m = 2
m = 3
m = 1
m = 2
(2,0)
-1.2E-4
8.5E-6
1.5E-6
8.4E-6
0
(2,2)
-1.6E-2
3.2E-3
l.OE-4 3.2E-3 -6.2E-6
(4,1)
2.5E-1
4.1E-2
4.0E-2 4.1E-2 -3.5E-5
(0,0)
1.3E-5 -1.2E-6 -4.2E-7 -9.6E-7 -3.7E-9
(*i,yi)
These four examples show t h a t the combination m e t h o d is very pow erful. Since the discrete m a t r i x of an integral equation is full, the amount of work and the storage for solving a full m a t r i x with mn unknowns are 0(m3n3 ) , on the contrary, those for solving m systems with n unknowns are 0(mn3). Furthermore, these m systems can be solved in parallel without communication among processors. R e m a r k 1. In principle, the combination m e t h o d can be applied to three-dimensional problems. However, the integral operators of threedimensional problems are weakly-singular, which have to be treated with special m e t h o d s . R e m a r k 2 . So far, the error estimation of boundary integral equations obtained from the finite element m e t h o d or the collocation-method is often conservative. In fact, the b o u n d a r y element m e t h o d has a very high order of accuracy. R e m a r k 3 . T h e accuracy of the collocation m e t h o d can also be im proved by using the combination m e t h o d . T h e collocation m e t h o d with piece wise constants is equivalent to the Nystrom approximation of the midrectangular rule, and the collocation m e t h o d with linear elements is equiv alent to the Nystrom approximation of the trapezoidal rule. Therefore, a simple average of the results by using the collocation m e t h o d with piecewise constants and the collocation method with piecewise linear functions can improve the accuracy. R e m a r k 4 . T h e Richardson m e t h o d m a y be invalid in some cases, since it is based on the polynomial expansion of errors, which is not available for analytic s m o o t h data. In these four examples, the errors decrease rapidly with n. It is known t h a t the computational complexity increases rapidly with the order of the full m a t r i x and hence the combination m e t h o d s are recommended to avoid this puzzle. R e m a r k 5 . For general integral equations, the double exponential formulae and the combination methods can also be applied.
Chapter VII SPARSE GRID METHODS AND COMBINATION TECHNIQUES
In the numerical solution of an s-dimensional partial differential equa tion, in order to obtain an error of order 0(h2) by using either finite ele ment methods or finite difference methods, the number of grids is 0(h~s). If s is large, the cost will be very high. For example, assuming that the band-width of a discrete matrix is 0 ( / i ~ s + 1 ) , a banded solver will have the computational complexity 0(h~3s+2) and storage complexity 0(h~2s+1). Even if better solvers are applied, the amount of work and storage needed by a problem with dimension greater than three are hard to bear. However, the recent development of multilevel algorithms makes the solution of mul tidimensional problems accessible. In 1990, C. Zenger proposed a sparse grid method (cf. [130]), and proved that only 0(h~1\\nh\s~1) nodes are needed to reach an accuracy of order 0 ( / i 2 | l n / i | 5 _ 1 ) . Comparing with full grid methods, the order of accuracy is slightly deteriorated by a logarithmic factor. The number of unknowns for a sparse grid method is almost inde pendent of the dimension. Unfortunately, sparse grid methods also have difficulties of their own. First, the coefficient matrix is no more band type and the condition number increases rapidly with the dimension. Second, for the solution of full grid matrices there are many available solvers, but little can be used for sparse grid matrices. M. Griebel, M. Schneider and C. Zenger^ y l proposed combination techniques based on the sparse grid prin ciple. U. Riidel 107 ! discussed the relationship between the combination techniques and the splitting extrapolation. The combination techniques mean that if the error has a splitting expansion, then by solving indepen dently some coarse grid equations and combining their solutions, one can obtain a solution with the same order of accuracy as that obtained from a globally refined grid. The combination principle is based on the approx imation theory of sparse grids, the union of these coarse grids is just the Zenger sparse grid. The difference of the combination techniques and the SEM shows that both are based on an asymptotic expansion of the error, 260
1. Sparse grids
261
b u t they differ in t h a t the combination techniques require the error expan sion in the splitting form while the SEM requires the expansion in a more strict form to enable the extrapolation values to reach a higher order of accuracy. T h e purpose of the combination techniques is by combining the coarse grid solutions to reach the same order of accuracy as t h a t obtained from the globally refined grid. In general, for smooth problems, the SEM is more efficient t h a n the combination techniques. But for singular problems, where the SEM m a y be ineffective, the combination techniques remain effi cient. M. Griebel and V. Thurnert 4 2 ] have obtained numerical results using the combination techniques for hydrodynamic problems and problems with complicated domains. T h e algorithms of the combination techniques are naturally parallel. All approximate equations on coarse grids are mutually independent and the order of their matrices are almost the same, hence, very little communi cation among processors is required when implementing on multiprocessor computers. In this Chapter, we discuss sparse grid m e t h o d s in §1 and the combina tion techniques with numerical examples in §2. We also introduce briefly the methods of implicit extrapolation which have been developing fast in recent years. 1. S p a r s e g r i d s
1.1. Multilevel
splitting
of finite
element
spaces
In the s t a n d a r d finite element method, each interpolation basis point Z{ corresponds to a basis function (fi satisfying
l
(7.1.1)
where n is the dimension of Sh, the space of basis functions. T h e G r a m m a t r i x [a((pi, ipjj\ ? of the basis function {<£>,} in the sense of energy inner product is called the stiffness matrix. It is a symmetric sparse m a t r i x , because the support of the basis function has a diameter of 0{h). The basic procedure of the s t a n d a r d finite element m e t h o d is to solve the linear equations with stiffness matrix. But the condition number of the stiffness m a t r i x is 0 ( / i ~ 2 ) which may cause a great difficulty in evaluation. O.C. Zienkiewicz and H. Yserentant proposed the concept of hierarchical basis based on the multilevel splitting techniques. Y s e r e n t a n t ^ y J proved t h a t the condition number of the stiffness m a t r i x constructed by hierarchical basis is only 0 ( | l o g / i | 2 ) , and by means of the conjugate gradient m e t h o d (CG), only 0{n log n log - ) operations will be required in order to make the e error of an approximate solution in energy norm less t h a n e. T h e multilevel
262
VII. Sparse grid methods and combination
techniques
theory is now an important branch of numerical mathematics, and a basis for studying the multi-grid methods as well as the domain decomposition methods. For details, see the monograph by J. Bramble^ 12 !. The concept of the hierarchical basis is introduced in the following. It is known that the automatic mesh refinement of the finite element method means that the initial partition Jo is subdivided automatically to generate the refined grid required. If Jo is a triangulation, then such a refinement, obtained by joining the three midpoints of each triangle to form four subtriangles, thus generates Ji, and so on. If Jo is a convex quadrilateral partition, then joining the midpoints of two opposite sides to form four sub quadrilaterals, generates J i , and so on. In general, for an s-dimensional domain A, if Jo is a partition consists of cuboid elements, then such a refine ment is to subdivide each cuboid into 2s sub-cuboids by the s hyperplanes through the center of element and perpendicular to the corresponding co ordinate axis. Denote by Afk the set of all vertices, and by Sk the subspace of functions which are continuous in Cl and piecewise s-linear on elements of Jk- Obviously, according to the definition, we have MkCXk+u SfcCW
(7.1.2)
For any u G C(Q), denote by Iku the piecewise s-linear interpolation function of u in Sk, i.e., Iku G Sk) and Jfcti(x) = u(x), VxGA/i.
(7.1.3)
In particular, if u G Skl then u = IjU, j > k. Hence, for a fixed n, under the assumption i_iti = 0, there is a decomposition n
u = £ ( J * t * - I t - i t i ) , VueSn,
(7.1.4)
Jb=0
The decomposition expression (7.1.4) is very important—it is easy to see that hu-h-xu G Sk\ and (Iku - I jb _ 1 ii)(x) = 0, Vx G -A4-1- Hence, if we define a sequence of subspace as follows: Tk = {veSk:
w(*) = 0, VxGA/]b-i}, k = lr..,n,
(7.1.5)
then Sn has a direct sum decomposition 5 n = 5 0 + T 1 + ...-hf n .
(7.1.6)
By (7.1.5), the hierarchical basis can be defined inductively as follows: First, take the ordinary nodal basis to be the hierarchical basis of So, and sec ond, suppose that the hierarchical basis on Sk has been defined, then the
I. Sparse grids
263
hierarchical basis on 5jt+i consists of the hierarchical basis on Sk and the nodal basis on Tfc+i. For the one-dimensional case, this concept can be illustrated as follows: Let Q = [0,1]. We denote by Sn a (2 n + 1)-dimensional piecewise linear function subspace with a partition of mesh width hn = 2~ n . Tn is the 2 n _ 1 dimensional subspace, and the functions in Tn vanish at the grid nodes corresponding to S n - i . It is easy to see that the support of the basis functions in TJ (i = l , - - - , n ) must belong to one of the intervals [(j — l ) / 2 , ' - 1 ) i / 2 * " 1 ] , 1 < j < 2*- 1 . The basis functions of TUT2 and T3 are shown in Fig. 1.1.
Fig. 1.1.
Basis functions of Ti, T 2 ,T 3 .
Take any u G C 2 [0,1]. u1 denotes the interpolation of u on Sn. By the standard error analysis, <e=
ti — t / '
n
-4" -
1
,d 2 u, dx2
(7.1.7)
Since u1 G 5 n , by (7.1.6), there is a direct sum decomposition n
u1 = uo 4- ^ u , - , w0 € S0l Ui eT{, i = 1, • -,n.
(7.1.8)
*=i
L e m m a l . l . t 1 3 0 ! / / t i G C 2 [0,1], then 1
.||d 2w
Nl-fij^-lsr
_
e4n-.+l
(7.1.9)
Proof. Obviously, U{ can be expressed by the basis function of 7}. Taking any basis function of Ti, one may assume that the support of this
VII. Sparse grid methods and combination techniques
264
basis function is (a - ft,-, a + ft,-), where ft,- = 2~'. Define the function ' -(hi - 0 / 2 ,
MO = < L
for£>0,
-(hi + 0 / 2 ,
(7.1.10)
for£<0.
j-0.5 . , Hence, for a = ———, i = 1, • • •, 21 \ we have 2«-i
/*' *,-(0«"(« + 0<*e = / ' -*i(0*'(« + *K i r°
l /^
= u(a) - [w(a + ft,) + u(a - hi)] / 2 = I,-ti(a) -
Ii-iu(a)
= iit-(a).
On the other hand, it is obvious that 1 / ' ki(t)u"(a + t)dt\ < |||ti"||oo / ' i , - « K = y | | « " | | o o . (7.1.11) Noting that the maximum values of U{(x) occur at the nodes of Tj, by (7.1.10) and (7.1.11), the proof of (7.1.9) is complete. D (7.1.9) shows that ti,- —► 0 as i —► oo, this is the main advantage of using the hierarchical basis approximation: when i becomes larger, the contribution of the subspace T} to u will be less. For a given £, there exists n > 0 such that for i > n, it is not necessary to evaluate 7}. In fact the error satisfies ll« J - «Hoo <
oo £ IMIoo »'=n+l
1
~
\\d2un
<- 2 Y 4 - ll
" Ai
^ 2 »~
i4
1 ,,d u.
23
1
= --4-"-
2
loo
4 which is deteriorated by a factor of — than the estimate in (7 .1.7). 3
265
1. Sparse grids
1.2. Two-dimensional
sparse
grids
Let Q = [0,1] x [0,1]. Divide Q into rectangular grids by the mesh widths h\ = 2~ n , /12 = 2 ~ m . Let 5 n j m denote the subspace of piecewise bilinear functions on the equidistant rectangular grid. Let 5° m = 5 n > m H HQ(Q), then the dimension of 5 ° m is (2 n - l)(2 m - 1). Let T n>m be the subspace of S„ m with functions vanishing on all nodes corresponding to the grids of S„ m_1 and SjJ_ ljm . T n>m is a 2 m + n _ 2 -dimensional subspace, and the support of its basis function is one of the intervals [ ( j - l ) / 2 n - 1 , j / 2 n - 1 ] x [(i - l)/2m-1,k/2m-1], 1 < j < 2 n " \ 1 < ib < 2 m " 1 . Fig. 1.2 shows some examples of the supports of the basis functions in T^*. Fig. 1.2.
T h e s u p p o r t of t h e basis functions in Tj f *.
For the bilinear finite element space S° n , there is a direct sum decom position
3U = Ei>.*> *=1
(7112)
Jk=l
i.e., each u £ S„ n can be uniquely decomposed into n
n
t = l
Jb=l
u = ^2J2Uik'
with u r t € 3i,k> l
(7.1.13)
In order to study the properties of u,i, define a function space X, which d4u is a set of functions on Q such that exists and is continuous. Define 8x2dy2
266
VII. Sparse grid methods and combination
techniques
a seminorm on X as follows: li/l — 1 u
dAu 2
,
(7.1.14)
2
\ \ - 1 0x dy 1
oo
For u € X, we denote by u1 the interpolation function in S„ n. By using (7.1.13), the unique decomposition of u1 is 7
n
n
u = 5 3 ]£!*.-*, UtteTU, l < « , i < n .
(7.1.15)
Lemma 1.2. If u E X, then there is MOP^-'-''-1!*!.
(7.1.16)
Proof. Let &,(£) be a function defined by (7.1.10), (x 0 ,i/o) be of TJ- j . Integrating by parts, one obtains
a n
°de
rhi rhj d4u(x0 + x,2/o + 2/) / / &i(x)fej(x) dxoh/
y. fci y_ fci
v
'
JV
5x25y2
= w(«o,yo)- r[ M ( x o,yo + ftj) + ti(so,!/o- A i) +ti(x 0 + ft,-, 2/o) + w(x0 - ft,-, yo)] 1 + - [u(x 0 4-ft,-,t/o + ftj) + u(xQ - fti, 2/0 + hj) 4 +u(x 0 + ft,-, yo - ftj) + u{xQ - ft,-, yo - hj)] = c.
(7.1.17)
Note that c = u,-;(x0,2/o), and |i/, ; (x, T/)| can attain its maxima only at the nodes of Ty. On the other hand, using (7.1.17), we have 1 fhi 1
/
fhi
/
*'(*)*i(*>
d4u(x0 + x, 2/0 4- y)
-***v < M—•
r^i
4
./-**J-fci dx2dy2 The proof of (7.1.16) is thus complete. Corollary. Ifu£X,
D
the following will be the estimate of Sn,n \\u-uJ\
1
W-
|oo<—1
18
(7.1.18)
1. Sparse grids
267
Proof. By using the estimate (7.1.16), one can derive oo
oo
n
Ui
n
ll« - "'lleo < I 5 3 X ) > - 1 3 1 3 UiJ II t=ij=i
»=ij=i oo
oo
<EEiKiiico+ E EIK»=lj=l
oo
oo
i=n+lj=l
oo
<2|«|£ ^ 4-<->"1 »=1j=n+l oo
oo
t=l
j=n+l
2 <
-4"n-1|ti|
9 = -hl\u\. 18
D
Using Lemma 1.2, the component of u1 in 7^*, denoted by i/,*, will decrease exponentially as i + k increases. Hence, the strategy of Zenger is to ignore the terms of £ ° n in TJ^ (i + k > n), and to replace S ° n by the following sparse grid space
t = EE^= »=1 fc = l
E
T
^-
(7119)
*+fc
The dimension of 5° n is (n — l)2 n -f 1, i.e., it is of order (^(/i" 1 log ft"1), however the dimension of 5° n is 0 ( / i ~ 2 ) . Furthermore the accuracy of the approximation of u at S„ n is only slightly deteriorated than that at S° n . In fact, we have the following T h e o r e m l . l . I 1 3 0 ! 7/ u G l , and u on S„ n , Men
denotes the interpolation of u
\W - S ' l U < - A » (log, A" 1 + - ) |«|. 48 6
(7.1.20)
Proof. Let n n —1+1 J
« = X) E «=i
t=i
"•>■ " o - e i i j .
(7.1.21)
VII. Sparse grid methods and combination
268
techniques
Hence, oo
oo
n
t=ij=i n
oo
^||E E i=lj=n-i+2 00 oo
oo
Uii+ 4
n—t-fl
»=i i = i oo
^ ^HL 1
i=n+lj=l oo
oo
* ( £ E "'-''- + EH*-'-*-->i t=lj=n-i+2
i=n+lj
1 4 = (n-4-n-2+-4-"-2)|u| o
=l
y
D = - ^ ( i o g 2 * » 1 + -)MWe denote by48Q n>n the full grid3 of S^n, and by Qsn>n the grid of S nj „, called the sparse grid. Fig. 1.3 shows the picture of a full grid and its corresponding sparse grid. (7.1.20) is only an estimate in L^ norm, and in Li norm the similar argument holds. Moreover, in the energy norm (H1 norm), the sparse grid method can reach the same order of accuracy, i.e., 0(h) as that by the full grid.
Fig. 1.3.
3 3 x 3 3 full grid and its c o r r e s p o n d i n g sparse grid
1.3. Higher dimensional
sparse
grids
The two-dimensional sparse grid can be generalized to higher dimensional cases with better efficiency. For an s-dimensional problem, the dimension of a full grid finite element space is 0(ft~ 5 ) while for the cor responding sparse grid, the dimension is reduced to 0(ft x (log ft"- i r n Moreover, the order of the interpolation errors is only reduced slightly from
1. Sparse
269
grids
0(h?) to 0 ( / i 2 l o g ( / i " 1 ) ) . Obviously, the higher the dimension of the prob lem, the more efficient the sparse grid method. Let Q = [0,1]* be a cube of Ms. We denote by Qn ... n the full grid with 1 ' ' mesh width — = / i n , and by 5 n , . ,n the corresponding piecewise s-linear finite element space. Let If' be a piecewise linear interpolation operator with respect to the variable x*. T h e interpolation basis points are taken at % — , i = 0, • • •, 2 J . Let X be a function space on ft, in which functions have 02su
continuous partial derivatives
\ii\
—
\u\ —
1
. Define a seminorm on X by
d2 u
'
1
U«? - - - a*?loo 1
(7.1.22)
Let u1 be the interpolation of u on 5 n > ... ) n . Obviously, u1 = l£)'-I{n)u. Define the subspace of Sn,-,n J-a
=
(7.1.23)
as follows:
-*Ofi, •••,<*•
^((^-eo-^-ei)). 0 < c*t- < n , i = 1 , . . . , 5 ,
(7.1.24) (k)
where 3R(A) denotes the range of the operator A. Again, define I)_{ = 0 (k = 1, • • •, s). It is easy to derive the following direct sum decomposition T Sn,..;n=c r E"-5-) «i.-.«.0 a,=0
(7-L25)
1=
In fact, suppose u E 5'nJ--,nj then « = «' = # > # > . . . # > «
= n[E(^ ) -e 1 )]« Jk = l
afc=0
= E--E(^ ) -e 1 )-(^ ) -ei)« a!=0 n
a,=0 n
or1
or,=0
i,•••)««> =
0
(7.1.26)
270
VII. Sparse grid methods and combination
techniques
and the decomposition of u is unique. By the estimate (7.1.9), it is easy to obtain an estimate of « a i ) ... ) t t |
IK,.,.. Iloo = | (/£> - i
OO
< -~4- , a , |w|, V u E X , V
(7.1.27)
it means that the components of u in Ta (|a| > n) can be omitted. Hence one needs only to consider the following sparse grid space 5 n ,.., n = ^
T«.
(7.1.28)
\a\
Let u1 denote the interpolation of u G X in S^.-.-.n- The following theorem gives an error estimate of the approximation in the sparse grid space. Theorem 1.2. If u G X, Men Mere eztsfe a constant Cs dependent on s, such that \\u - S'Hoo < CshKlogh-1)-1^.
(7.1.29)
Proof. By (7.1.27),
|TX — G^Hoo = || 5 3
Ua
W°°
\a\>n
< £ Halloo |a|>n
oo
4-fc
s ( E - E I)M fc=n + l *
|or|=fc
4
£ E T ( t i l ')M <
4-nn-1C,|u|
^C/i^log/i-1)5"1^!.
D
(7.1.30)
We now estimate the dimension of S n , .,n- Obviously, Qp C ft<>, A < a,- (i = 1, • • • ,5), hence,
dim5n,...,n < 5 3 diimla \a\=n
1. Sparse
271
grids
= 2 " £ l = 2»(" + *T 1 ) = 0(2nn'-1)
= Oih-^logh-1)'-1),
(7.1.31)
which shows that the dimension of the sparse grid space is much less than the dimension of the full grid space 0 ( / i ~ 5 ) , while the order of accuracy in Loo norm is only slightly deteriorated by a logarithmic factor. Moreover, the order of accuracy in H1 norm is the same as that of the full grid. For details, see [18]. 1.4. Finite
element
equations
on a sparse
grid
Let the initial mesh widths be hx and /i y , and let Clij denote the rect angular grid with mesh widths hx2~% and /i y 2~ J . Obviously,
«* -
k
fc+l-t
U
U n<j
t=l
(7-1.32)
j=l
is a sparse grid with k refinements. k
k + l-i
k
k+l-mNij
^=EE ^ = £ £ E^>. i=i
j=i
»=i
j=i
( 7 - 133 )
i=i
is a sparse grid space. Vij)Zl =sp&n{(pj } denotes the one-dimensional space spanned by the nodal basis functions corresponding to the node z\. Nij is the number of nodes of fiij, but {^/ : i + j
for i = j = 1,
{ ^ € ^ 1 , ^ ^ 0 , - 1 , 1 } ,
for i > 1, j = 1,
{^GBij^^fiij-i},
for i = 1, j > 1,
{^J)Gflij,^jfflMjU".j-i}.
for i > 1, j > 1,
Bij = {
(7.1.34) where Bij are the sets of the standard nodal basis functions of Qij. Take the hierarchical basis of Sk as k
k+l-i
„
m ~ u u B^. t=l
j=l
(7.1.35)
VII. Sparse grid methods
272
and combination
The finite element equation on the sparse grid becomes:
techniques
Find u e
Sk
satisfying
, v«effj,
(7.1.36)
a(u ,«) = / ( « ) .
or equivalently, the stiffness equation for the sparse; grid L»Su
fHS
=
(7.1.37)
with L
K
*)» V y m > V n eJTjE, J m , n= a(
(f?s)m
= (f,
Fig. 1.4. Sparsity pattern for sparse grid stiffness matrix
Table 1.1. Eigenvalues and condition numbers of sparse grid matrices *
0
1
dimension
1
5
2
4
Ml/2,l/2v
8.000
12.162
Ml/2,l/2x
8.000
5.837
2.208
1.000
2.083
8.272
17
5
49
6
129
m a t r i x w i t h o u t scaling A A
.
/2 1/2
*K * )
18.272
32.715
64.290
1.785 18.322
128.134
1.469 1.199 43.739
106.857
m a t r i x after diagonal scaling U ^ x C P - 1 / ^ ^ ' 1 / ^ - 1 ^ ) 1.000
1.335
1.769 2.115 2.553
1.000
0.664
0.248
1.000
2.009
7.108 11.352
UMID-WAI'WD-W) [
KiD-^Al'^^D-1'2)
0.186
2.903
0.135 0.098 18.890
|
321 |
29.482
273
2. Combination techniques
But the stiffness matrix of (7.1.37) has complicated structure. Fig. 1.4 plots the sparsity pattern of a 49x49 stiffness matrix for a sparse grid' 1 ^. In addition, unlike the hierarchical basis stiffness matrix for a full grid, the condition number of the stiffness matrix for a sparse grid increases rapidly with k. Table 1.1^^ '^ shows that the condition number of the sparse grid increases as fast as 0(2 f c / 2 ), it is faster than 0(k)y the condition number of the full stiffness matrix for the hierarchical basisl 129 !. Hence, direct appli cation of the iterative methods (e.g., the Gauss-Seidel method, CG method) cannot attain the optimal complexity. By means of the Bramble-PasciakXu (BPX) multilevel method, M. Griebel, C. Zenger and S. Zimmert41^ suggested the multilevel Gauss-Seidel algorithms, where the number of iter ations is independent of k. Detailed computational methods and numerical results can be found, e.g., in [41]. 2. C o m b i n a t i o n techniques Although the dimension of the sparse grid space is reduced from 0(h~s) to 0(/i~ 1 (log/i~ 1 ) 5 - 1 ), the interpolation error only slightly deteriorates from 0(h„) to © ( / ^ ( l o g / i " 1 ) 5 - 1 ) . However, the sparsity of the stiffness matrix generated by the sparse grid is destroyed, and the condition num ber increases exponentially with k. Furthermore, there are many efficient software packages available for the standard finite element methods, but so far, good software packages for solving equations with sparse grid stiffness matrix derived from sparse grid methods are still lacking. For this reason, M. Griebel, M. Schneider and C. Zenger^^ proposed the combination tech niques for solving the sparse grid equations. The combination techniques are based on the following relation between the sparse grid and the full grid interpolations uTn n and u\ -:
«».» = E
i+j=n + l
( 721 )
This relation shows that the interpolation u„ n in the sparse grid space 5° n can be regarded as the linear combination of interpolations ujj in the full grid spaces Sij (i+j — n, n + 1). In two-dimensional case, this combination is shown in Fig. 2.1. We denote by Uij the finite element approximation on the full grid space
S?j. Let
"n,» = £ t+j=n+l
"U - £ "V i+j=n
(7'2-2)
274
VII. Sparse grid methods and combination
techniques
be a combined approximation. We shall now prove that if the solution is sufficiently smooth, and the error of the discrete solution Uij has the following expansion « u - uij uu = = d(hi)h? d(hi)hl
+ C22{hj)h] (hj)h] + + Dihuhfihlh], D(hi: hj)h?h],
(7.2.3)
then the error u — G£ n is of the same order as that of u — u„ n . Here, the coefficients C\{hi),C2{hj) and D{hi,hj) are bounded functions of x,y and the parameters hi (or/and /ij), i.e., there exists a constant K such that \Cx(hi)\ < K, f \Ci(hi)\ { I \D(hi,hj)\
\C2(hj)\ < K, \d{hj)\
(7.2.4)
Fig. 2 . 1 . I n d e x diagram of linear c o m b i n a t i o n for t h e two-dimensional case
2.1. Two-dimensional
combination
techniques
39
T h e o r e m 2.1.[ 1 If the error of the discrete solution Uij has an expansion of the form (7.2.3), and the coefficients satisfy (7.2.4), then the error of the combined approximation u„ n has the following estimate lltl--K,n\\0o
~K,n ~ ucn,n—=U ti---
u■
2sJ2 i+j=n+l »+j=n+l
UiW
1>
» ++ Yl -M
Ui
>i ,3
i+j=n
ui ) fa -~ < Jj) =E = S fa(M -- uuij) u) -- J2 £ fa = E (CiMhj+ 02^)1$+ Dih hi)tihj) h )h?h]) =«'+i=n+l E M
*+j=n + lL
»+j=n i+j=n
it j
i+i=n+lL
(7.2.5)
2. Combination
275 275
techniques techniques
£ (Ci^w{Cxih^hl YC + C^h^ 2{hj)h) - E
++ Dihuh^hfh)) D{hi,hj)hfh'])
i+j=n i+j=n
= Eci(W»=1 »=i
n-l
n
h{hi)h? + Ec 2 (Ai)ftJ
1=1 1= 1
3j = = l1
n-1
-Ec2(ft,-)fe? + E 2 i =l
+j=n+l »'+.;'=n+l
ft D(hi>ithj)h]h] D(h hj)h?h]-'£ - E ^(DfahjWh] .- ,A;)W «+j=n
1 hj)]hl (huhj)- £E ^C D^.fc,-)]^■- »-+ Ej = nE+ l D^c^o*+j=n
= = [Ci(/i„) ++ CC2(M + 2 (A„)-f
44
»+j=n+l
t+j=n
Applying (7.2.4), it leads to an estimate 2
W-K,n\<\Cl(hn)\h2n W-K,n\<\Cl(hn)\h I
1
+-
n
+ \C,(h \C,(hn)\hl )\hl
^
^
.
D h h h DWi,N)h h E E ( i> i)\ l +|- *+j=n E + l ( i, >)- E i+j=n ^ . ^ l ^
*+j=n + l \c (h )\hl < |C!(A„)|AS + \C22(hnn)\hl
< \Cl(hn)\h2n +
»+j=n
1
+- E 4
x+j=n+l
\D(hi,hj)\h2 + E p(^,fti)^+
-|-j=n t*+j=n
w^hMhi *i)l*n
1 2 1 < Kh Khl + -ifnft^ Khl + tfft* + A"(n.-1)A» - l)h2n + 4 -Knhl+K(r 5 = Ki a/ £*((l l ++ - n ) 4 5 = **»(! ia*(i + o g2(fc„ ). 2 ( 0J ))+ -- ilog 4 4 Taking the maximum norm and the proof of (7.2.5) is complete.
(7.2.6) (7-52.6) D
Theorem 2.1 shows that the terms depending onft,-or hj with i,j / n are cancelled, and just the ftn-terms remain. However, the error of the combined approximation is only slightly deteriorated by a logarithmic factor than that of the full grid approximation. The discrete error U{j has an expansion of the form (7.2.3) if and only if u is sufficiently smooth. In general, instead of (7.2.3), we assume that the discrete approximation satisfies the following expansion uitj = = dihiW Ci(hi)h? + + C*{hj)h) C2{hj)h) + + D(hi, D(hi, hjWq, u - UiJ hj)h\h),
(7.2.7)
where v is a positive number which is independent offt,-,hj, x and y, and the coefficients satisfy the condition (7.2.4). Hence, by repeating the proof of
276
VII. Sparse grid methods and combination
techniques
Theorem 2.1, the combination solution defined by (7.2.2) has the following error estimate 2" + l \\u
U
n,n\\oo
v. iVAl n ^l -f
!Og2^n
;;
= 0(Klog(h-1)).
(7.2.8)
By comparing this estimate with \\u — i/n,n||oo = 0(h„), the difference is only a logarithmic factor. Note that the combination techniques have an advantage that the com bination pattern is independent of the exponent v in the asymptotic expan sion. Therefore, in the expansion (7.2.7), the exponent v is not necessary to be exactly known and hence comparing with the extrapolation methods, the combination techniques are more suitable for non-smooth problems, since the extrapolation methods are applicable only when the value of v is known exactly. 2.2. Three-dimensional
combination
techniques
Two-dimensional combination techniques can be generalized to threedimensional case. Let Q be a unit cube, /i, = 2~\hj = 2~J', hk = 2~k be mesh widths in the x-, y- and ^-directions respectively. We denote by Uitjtk the finite element approximate solution at the grid Qij}k. Suppose that the solution is sufficiently smooth, and the error has an expansion of the following form u - ui)j}k = Ci(A,-)A,? 4- C2(hj)h] +C3(hk)h2k + D^hi.h^hfh] +D3(hj1hk)h]hl
+ D2(hi, hk)h?h2k (7.2.9)
+ E(hiyhjihk)h?h]hl
where the coefficients are bounded, i.e., there exists a positive constant K independent of ft,, hj,hk and x, y, z, such that |A(/i,>i)|<^,
\G(hi)\
/ = 1,2,3,
and (7.2.10)
\E{hi,hhhk)\
T h e o r e m 2.2.^ y l Assume that the error of a full grid solution itt,j,fc has an expansion as (7.2.9), and the coefficients satisfy (7.2.10), then the combined approximation
"ntn,n =
Ui
>3>k
J2 i+j+k=n+2
-2
£ »+j+fc=n+l
«U,*+
£ ,+j+fc=n
W
M>*
(7-211)
2. Combination techniques
277 211
has the following error estimate
\\ul l « -
=r
) i n ^ e g ra -l solutions of the Diophantine
2
n 1 i == ( n +2 )4\ J- --(2 ( 2
4
2
(7.2.12) (7.2 .12)
n
3
2 n 2) +J (+ l2 2)" >) ■
then W U
-
U
n,n,n
U ui k 53 "•".)>,*) 53 ((u - ~ J> i+j+k=n+2 i+j+k=n+2
--2 2
£YL
t+j+Jb=n+l
(u-uij,*)+ + 5533 ((*w - wW.\i.*). »'J.*)0*-^,*) i+j+k=n t+j+fc=n
Substituting (7 (7.2.9) (7.2.13), Substituting .2.9) into (7.2 .13), we get
uU -—S nu° ,n,n n
n
n
J^(n -- * i ++ l)d(MA? + E ( n - i' ++ ^i)cC(fci)*? W l)Ci(M*. +X> == 53( t=i ii==ii ?
2
n-l
+ 5 > -- ** + [ j > --i)Ci(hi)hf i)Ci(ft,-)/i? + l)C l)C33(^)/il (^)/il -- 22[53(n fc=i n-l
»= 1
n-l
- JMhfih) + 53(n - *)C3(fcfc)*2] fc=l
n-2
+5>1=1
n-2
- » + l)Ci(A.-)A? + E ( " " J J=l
t= l n-2
l c
) *ihi)h}
h h -k-l)C3(h I>i(fc.-,fchj) k)hl+ + 53 i)ft?ft? + 53(n-*-l)C E Di(hi> l] s (M*k « +i+j
k=l Jfe = l
+ J2 x+Jb
+ D (hi,h k)hjhl+ MW
2 I»2(/»i
2 53 (hj,hk)h]h 2 DD33(hj,h k)h]hik j+k
E + E a(fci,fc *)W] + E i,M^fc*]+ + 53 E ^i>s(ft -2[ 2
ti+j
j+k
++^53 D )h}hi D1(hi,hft>)ft?A? Dz{h h**)W I»2(^,i 2(h{u,h j)h^hj+ k)h}hl I>i(A,-, *iW i+k
■1 i+j
,/.,)/»?/»? Mhi,hj)h]h]
7 2 13 ((7.2.13) -- )
278
VII. Sparse VJI. Sparse grid grid methods and combination
techniques
Yl DD(h(hi,h )h?h ++ E 53 D {h,ft*)ft| h )h]h\ hi ,MW E tt+fc
+
22 i:
2
k
k
3
22
k
2
k
2
jt k
* T J T * — T*T*
+ £ + £
h k
■1
■l
*-rj -rn,—T*TX
E(hi9Eihi^M^hlhl hjthk)h^h]hl
(7.2.14)
i+j+k=n
In the above derivation, the equalities n 53 d(Mtf = £ > -- t + l)Ci(A,-)ft?, «'+J+*=n+2 «=1 t+i+*=n+2
and and
t=l
h)h]h) A,-)fc?A? £ Di(hi, D (h ,h )h]h == = E J2 i?i(A*, DifcWih] E i+j+Jb=n+2 i+i+fc=n+2 » + j < n + lL »+i
i
j
2
j
are applied. Combining all the Ci-terms in (7.2.14), we have n
5 ^ ( n -- tt + i)Ci(fcO&? l)Ci(fci)A? 5>n-l
n-2
-2 53(n - i)Ci(ft,-)ft? + 53(n - i - l)Ci(fc)*? ««=1 = 1
»»== 11
n-2
? = *' +J!) 1)] d(hi)h} = £E i(Knn "~*+ ) ~" 22(« ( n "~0o++ (n ( »-- *t ---l)]Ci(A,-)A, tt = = ll
l))]C 1 (/i„_ 1 )^_ 1 -mc^h^hi^
+ ( n_- (( „n_- il )) + ( n --- ((„ » -+ [[(„ + il )) _- 22 („ +(n n+ + +(n -- n C!(hn)hl =■■ C!(hn)hl.
l)Ci(hn)hl l)Ci(hn)hl
(7.2 1.15) (7.2.15)
Therefore, all terms with hi, h{, hj, h /i*k (i, ( t ,jj, ,fc k ^^ n) n) are are cancelled cancelled in in this this comcombining procedure. Similarly, combining all the £)i-terms, Di-terms, we obtain Di(hi, D {h ,hhj)hlh] )hih E
53
l
i
j
2
j
E
-2 53 Dx{hi Diihi^hfh] ,hj)h?h] + + 53 -2E
i + j < n _ Ii *'+i
i+j
.*i)W
D^hi DtfahjWh]
( A,A,hj)h?h] 5 )A?A?++ Ea5 3 {l-2)D {h .A,)*?*? h )h h - 2)D (ft i - - 2 + l ) D(A - *+>
=
11
it
i+i=n+l
i il i
11
u j
2
i
2
j
2. Combination techniques
=
£
279
D^hi^hfh) - £
»+j=n+l
= [-
Dxihi^h^h]
i+j=n
£
£
D1(hi,hj)-
*+j=n + l
i?i(A.-,A,-)]^
»+j=n
1 < [-ntf + (n - l)tf]A*. (7.2.16) 4 Similarly, C2-, C3-, A r and Z)3-terms can be combined and we have the same results as those for Ci-terms and Di-terms. Finally, for 2?i-terms, we have the following estimate:
I
£
Eihi.hjMWtfhW
i+j+k=n+2
£
<
|^(Ai,fci,M|An+2
i+j+k=n+2
n(n + l) < K
(7.2.17)
hl+2.
Substituting the results of (7.2.15)—(7.2.17) into (7.2.14), it follows 1
l« - «n,n,nl < ?>Khl + *Khl ("« + « ~ l) 4
ln(n+l)
+Kh* [
l(n-l)n
+
(n - 2)(n - 1)
+
16 2 2 2 65 25 = Kh2n(l+— n+—n2) 32 32 65 25 = Khl(l+-log2(h-l)+-(\og2(h-1))2). The proof of Theorem 2.2 is complete.
] 2
(7.2.18)
□
Corollary. Instead of expansion (7.2.9), if the error ofuij^
satisfies
u - uiJ}k = Ci(/i,-)fc? + C2(hj)h) + Ca(hk)hl
+D1(hi,hj)h»h"j + D2(hit hk)h?hl + +E(hi,hj,hk)hWhl,
D^hjMWjK (7.2.19)
where v is a positive real number, the coefficients Ci,Di and E satisfy the condition (7.2.10), then, the combined approximation defined by (7.2.11) has an error estimate ll"-
(7.2.20)
VII. Sparse grid methods and combination
280
techniques
Since the coefficients of the combination will not change when the ex ponent v varies, the combination method is more effective than the extrap olation method for nonsmooth problems. R e m a r k 1. By means of the following identity^
gc-tfCj1)^!1)-1it is easy to derive the s-dimensional combined approximation
<••■» ='£(-#(•I1) £ «J=0
V
' |a|=n+*-j-l
R e m a r k 2. The combination techniques are also successful in evalua tion of multiple integrals. For details, see Basenzenski and Delves [6]. R e m a r k 3. The number of independent mesh parameters for combi nation techniques can be larger than the dimension of the domain. This is similar to the multi-parametric expansion considered in §3.4, Chapter V. R e m a r k 4. The exponent v in the expansion (7.2.19) may depend on the nodes (x,y, z) (e.g., in the case that the solution has singular points), but the combined approximation can still be almost as accurate as the solution obtained from a full grid method.
2.3. Numerical
examples
For two-dimensional problems, the idea of the combination technique is to solve independently n problems with 2 n unknowns and (n — 1) problems with 2 n _ 1 unknowns. And for three-dimensional problems, it is to solve in dependently n(n + l ) / 2 problems with 2 n unknowns, (n — l ) n / 2 problems with 2 n _ 1 unknowns, and (n - l)(n - 2)/2 problems with 2 n ~ 2 unknowns. All these problems can be solved in parallel. Moreover, the size of each problem is dependent only on the parameter n, and independent of the di mension of the domain Q. Hence the combination technique is suitable for working on multiprocessor computers. The results from different proces sors will be combined without any communication among these processors. The detailed discussion can be found in the work of M. Griebel et a / . ^ . The comparison between the errors of the full grid approximation and the combined approximation is given in the following examples. E x a m p l e 2 . 1 . ( S m o o t h s o l u t i o n ) ^ Consider the Laplace equation Au = 0,
infi = (0,1) x (0,1),
2. Combination
techniques
281
with Dirichlet's boundary condition. The exact solution equals u = sin(7rz/) •sinh(7r(l — x))/sinh (7r). Table 2.1 shows the errors of the full grid so lution and the combined solution at points Pi = (0.5,0.5) and Pi — (0.25,0.25). Table 2.2 shows the errors in Li norm and Loo norm. The ratio | e n _ i ) n _ i | / | e n ) n | shows that the error is of 0(h?) for the full grid solution and is of O (h2\ogh~1) for the combined solution. Table 2.1. n 1 2 3 4 5 6 7 8 9 10 11 12
e
P o i n t w i s e e r r o r , full g r i d o n t h e l e f t , c o m b i n e d s o l u t i o n o n t h e r i g h t .
i , n ( j » j ) ~ l«n,nl
-
7.4268E-2 1.5498E-2 3.7333E-3 9.2504E-4 2.3075E-4 5.7655E-5 1.4412E-5 3.6028E-6 9.0070E-7 2.2517E-7
4.792 4.151 4.036 4.009 4.002 4.001 4.000 4.000 4.000
-l,n-ll -cn,n\%(I J)W' nl*n,nl e
}
-
1.3252E-2 3.1526E-3 7.7889E-4 1.9415E-4 4.8503E-5 1.2123E-5 3.0307E-6 7.5768E-7 1.8942E-7
4.204 4.048 4.012 4.003 4.001 4.000 4.000 4.000
[l3
<,.(*.*)- n - l , n - l I-1.J 7.4268E-2 9.9913E-3 1.2929E-3 5.5787E-5 -5.1448E-5 -2.9150E-5 -1.1357E-5 -3.8568E-6 -1.2186E-6 -3.6824E-7 -1.0796E-7 -3.0964E-8 -8.7346E-9
-
7.433 7.727 23.176 1.084 1.764 2.566 2.944 3.165 3.309 3.410 3.486 3.544
-9.3480E-3 6.1846E-3 1.2064E-3 2.2655E-4 3.8414E-5 5.0772E-6 1.3935E-7 -2.4756E-7 -1.3249E-7 -5.0770E-8 -1.7105E-8 -5.3792E-9
ZIZ^J] K.J 1.511 5.126 5.325 5.897 7.566 3.643 0.562 1.868 2.609 2.968 3.179
T a b l e 2 . 2 . E r r o r s i n L2- a n d L > full g r i d o n t h e l e f t , c o m b i n e d s o l u t i o n o n t h e r i g h t . n 1 2 3 4 5 6 7 8 9 10
1
,
l e n - l , n - l 2 1| l*n - l , n - l l o o |Cn,n|oo |cn,nloo |«n,»| 2
I6*'*'2 7.4268E-2 1.2047E-2 2.5072E-3 5.7950E-4 1.3985E-4 3.4385E-5 8.5272E-6 2.1234E-6 5.2980E-7 1.3232E-7
-
6.165 4.805 4.326 4.144 4.067 4.032 4.016 4.008 4.004
7.4268E-2 1.8742E-2 4.4585E-3 1.1169E-3 2.7847E-4 6.9659E-5 1.7412E-5 4.3528E-6 1.0882E-6 2.7205E-7
-
3.963 4.204 3.992 4.011 3.998 4.001 4.001 4.000 4.000
- l , n - l * ' \rc 1 lCn,nl«>
7.4268E-2 1.4123E-2 3.2023E-3 7.9926E-4 2.1322E-4 5.9058E-5 1.6610E-5 4.6816E-6 1.3141E-6 3.6642E-7
-
7.4268E-2 3.2387E-2 1.2887E-2 4.5351E-3 1.4838E-3 5.0056E-4 1.5518E-4 4.6647E-5 1.3667E-5 3.9669E-6
5.258 4.410 4.007 3.748 3.610 3.556 3.548 3.563 3.586
n
-l,n-l|o° | an.nl°°
-
2.293 2.513 2.841 3.056 2.964 3.226 3.327 3.413 3.445
E x a m p l e 2.2. (Singular s o l u t i o n ) ^ ' Consider the Dirichlet prob lem of the Laplace equation Ati = 0,
i n O = ( 0 , l ) x (0.5,0.5),
with an exact solution u = (x2 -f y2)
cos ( a r c t a n ( - ) / 2 ) , which has a y
singular point (0,0). Table 2.3 shows the comparison between the errors of the full grid solution and the combined solution at points Pi = (0.5,0.0) and P 2 = (0.25,-0.25). Table 2.4 compares the L2 norm and Loo norm
282
VII. Sparse
grid methods
and combination
techniques
of these errors. Table 2.3 shows |e n _i > n _i
l/|c»,n| — 2 3 / 2 «2.828, hence, the full grid error is of 0(h / ) at points far away from the singular point. Table 2.4 shows len-i^-iloo/le^nloo —► 2 1 / 2 «1.414, hence, the error is only of Ofi1/2) in the neighborhood of the singular point. The theory of the combination techniques for nonsmooth problems is not yet perfect, but Tables 2.3 and 2.4 illustrate that, similar to the case of smooth solutions, the error of the combined solution is only slightly deteriorated than that of the full grid solution by a logarithmic factor \ogh~1. 3 2
Table 2.3.
P o i n t w i s e e r r o r , full g r i d o n t h e l e f t , c o m b i n e d s o l u t i o n o n t h e r i g h t .
n
en,n(h°)-
1 2 3 4 5 6 7 8 9 10 11 12 13 14
3.9695E-3 6.3484E-3 2.7436E-3 1.0995E-3 4.1708E-4 1.5408E-4 5.6081E-5 2.0225E-5 7.2498E-6 2.5879E-6
c
en,n§-\)]-
n-l.n-l
l«n,nl
-
l«n,nl
-
0.625 2.314 2.495 2.674 2.707 2.747 2.773 2.789 2.801
n-l,n-ll
-
1.2359E-2 3.9539E-3 1.3877E-3 5.0905E-4 1.8525E-4 6.6799E-5 2.3929E-5 8.5351E-6 3.0357E-6
3.126 2.849 2.726 2.748 2.773 2.792 2.804 2.812
115
1 2 3 4 5 6 7 8 9
110
i-l,n-l
i « - . . | a ^ I e n,nl2 3.9695E-3 1.1000E-2 4.6957E-3 1.9065E-3 7.5691E-4 2.9572E-4 1.1405E-4 4.3523E-5 1.6463E-5 6.1820E-6
1 n - l , n - l 's n,nl>T'~T' I'n.nl
n,nVjiUJ
3.9695E-3 -7.5628E-3 0.5248 -1.4178E-3 -3.4260E-3 2.2075 1.1336E-2 1.3736E-3 5.7404E-4 5.9681 7.4114E-4 1.2047E-3 0.4765 7.6178E-4 1.5814 -2.2430E-6 4.1336E-4 1.8429 4.5173E-6 1.8820E-4 2.1964 -5.4202E-6 7.7368E-5 2.4325 -3.1142E-6 3.0123E-5 2.5684 -1.4430E-6 1.1331E-5 2.6585 -5.9843E-7 4.1710E-6 2.7165 -2.3432E-7 1.5146E-6 2.7538 -8.8687E-8 5.4518E-7 2.7781 -3.2851E-8 1.9512E-7 2.7940 -1.1997E-8
n-l.n-l1 1
l-l.nl
-
0.125 8 1 330.429 0 0. 1. 2 2.411 2.554 2.642 2.700 2.738
E r r o r s i n L>2- a n d L o o - n o r m , full g r i d o n t h e l e f t , combined solution on the right.
Table 2.4.
n
\*1:
i e
2
0.361 2.343 2.463 2.519 2.559 2.593 2.621 2.644 2.663
lc
1
-l,n-lloo 1cn,nloo
|Cn
\^n,n\oo
3.9695E-3 2.7093E-2 2.1732E-2 1.5744E-2 1.1195E-2 7.9270E-3 5.6071E-3 3.9651E-3 2.8038E-3 1.9826E-3
-
0.147 1.247 1.380 1.406 1.413 1.414 1.414 1.414 1.414
C
\r lL l c n,nl2
I« C - l , n - l " K.nb
3.9695E-3 1.9356E-2 1.2190E-2 7.5137E-3 4.6105E-3 2.8254E-3 1.7246E-3 1.0491E-3 6.3508E-4 3.8318E-4
-
0.205 1.588 1.622 1.629 1.632 1.638 1.644 1.652 1.657
2
\rc 1 I'-n.nloo
l«°n - l , n - l » ~ 1 c n.nloo
3.9695E-3 3.6331E-2 5.3937E-2 5.7539E-2 5.3173E-2 4.5382E-2 3.7325E-2 3.1997E-2 2.6971E-2 2.2621E-2
-
0.109 0.674 0.937 1.082 1.172 1.216 1.167 1.186 1.192
E x a m p l e 2.3. (Nonlinear h e a t t r a n s f e r ) ^ ! Consider the Dirichlet boundary problem of the nonlinear heat conduction equation du
d dx
w
ft .
+
■ fe> 5<'-
<9ix
oy
= /,
i n n = (0,l)x(0,l),
2. Combination
techniques
283
where t h e material property function is
r I-^-I^)1^ eQ = <
1,
u>
1/2
ti = l / 2
I l + (l/2- u )
1/a
,
u < 1/2.
sinh(7r(l — . — has a singular point at ( 0 , 0 ) . ; sinh(7r) In order to find a combined solution, it is necessary to solve the nonlinear discrete equation T h e exact solution u = sin(7ry)
= fij , l -f J = n , n - h 1.
LijUij
Table 2.5 lists the error in Li n o r m and L{ x) n o r m for the full grid solution and the combined solution. Table 2.5. E r r o r s in L^- and Loo-norm, full grid on t h e left, combined solution on t h e r i g h t . n
• |Cn
| l«n-l,n-lb| | 'n|2 l«n,nl2 \Cn>n*°°
a = 0.125 3 2.3171E-3 4 5.4442E-4 5 6 7 8 a = 0.25 3 1.2328E-3 4 3.2187E-4 5 6 7 8 a = 0.5 3 1.7520E-3 4 3.9924E-4 5 6 7 8 a = 0.75 3 2.2734E-3 4 6.3935E-4 5 6 7 8 a - 1.0 3 1.5987E-3 4 3.7171E-4 5 6 7 8
l«n-l,n-lloo |Cn,nloo
, c , ' n>nl2
|c
n-l,n-l'2 \<,J2 ^n,n\oo
I ^ l V n ^ 0 ^ ] ^ J ^
4.696 4.256
4.1210E-3 1.0569E-3
3.978 3.899
3.4773E-3 8.5692E-4 2.2023E-4 6.0845E-5 1.7488E-5 5.0685E-6
4.062 4.058 3.891 3.619 3.479 3.450
1.1681E-2 4.0742E-3 1.4670E-3 4.5330E-4 1.3810E-4 4.1357E-5
2.496 2.876 2.777 3.236 3.282 3.339
3.408 3.830
2.3066E-3 6.5162E-4
3.351 3.540
5.1882E-3 1.4933E-3 5.1843E-4 1.7537E-4 5.6677E-5 1.7737E-5
3.594 3.474 2.880 2.956 3.094 3.195
1.2641E-2 5.9064E-3 2.3389E-3 7.6980E-4 2.7278E-4 8.9370E-5
2.267 2.140 2.525 3.038 2.822 3.052
4.886 4.388
5.9662E-3 1.3116E-3
3.077 4.549
8.5599E-3 2.7436E-3 1.0472E-3 3.3565E-4 1.3102E-4 4.4540E-5
2.735 3.120 2.620 3.120 2.562 2.942
2.8296E-2 1.1184E-2 4.3777E-3 1.6041E-3 6.0695E-4 2.2305E-4
1.411 2.530 2.555 2.729 2.643 2.721
4.301 3.556
7.1804E-3 1.8969E-3
2.923 3.785
1.0241E-2 2.6358E-3 1.0987E-3 3.6211E-4 1.9252E-4 1.7071E-4
2.078 3.885 2.399 3.034 1.881 1.128
3.0782E-2 1.1040E-2 4.4042E-3 1.8599E-3 7.4955E-4 9.6088E-4
1.365 2.788 2.507 2.368 2.481 0.780
4.360 4.301
5.0857E-3 1.2175E-3
3.151 4.177
7.9072E-3 2.8952E-3 1.0115E-3 3.3684E-4 1.0732E-4 3.2960E-5
2.567 2.731 2.862 3.003 3.138 3.256
2.2445E-2 9.1664E-3 3.9667E-3 1.7222E-3 6.3854E-4 2.2181E-4
1.809 2.449 2.311 2.303 2.697 2.879
VII. Sparse grid methods and combination
284
techniques
Applications of the combination techniques to b o u n d a r y layer problems and problems with non-rectangular domains can be found in [39]. There are also applications to hydrodynamics in the work by M. Griebel and V. T h u r n e r ^ 4 ^ . T h e numerical experiments of the combination techniques under various conditions can also be found in these references.
2.4. Comparison sparse grid
of combination methods
techniques,
SEM
and
U. Rude expounded in [107] and [106] the relation between the combi nation techniques and the SEM, and compared the numerical results. First, it is obvious t h a t both are based on the asymptotic expansion of the error, with the aim of eliminating as m a n y terms as possible. Second, there is also a big difference between them: the former tries to obtain an approximate solution of the fine grid, usually the order of accuracy is 0(h2), therefore this is a 0(h2) m e t h o d in essence, while the latter can have an arbitrary order of accuracy, provided there is a suitable asymptotic expansion. Of course, this higher order of accuracy is limited on the coarsest grids, b u t it can be extended to other points by interpolation (an algorithm for ob taining a higher order accuracy on the finer grids is given in Appendix III). Hence, as far as the smooth solution is concerned, the SEM is better t h a n the combination techniques. But for singular problems, the combination techniques may be more popular because they require weaker conditions for error expansion. Furthermore, the coefficients of the combination do not change when the exponents of the expansion vary. Hence, for singu lar problems, the combination techniques may be more effective t h a n the SEM. Finally, b o t h methods split the problem into mutually independent subproblems, and are suitable for parallel computation. T h e higher the dimension, the better the parallelism. T h e following examples are quoted from U. Riide's works [106] and [109]. E x a m p l e 2 . 4 . Consider the Dirichlet problem of Poisson's equation f -Au |^
= f(x,y), u = 0,
infi = (0,l)2, on 3fi,
(7.2.21)
with an exact solution 7rx wy u = x(l — x ) c o s ( — ) y ( l — y)cos ( — ) . z z T h e numerical results by the Richardson extrapolation, the SEM, the combination techniques and the sparse grid m e t h o d are listed in the follow ing four tables.
2. Combination
techniques
285
Table 2.6. Richardson extrapolation for a smooth model problem h Error u2 "0 "1 "3 "4 1 Point 5.916E-3 -1.163E-4 3.093E-7 -9.569E-11 -5.121E-11 2.020E-1 4.918E-3 3.994E-5 4.656E-8 1.363E-7 1/4 L2 Energy 2.121E-1 5.894E-3 5.768E-5 6.438E-8 1.850E-7 Point 1.391E-3 -6.979E-6 4.739E-9 -5.139E-11 4.739E-2 2.949E-4 9.185E-7 1/8 L2 1.737E-7 \ Energy 5.076E-2 3.995E-4 2.136E-6 5.116E-7 Point 3.427E-4 -4.317E-7 1.Q97E-10 1/16 L2 1.164E-2 1.807E-5 2.423E-8 1 Energy 1.254E-2 2.642E-5 1.118E-7 Point 8.535E-5 -2.696E-8 1/32 L2 2.899E-3 1.140E-6 1 Energy 3.127E-3 1.911E-6 Point 2.131E-5 7.241E-4 1/64 L2 1 Energy 7.813E-4
Table 2.7. Splitting extrapolation for a smooth model problem Error Point 1/4 L2 Energy Point 1/8 L2 Energy Point 1/16 L2 \ Energy Point 1/32 L2 1 Energy Point 1/64 L2 1 Energy h
"0
5.916E-3 2.020E-1 2.121E-1 1.391E-3 4.739E-2 5.076E-2 3.427E-4 1.164E-2 1.254E-2 8.535E-5 2.899E-3 3.127E-3 2.131E-5 7.241E-4 7.813E-4
"1
-3.270E-4 1.397E-2 1.683E-2 -1.927E-5 8.097E-4 1.095E-3 -1.187E-6 4.918E-5 7.125E-5 -7.393E-8 3.047E-6 4.633E-6
u2 u3 1 5.118E-6 -6.066E-8 7.938E-5 9.439E-5 1.157E-3 1.396E-4 | 7.527E-8-1.354E-10 1.993E-5 3.046E-6 5.224E-5 8.579E-6 1.162E-9 5.683E-7 2.966E-6
_ ..
i
286
VII. Sparse grid methods and combination
techniques
Table 2.8. Combination technique for a smooth model problem hx = hy
Error Point L2 Energy Point L2 Energy Point L2 Energy Point L2 Energy Point L2 Energy Point L2 Energy
1/2
1/4 1/8
1/16 1 1/32 1 1/64 1
tin 3.008E-2 5.299E-1 8.189E-1 5.916E-3 1.096E-1 3.618E-1 1.391E-3 2.635E-2 1.711E-1 3.427E-4 6.943E-3 8.236E-2 8.535E-5 2.096E-3 3.671E-2 2.131E-5 7.241E-4 7.813E-4
tli "2 "3 "4 u$ 2.717E-3 -1.840E-4 -2.439E-4 -1.092E-4 -3.935E-5 1.404E-1 3.491E-2 9.575E-3 3.051E-3 1.124E-3 4.235E-1 2.086E-1 1.013E-1 4.703E-2 1.470E-2 1.233E-3 2.571E-4 5.207E-5 1.000E-5 2.527E-2 6.408E-3 1.958E-3 7.130E-4 1.724E-1 8.326E-2 3.724E-2 3.280E-3 3.335E-4 8.021E-5 1.929E-5 6.864E-3 2.063E-3 7.170E-4 8.239E-2 3.673E-2 1.085E-3 8.479E-5 2.100E-5 2.092E-3 7.228E-4 3.671E-2 8.015E-4 2.128E-5 7.239E-4 7.823E-4
Table 2.9. Method of sparse grid for a smooth model problem hx = hy
1/2
1/4
1/8 \
1/16 1
1/32 \ 1 1/64 |
Error size Point L2 Energy size Point L2 Energy size Point L2 Energy size Point L2 Energy size Point L2 Energy size Point L2 Energy
qh,X,hy
"Vo
1 6.407E-3 4.488E-1 6.739E-1 9 1.663E-3 1.138E-1 3.379E-1 49 4.247E-4 2.818E-2 1.679E-1 225 1.171E-4 6.836E-3 8.193E-2 961 4.048E-5 1.622E-3 3.664E-2 3969 2.131E-5 7.241E-4 7.813E-4
q^x,hy
*Vl
5 1.045E-3 1.482E-1 3.732E-1 33 3.682E-4 2.849E-2 1.686E-1 161 1.138E-4 6.839E-3 8.194E-2 705 1.078E-4 2.626E-3 4.658E-2 2945 2.130E-5 7.241E-4 7.816E-4
Q^Xj^y
^0.2
Q^Xt^y
^0.3
Q^Xj^y
^0.4
17 49 129 7.211E-6 -6.145E-5 -2.215E-5 4.157E-2 1.081E-2 2.693E-3 1.917E-1 9.493E-2 4.390E-2 97 2 641 8.590E-5 2.829E-5 1.717E-5 6.956E-3 1.658E-3 7.288E-4 8.241E-2 3.693E-2 2.439E-3 449 1153 3.862E-5 2.058E-5 1.623E-3 7.243E-4 3.665E-2 9.232E-4 1921 2.121E-5 7.241E-4 7.889E-4
Q^Xt^y
^0.5
1
321 1.259E-6 8.816E-4 1.213E-2|
|
2. Combination
techniques
287
Table 2.6 lists the errors of the Richardson extrapolation in the L2 norm, the energy norm and the error at point P = ( - , - ) . Here, uk (k = 0,1, • • •, 4) denote the results after k extrapolations. Table 2.7 lists the errors of Type 1 SEM, where uk (k = 1, 2, 3) stands for the result after k split. Table 2.8 lists the errors of the combination techniques, where UQ stands for the result evaluated on the full grids, and the results of the combination k
techniques are uk = ^T uhx2-i}hy2i-k 1=0
k-1
- ] T uhx2-i)hy2i-k+i
(k = 1, • • •, 5).
»=o
Table 2.9 lists the errors of the method of sparse grid, where S0xk'
y
=
k
U 5°. x_i ofc_ijB_i denotes the sparse grid, and the "size" denotes the order y
i=o
'
of the stiffness matrix. These tables show that the Richardson extrapolation converges fastest but has the highest cost; the method of sparse grid are slightly better than the combination techniques and is unsuitable for performing in parallel; the SEM has lower cost and higher degree of parallelism, furthermore, it has the same order of error as the Richardson extrapolation (only with larger coefficients). Therefore it seems that, among these methods, SEM is superlative. E x a m p l e 2.5. Consider again the problem (7.2.21) with another solu tion u = sin(16x 4- 16y)x(l — x)y(l — y) which is an oscillatory function. The numerical results from the four dif ferent methods are listed in Tables 2 . 1 0 - 2 . 1 3 . It can be seen that when the mesh width is larger than the oscillatory period, the result obtained by a few steps of Richardson's extrapolation is not as good as the combina tion techniques. The reason is that when the mesh width is large, the high frequency oscillations cannot be correctly reflected. But when the grids are refined step by step, the accuracy of the Richardson extrapolation and SEM will be rapidly improved. The results of Type 1 SEM methods are listed in Table 2.11. Comparing with Type 1 methods, the amount of work and storage of Type 2 methods are more economical and the accuracy is only slightly worse, which are verified by the examples shown in Chapter V.
288
VIL Sparse grid methods and combination
techniques
Table 2.10. The Richardson extrapolation for an oscillatory model problem Error u0 ui u2 ti 3 -0.1910 2.163E-2 2.349E-3 -9.073E-5 Point 10.7775 1.028E+0 1.788E-1 6.366E-3 1/4 L2 Energy 1.061E+1 9.919E-1 1.771E-1 6.264E-3 -0.0315 3.554E-3 -5.260E-5 -6.584E-S Point 1.8661 2.182E-1 3.085E-3 1.483E-5 1/8 L2 1.505E-5 \ Energy 1.872E+0 2.196E-1 3.092E-3 Point -5.215E-3 1.728E-4 -8.870E-7 1/16 L2 3.066E-1 1.103E-2 6.186E-5 1 Energy 3.201E-1 1.184E-2 6.854E-5 Point -1.174E-3 9.970E-6 1/32 L2 6.858E-2 6.346E-4 1 Energy 7.222E-2 6.953E-4 Point -2.860E-4 1/64 L2 1.667E-2 1 Energy 1.759E-2 h
Table 2.11. Splitting extrapolation for an oscillatory h Error UQ ui U2 Point -0.1910 1.483E-1 -4.577E-2 1/4 L2 10.7775 8.696E+0 3.358E+0 Energy 1.061E+1 8.277E+0 3.063E+0 Point -3.150E-2 1.077E-2 -9.635E-4 1/8 L2 1.866E+0 6.669E-1 5.847E-2 Energy 1.872E+0 6.691E-1 5.857E-2 Point -5.215E-3 4.672E-4 -1.348E-5 3.066E-1 3.004E-2 9.573E-4 1/16 L2 1.061E-3 Energy 3.201E-1 3.234E-2 Point -1.174E-3 2.607E-5 1/32 L2 6.858E-2 1.665E-3 Energy 7.222E-2 1.831E-3 Point -2.860E-4 1/64 L2 1.667E-2 Energy 1.759E-2
u4 2.897E-7 1.868E-5 1.833E-5
model problem U3 ] 1.590E-2 1.229E+0 1.146E+0 2.085E-5 1.297E-3 1.303E-3
2. Combination techniques Table 2.12. hx = hy Error Point 1/2 L2 Energy Point 1/4 L2 Energy Point 1/8 L2 1 Energy Point 1/16 L2 \ Energy Point 1/32 L2 1 Energy Point 1/64 L2 1 Energy
289
Combination technique for an oscillatory model problem UQ
-8.727E-1 1.264E+1 2.876E+0 -1.910E-1 6.258E+0 3.372E+0 -3.152E-2 6.960E-1 1.131E+0 -5.215E-3 1.377E-1 3.675E-1 -1.174E-3 4.223E-2 1.466E-1 -2.860E-4 1.667E-2 1.759E-2
ui t*2 u3 u4 2.298E-1 3.345E-1 1.050E-1 2.254E-2 1.719E+1 1.396E+1 5.778E+0 9.796E-1 5.544E+0 4.988E+0 3.306E+0 1.640E+0 6.349E-2 1.439E-2 9.538E-3 3.441E-3 5.722E+0 9.649E-1 2.274E-1 6.492E-2 3.075E+0 1.523E+0 6.578E-1 2.984E-1 1.980E-4 1.230E-3 5.601E-4 1.851E-1 4.947E-2 1.613E-2 4.550E-1 1.904E-1 6.072E-2 -9.534E-4 -1.707E-4 4.069E-2 1.558E-2 1.470E-1 1.959E-2 -2.740E-^ 1.654E-2 1.757E-2
us 1 1.146E-2 2.337E-1 7.213E-l|
|
Table 2.13. Sparse grid method for an oscillatory model problem hx = hy
1/2 1
1/4 1
1/8 1 1 1/16 1 1 1/32 j
1/64 \
Error width Point L2 Energy width Point L2 Energy width Point L2 Energy width Point L2 Energy width Point L2 Energy width Point L2 Energy
ofti>hy
•Vo
1 1.746E-2 1.000E+0 9.999E-1 9 9.550E-3 1.001E+0 9.947E-1 49 -5.882E-3 4.442E-1 6.183E-1 225 -1.704E-3 1.059E-1 3.035E-1 961 -5.647E-4 2.518E-2 1.374E-1 3969 -2.860E-4 1.667E-2 1.759E-2
Q f c i» f t y
^0.1
ch*>hy
^0.2
cfcx,fcy
^0.3
Qhx>hy
"^0.4
Q^x^y
1
"^0.5 5 17 49 129 321 1.617E-2 9.183E-3 2.881E-4 -4.409E-3 -4.374E-4 1.000E+0 1.002E+0 8.671E-1 4.101E-1 1.308E-1 9.998E-1 9.934E-1 8.895E-1 5.817E-1 3.178E-l| 33 97 257 641 2.674E-4 -4.416E-3 -4.395E-4 1.661E-4 8.669E-1 4.101E-1 1.308E-1 3.359E-2 8.898E-1 5.817E-1 3.179E-1 1.501E-1 | 161 449 1153 -9.617E-4 2.927E-5 -3.083E-5 1.373E-1 3.469E-2 1.712E-2 3.336E-1 1.587E-1 4.422E-2 705 1921 -4.877E-4 -2.425E-4 2.538E-2 1.669E-2 1.380E-1 1.907E-2 2945 -2.813E-4 1.667E-2 1.765E-2
290
VII. Sparse grid methods and combination
techniques
3. Outline of t h e implicit extrapolation T h e Richardson extrapolation, the SEM, the sparse grid m e t h o d s and the combination techniques have a common characteristic, i.e., combine ap proximate solutions obtained on different grids to obtain a solution of higher order of accuracy. T h e theoretical foundation of this idea comes from the existence of an asymptotic expansion of the error, which are called the ex plicit extrapolation methods by [18], [107]. Recently, a kind of implicit extrapolation methods was studied. For implicit extrapolation, the idea of extrapolation is applied to get approximate equations with higher order ac curacy. Hence, neither a global expansion of the discrete error nor a uniform mesh is required. Problems with nonuniform meshes, locally refined grids, or the related h- and p-adaptive methods, can all be treated by the frame work of the implicit extrapolation m e t h o d s . At present, the research on m e t h o d s such as truncation error extrapolation, energy extrapolation and T-extrapolation is very active. In the following we shall introduce briefly the development of the implicit extrapolation m e t h o d s . • r - e x t r a p o l a t i o n . Many optimal algorithms for solving partial dif ferential equations, such as the multigrid method, can be considered as applications of the multilevel principle. These m e t h o d s require evaluations on various grids with different widths, and extrapolation will be imple mented economically. A. B r a n d t ^ 3 ) called this kind of extrapolations on multigrid method as r-extrapolation. For details, see W . Hackbuscht 4 3 ! and U. R i i d e l 1 0 3 ] . However, all these extrapolations basically belong to 113 the Richardson type. A. Schiiller and Q. iw i applied SEM to multigrid m e t h o d and constructed a new form of r-extrapolation. • T r u n c a t i o n error e x t r a p o l a t i o n . T h e truncation error extrapola tion is based on the asymptotic expansion of the discrete truncation error. By means of the idea of extrapolation, several discrete equations can be combined into an equation with higher order of error after eliminating the truncation error terms with lower order. Of course, in order to make such a combination possible, it is necessary to transfer the truncation errors on different grids to a common space, at the same time, it is also necessary to transfer the extrapolation solution to the original grid space. T h e imple mentation and numerical experiments of the truncation error extrapolation can be found in papers by U. Ridel 107 ! , and the application of truncation error extrapolation in multigrid m e t h o d can be found in the monograph by W . Hackbuscht 4 3 ). • E n e r g y e x t r a p o l a t i o n . T h e finite element solution can be ex plained as the approximation of a minimization problem. For example, the homogeneous Dirichlet problem of the Poisson equation is equivalent to the
3. Outline of the implicit
extrapolation
291
minimization problem of the quadratic functional u e
^n)(^(")"(«./)).
(7-3.1)
where En(u) = - I{Vufdxdy. (7.3.2) 2 Jn Denote by SQ the space of test functions. In order to evaluate approxi mately (7.3.2), replace u by the interpolation function u1, and correspond ingly, define the approximate energy by E&u) = E^u1)
= \vFAhu,
(7.3.3)
where u is the vector composed by the nodal values of u and Ah is the stiff ness matrix. U. Riidet 109 ! proved that under an appropriate assumption, the following asymptotic expansion exists, E&(u) - En(u) = /i 2 ei + h4e2 + .. •.
(7.3.4)
Hence, carrying out one extrapolation, one gets the approximate functional
££(«) = ^n / 2 («)-^fc(«).
(7-3.5)
which approximates En(u) to order 0 ( / i 4 ) . The stiffness matrix given by
£»(«) is ^
= ^"
/ 2
-^«/2)
T
^^/2.
(7-3-6)
where l\,2 is the restricted operator. Thus the numerical examplefluy^ has an order of error 0 ( / i 4 ) , and the method is simpler than the finite element method with piecewise polynomials of higher degree. It is also possible to derive methods of higher order approximation in the sense of energy extrapolation by applying the multivariate expansion of mesh parameters and using the coefficients of SEM. The numerical examples in [109] show that this is very effective. Apart from the implicit extrapolation methods discussed above, the ex trapolation method for hierarchical basis is very attractive. It is the special case of the energy extrapolation under the framework of finite element with hierarchical basis, and its idea is based on a purely local element-by-element analysis. Hence, it can be applied to irregular and unstructural grids. More over, it has special effect for the composite grids generated by the adaptive
292
VII. Sparse grid methods and combination
techniques
local refinement. For details, see U. Rude' ' and S. McCormick and U. Riidef91!. However, the research on the theoretical aspect of the implicit extrapo lation methods is far from perfect. Up to now, there are only results from the numerical experiments on the model problems. Further research on the link in the extrapolation techniques, multilevel methods and the fast adaptive composite grid methods (FAC) is recommended.
Appendix I TABLES OF SPLITTING EXTRAPOLATION COEFFICIENTS s m
P
Type 1
1
Type 2
1
2 2 (0,0)
37/45
31/36
(1.0)
-20/9
-128/45
(1.1)
16/9
16/9
(2,0)
64/45
81/40
2 3 (0,0)
-97/567
-41/180
(1.0)
148/135
41/18
(1.1)
-80/27
-208/45
(2,0)
-64/27
-297/56
(2,1)
256/135
27/10
(3,0)
4096/2835
1024/315
1
(o.o)
11749/722925
107/2880
1
(1,0)
-388/1701
-946/945
(1,1)
592/405
3328/675
(2,0)
2368/2025
729/128
(2,1)
-256/81
-1458/175
(2,2)
4096/2025
6561/1600
(3,0)
-4096/1701
-4096/405
;(3,1)
16384/8505
4096/945
(4,0)
1048576/722925
390625/72576
|2 4
2 5 (0,0)
-110689/147910455
-3751/907200
(1,0)
46996/2168775
9181/32400
(1,1)
-1552/5103
-8576/2835
(2,0)
-6208/25515
-11223/3200
(2,1)
9472/6075
1773/160
(2,2)
-4096/1215
-6561/448
(3,0)
151552/127575
82304/6075
(3,1) (3,2)I
-94208/6075
-16384/5103
1
262144/127575
293
1152/175
1
294
s\m ,2| 5
2 6
Appendix
P
Typel
Type 2
1
(4,0)
-1048576/433755
-6640625/342144
1
(4,1)
4194304/2168775
390625/54432
(5,0)
1073741824/739552275
17496/1925
(0,0)
50844133/3028466566125
2431/7257600
(1.0)
-442756/443731365
-2402/42525
(1,1) (2,0)
187984/6506325
832/675
751936/32531625
64131/44800
(2,1)
-24832/76545
-1527/175
(2,2)
151552/91125
14585103/627200
(3,0)
-397312/1607445
-557056/51975
(3,1)
606208/382725
1048576/42525
(3,2)
-262144/76545
-32768/1225
(3,3)
16777216/8037225
1048576/99225
(4,0)
38797312/32531625
49609375/1596672
(4,1)
-4194304/1301265
-78125/2673
(4,2)
67108864/32531625
78125/7168
(5,0)
-1073741824/443731365
-186624/5005
(5,1)
4294967296/2218656825
23328/1925
(6,0)
4398046511104/3028466566125
13841287201/889574400
3 2 (0,0,0)
17/5
83/24
(1,0,0)
-4/1
-208/45
(1,1,0)
16/9
16/9
(2,0,0)
64/45
81/40
-593/315
-2209/1080
3 3 (0,0,0) (1,0,0)
68/15
887/135
(1,1,0)
-16/3
-944/135
(1,1,1)
64/27
64/27
(2,0,0)
-64/15
-2241/280
(2,1,0)
256/135
27/10
(3,0,0)
4096/2835
1024/315
3 4 (0,0,0)
5129/8925
20/27
(1,0,0)
-2372/945
-976/189
(1,1,0)
272/45
294/25
(1,1,1)
-64/9
-1408/135
[(2,0,0)
1088/225
60597/4480
(2,1,0)
-256/45
-2088/175
[(2,1,1)
1024/405
18/5
|
i
Appendix
295
s \m\
P
Type 1
Type 2
3 4
(2,2,0)
4096/2025
6561/1600
(3,0,0)
-4096/945
-8192/567
(3,1,0)
16384/8505
4096/945
(4,0,0) 1
1048576/722925
390625/72576
(0,0,0)
-934921/9130275
-2297/12600
(1,0,0)
20516/26775
196753/75600
(1,1,0)
-9488/2835
-52918/4725
(1,1,1)
1088/135
4544/225
(2,0,0)
-37952/14175
-295227/22400
(2,1,0)
4352/675
129933/5600
(2,1,1)
-1024/135
-3078/175
3 5
(2,2,0)
-4096/675
-225261/11200
(2,2,1)
16384/6075
2187/400
(3,0,0)
69632/14175
1201408/42525
(3,1,0)
-16384/2835
-905216/42525
(3,2,0)
262144/127575
1152/175
(4,0,0)
-1048576/240975
-63671875/2395008
(4,1,0)
4194304/2168775
390625/54432
(5,0,0)
1073741824/739552275
17496/1925
349/45
47/6
(1,0,0,0)
-52/9
-32/5
(1,1,0,0)
16/9
16/9
(2,0,0,0)
64/45
81/40
-21277/2835
-2113/270
4 2 (0,0,0,0)
4 3 (0,0,0,0) (1,0,0,0)
1396/135
397/30
(1,1,0,0)
-208/27
-1264/135
(1,1,1,0)
64/27
64/27
(2,0,0,0)
-832/135
-2997/280
(2,1,0,0)
256/135
27/10
(3,0,0,0)
4096/2835
1024/315
3153181/722925
63517/12960
(1,0,0,0)
-85108/8505
-44558/2835
(1,1,0,0)
5584/405
44044/2025
(1,1,1,0)
-832/81
-5504/405
(1,1,1,1)
256/81
256/81
(2,0,0,0)
22336/2025
111807/4480
(2,1,0,0)
-3328/405
-2718/175
(2,1,1,0)
1024/405
18/5
4 4 (0,0,0,0)
1
|
|
1
Appendix
296 s m
P
Type 1
Type 2
4 4
(2,2,0,0)
4096/2025
6561/1600
(3,0,0,0)
-53248/8505
-53248/2835
(3,1,0,0)
16384/8505
4096/945
(4,0,0,0)
1048576/722925
390625/72576
(0,0,0,0,0)
125/9
1007/72 -368/45
5 2
5 3
6 2
(1,0,0,0,0)
-68/9
(1,1,0,0,0)
16/9
16/9
(2,0,0,0,0)
64/45
81/40
(0,0,0,0,0)
-11005/567
-4307/216
(1,0,0,0,0)
500/27
334/15
(1,1,0,0,0)
-272/27
-176/15
(1,1,1,0,0)
64/27
64/27
(2,0,0,0,0)
-1088/135
-3753/280
(2,1,0,0,0)
256/135
27/10
(3,0,0,0,0)
4096/2835
1024/315
(0,0,0,0,0)
109/5
263/12
(1,0,0,0,0)
-28/3
-448/45
(1,1,0,0,0)
16/9
16/9
(2,0,0,0,0)
64/45
81/40
-12589/315
-22009/540
6 3 (0,0,0,0,0,0) (1,0,0,0,0,0)
436/15
9091/270
(1,1,0,0,0,0)
-112/9
-1904/135
(1,1,1,0,0,0)
64/27
64/27
(2,0,0,0,0,0)
-448/45
-4509/280
(2,1,0,0,0,0)
256/135
27/10
(3,0,0,0,0,0)
4096/2835
!
1
|
1
1024/315
where s, m and /? denote the dimension, splitting number and splitting index, respectively. Remark 1. For m = 1,/? = (1,0, • • • ,0),ap = 4/3;/? = (0,0, • • - , 0 ) , ^ = -{As - 3)/3. Remark 2. ap = a 7 if /? and 7 have the same component but different permutations.
Appendix
297
Appendix II TABLES OF SUCCESSIVE SPLITTING EXTRAPOLATION COEFFICIENTS Type 1 s \m 2 1
2 2
Coefficients
P
(1 + 2P)/(1 - 2P)
(0,0) | (0,1)
2 P / ( - l + 2p)
(o,o)
l / ( l - 22P) 2
(1.1)
2 3
2P/(_I +
2 2P)
(0,2)
0
(o,o)
{(1 + 2 * ) 3 ( - l + 2P - 2 2 P)(1 + 2 2 P)} / { - 1 - 2(2P) - 2(2 2 P) 3
4
7
9
10
1
n
-3(2 P) - 3(2 P) -|- 3(2 P) + 4(2 P) + 2 ( 2 P ) + 2 ( 2 P ) } (0,3)
{ 2 8 P } / { l + 2(2P) + 2(2 2 P) + 3(2 3 P) + 3(2 4 P) -3(2 7 P) - 4(2 9 P) - 2(2 10 P) - 2 ( 2 n P ) }
2 4
2 5P(X
_ 2 P + 2 2 P)(1 + 2P + 22P)
(2,1)
- 1 _ 2 P - 2(2 3 P) - 24P + 3(2 5 P) - 2(2 6 P) + 2(2 7 P) + 2(2 9 P)
(o,o)
l / ( l - 28P)
(2,2)
2»P/(_1 +
0
(1,3)
(0,0,1) 3 2 (0,0,0) (0,0,2) ' (0,1,1)
28P)
0
(0,4)
3 1 (0,0,0)
|
(1 + 2 P
+1
)/(1-2P)
2 P / ( - l + 2P) ( - 1 - 3(2P) + 2 J P)(1 + 2P + 22P) - 1 - 4(2P) + 4(2 3 P) + 2 4 p 24P 1 + 4(2P) - 4(2 3 P) - 2 4 p 2 2 p ( l + 2P) - 1 - 3(2P) + 3(2 2 P) + 23P
'
Appendix
298
s m
P
Coefficients
4 1
(0,0,0,0)
(1 + 3 ( 2 P ) ) / ( 1 - 2P) 2 P / ( - l + 2P)
(0,0,0,1) p
14 2
(0,0,0,0) (0,0,0,2) (0,0,1,1)
5 1 (0,0,0,0,0)
(0,0,0,0,1) 5 2 (0,0,0,0,0)
(0,0,0,0,2) (0,0,0,1,1)
- 1 - 6(2 ) - 6(2 2 p ) - 6(2 3 P) + 3(2 4 P) - 1 -- 6(2P) + 6(2 3 P) + 24P 2(2 4 P) 1 + 6(2P) - 6(2 3 P) - 24P 2 2 P(1 + 2P) - 1 - 5(2P) + 5(2 2 P) + 23P (1 + 4 ( 2 P ) ) / ( 1 - 2 P ) 2 P / ( - l -I- 2P) - 1 - 8(2P) - 10(2 J P) - 12(2 3 P) + 6(2 4 P) - 1 - 8(2 p ) + 8(2 3 P) + 24P 3(2*P) 1 + 8(2P) - 8(2 3 P) - 24P 2 2 P(1 + 2P) - 1 - 7(2P) + 7(2'2P) -|- 2 3 p
Type 2 s TO
P
2 2 (0,0)
(1.1) (0,2) 12 3 (0,0)
Coefficients l / ( l - 22P) 22P/(-1 +
2 2 P)
0 | ( 2 P + 3P)(1 + 2 p )(-2P3P + 24P + 2 2 P3 2 P - 2 5 P3P)} / { ( - 1 + 2P)(22P3P + 2P32P - 2(2 3 P3 3 P) - 25P -f 24P3P + 2 3 p 3 2 P _ 2 2p 3 3p _ 2 (2 7 P) + 26P3P + 2 5 p 3 2 P - 2(2 4 p 3 p ) + 2(2 7 p 3 3 P)}
(0,3)
{ 2 6 P ( - 2 P + 3P) 2 (2P + 3P)} / {(1 - 2 2 p )(2 2 P3 p -|- 2 p 3 2 p - 2(2 3 P3 3 P) - 25P + 2 4 p 3P + 2 3 P3 2 P _ 2 2p 3 3p _ 2(2 7 P) + 2 6 P3 p + 2 5 p 3 2 P - 2(2 4 P3 p ) -f 2(2 7 P3 3 P)}
(2,1)
{2 3 P3 3 P(-1 + 2 2 P)(1 + 2 2 P } / / 2 2 p 3 p _j_ 2P32P - 2(2 3 p 3 3 p ) - 2 5 P + 2 4 P3 P + 2 3 p 3 2 p __22p33p _ 2(2 7 P) + 26P3P + 2 5 p 3 2 P - 2(2 4 p 3 p ) + 2(2 7 P3 3 P)}
2 4 (0,0)
l / ( l - 3 4 P)
(0,4)
0
(1-3)
0
(2,2)
34P/(-14-34P)
Appendix
5
771
13 2
299
£
Coefficients ( - 1 + 2*3*)(l + 2(2?) - 4(3^) + 2P3P) ( - 1 + 2 P ) ( - 1 + 3 P ) ( - 1 - 3(2^) + 3(3^) + 2P3P) 32P(-1 + 2P) (1 - 3 P ) ( - 1 - 3(2^) + 3(3^) + 2P3P) 2 2 P ( - 1 + 3 P) ( - 1 + 2P)(-1 - 3(2P) + 3(3^) + 2P3P)
(0,0,0) (0,0,2) (0,1,1)
3 3
l / ( l - 2 P)
(14,1)
3
3 P/(-1 +
(0,0,3)
23P)
0
(0,1,2) 4 2
\
3
(0,0,0)
0 2
{ - 1 - 4(2P) - 2 P + 6(3P) + 6(2P3P) + 3(3 2 P) - 12(2 2 P3 2 P)
(0,0,0,0)
+3(2 2 P3 2 P)} / { ( - l + 2P)(-1 + 3 P ) ( - 1 - 5(2^) + 5(3^) + 2 ^ ) } 2 ( 3 2 p ) ( " l + 2P) (1 - 3 P ) ( - 1 - 5(2P) + 5(3P) + 2P3P) 2 2 P ( - 1 + 3 P) ( - 1 + 2P)(-1 - 5(2P) + 5(3P) + 2P3P)
(0,0,0,2) (0,0,1,1) 5 2
|
{ - 1 - 6(2P) - 3(2 2 P) + 8(3P) + 12(22P3P) + 8(3^) - 24(2^32P)
(0,0,0,0,0)
+6(2 2 P3 2 P)} / { ( - l + 2P)(-1 + 3P)(-1 - 7(2P) + 7(3P) + 2P3P)} 3 ( 3 2 P ) ( - l - f 2P) (1 - 3 P ) ( - 1 - 7(2P) + 7(3P) + 2P3P)
(0,0,0,0,2)
2
(0,0,0,0,1)
2P(_I +
3 P)
( - 1 + 2P)(-1 - 7(2P) + 7(3P) + 2P3P) { - 1 - 8(2P) - 6(2 2 P) + 10(3P) + 20(22P3P) + 15(3P) - 40(2P32P)
6 2 (0,0,0,0,0,0)
+10(2 2 P3 2 P)} / { ( - l + 2P)(-1 + 3 P ) ( - 1 - 9(2P) + 9(3P) + 2P3P)} (0,0,0,0,0,2) (0,0,0,0,1,1)
4(3 2 P)(-1 + 2P) (1 - 3 ) ( - l - 9(2P) + 9(3 p ) + 2P3P) 2 2 P ( - 1 + 3P) ( - 1 + 2P)(-1 - 9(2P) + 9(3P) + 2P3P) p
1
Remark 3. There are some terms with coefficient 0, because when both of s, m are even or odd, the homogeneous terms can be eliminated only by global refinement. Remark 4. The successive splitting extrapolation is much less than the Romberg extrapolation but is larger than SEM in the amount of work. The successive splitting extrapolation can be applied to the expansion with fractional power, and has a large scope of applications.
Appendix
300
Appendix III EXTRAPOLATION AT FINE GRID POINTS
It is well known that the extrapolation methods only offer the values of high order accuracy at coarse grid points and, in general, by using interpo lation formulae of higher degree, one can obtain higher order accuracy at finer grid points. C M . Chen and Q. proposed a fine grid extrapola tion formula, which can give directly the higher order accuracy at finer grid points, and can also be applied to SEM. The algorithm will be introduced briefly as follows: Let Jh be a triangulation of Q C M2, K be a triangular element, and z be a vertex of K. Suppose that for the error of finite element approximate solution there is an asymptotic expansion (ti* - u){z) = h2$(z) + 0(/i 4 |log/i|).
(1)
Let Z12 be the midpoint of the line joined two vertices, z\ and 2 2 , of K as well as a vertex of the refined partition. The extrapolation formula, with z\2 as the finer grid point, is «*(*«) = « h / 2 (zi 2 ) +-(uhl\Zl) - uh(Zl)) + -{uhl\z2) - uh(z2)). 6 6 It can be proved that for any e > 0, there exists an error estimate ""0*12)- u{z12) = 0 ( / i 4 - ) .
(2)
(3)
In fact, by (1), 9(zt) = 3 ^ ( « " - «*/»)(*) + 0(h*\\ogh\),
i = 1,2.
(4)
Applying the error estimate of the linear interpolation of $,
*(*„) = !*(*!) + |*(z 2 ) + 0(fc2-*),
(5)
Appendix
301
hence, we have * ( * » ) = ^ j ( u f c - n fc/2 )(2n) + ^ - Z( n * - ufc/a)(Z2) + 0(h*-). on
(6)
3AI
Since uh'\zl2)
= ti(z ia ) + £ * ( * « ) + 0(/» 4 | log&l),
(7)
by using (6), the formula (2) and estimate (3) are obtained. For a rectangular partition, the bilinear finite element approximate so lution also has an asymptotic expansion of the form (1). In Fig. 1, Pi (i — 1,2,3,4) are coarse grid points, M,- (i = 0, • • •, 4) are refined grid points, and the extrapolation formula of refined grid points is f « h (Mi) = u h / 2 (Mi) + \\(u*l\Px) «"(M 0 ) = uhl\M0)
- u\Px))
+ ^ £
+ ( U "/2(p 2 ) _
(uh'2(pi)
uh(P2))},
(8)
~ «*(*))•
1= 1
The proof follows immediately by replacing $ in expansion (1) with the bilinear interpolation $ 7 of $.
Fig. 1
Fig. 2
We shall derive the results at refined grid points of SEM, by taking twodimensional case as an example. Suppose that with two independent mesh widths hi and h2, the asymptotic expansion of the bilinear finite element approximation is uhlM{z)-u{z)
= $! {z)h\ + $2(2)ft2 + o (AJ) , z e ahl M,
(9)
Appendix
302
where 2z is a grid point. Refining the mesh width h\, hi, « * ! / * . * » ( * ) - tUi ( zZ)) «*!/*.*»(*)
A? = # ! ( * ) — + *2(z)hl 4
+ 0(hi),
2: 6G f i&h W1/2,h 2 ,2,
(10)
hence, ** li(W «)
= = Similarly, Similarly,
A4 ( ^( «ik^l , h(az( )z ) >--«»»/».»»(z))+0(^), uh*'2>h>(z))+o(ht), ^ ze fel hni .fcl|fca fcj. .
( i i(ii) )
3/if *2(*) *a(*) = — j ( « h l , f c a W - uhlM,2(zj)+0(ht), 3/»l 3/15
z G n fcl|fca .
(12)
Let 3>f (z) (i = 1, 2) be bilinear interpolation functions of $,(2). $i(z). In the SEM, only Mi (i = 1, • • •, 4) are refined grid points obtained after refinement (see Fig. 2). Hence, 4 i ( f 2t )))) *{(Jfl) *{(M 1 ) ==: --( (*$i i(Pi) ( P 1 ) + *i(P 2V 2 1 2 u W 2 p w h l , hh l 2
== _lJ_V( - ^ E ( u"i/2,i( ^(Pi) -- « '(* ^" ( *)))) ,, ha
<> " 1
*£(Mi) =
t=i
™A£(« hi,fca/a (*)-« hi,ka (^))3 Ai2
t = 1
Thus, h2 a U(h(M!) ^ M i ) == ufc./a.i t i f c l ' 2 » f c(Ma) (Mi)
1 2 hl 1/2,) k a + " £ (" f c l / 2 , *»(p*)(^) - - utihl'Mfca(pi)) (P<))
+-E(«" .=i 6
«=i
2 2 kl,ka/2 a +- E D (« (uhlM,2 (Pi)(#) - « tihlfcl'* ' fca (P0). + (pi))=1 3 » i=i
(13)
3
Note that for M 2 G tthl)h2/2, there is a similar extrapolation formula. In other words, for any point in the refined grid ^ ^ ^ U Q ^ ^ , the higher order accuracy can be obtained directly.
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Index
collectively compact convergent 102, 143 collocation method 105, 241 combination technique 260 combination principle 233, 260 compact operator 99, 125, 145 comparison theorem 176 complexity 44, 103, 259, 273 composite grid 291 conjugate Banach space 149 consistency condition 170 continuous kernel 99 coarse grid solution 260 coherence condition 190
A algebraic precision 24, 31, 69, 124 Ascoli-Arzela Theorem 100 asymptotic expansion 2
B Banach-Steinhaus Theorem 104, 145 basis function 3, 64, 145, 163, 203, 261 Bernoulli number 21 Bernoulli polynomial 20, 74 bilinear finite element space 202 bordered equation 258 boundary element method 33 boundary integral equation 118 BPX multilevel method 273 Bramble-Hilbert Lemma 157 Bulirsch-Stoer extrapolation algorithm 12
D degree of parallelism 44, 118, 198, 287 direct sum decomposition 262 Dirichlet problem 170, 204, 247 256, 281 discontinuous kernel 99, 253 discrete maximum principle 172 discrete Green's function 210 double exponential formula 80, 253
c central difference operator 179
313
Index
314
double layer potential theory 247 Duffy transform 83
E energy extrapolation 290 energy norm 261, 268, 287 Euler-Maclaurin asymptotic ex pansion 2, 136 explicit extrapolation 290 extrapolation coefficients 9, 51, 234
F FAC 292 Fibonacci number 14, 50, 75 finite element extrapolation 43 Fredholm integral equation of the second kind 99 fractional power expansion 61 full grid 260, 268 full grid finite element space 268
H hierarchical basis 261 homogeneous decomposition 37 homogeneous function 33, 83 homogeneous splitting extrapo lation Q6 Holder continuous 191
I implicit extrapolation 261, 290 improper integral 61, 79 interpolation polynomial 2, 47, 63, 179 interpolation projection 144 inverse estimate 147, 205 irregular grid 173, 179 isometric partition 169 iterated kernel 101 iterative solution 146
K
G Galerkin method 105, 144 Gauss-Seidel algorithm 273 GC condition 64, Q6 generalized Euler-Maclaurin ex pansion 25 global expansion 290 gradual partition 169 Green's function 162, 210, 244
kernel of the integral operator 99, 256
L Lagrange interpolation formula 3, 47, 173 locally refined grid 290 logarithmic singular 31, 92, 143 Lax-Milgram theorem 202
Index
315
M
0
mean value theorem 5, 29, 163, 246 mesh parameter 6, 44, 117 169, 200, 223, 280, 291 modified quadrature method 141 Monte-Carlo method 69 multidimensional improper inte grals 61, 79 multidimensional weaklysingular integral equation 99 multigrid method 290 multilevel algorithm 260 multilevel splitting 261 multi-parameter asymptotic ex pansion 44 multiprocessor computer 44,135, 160, 198, 280 multivariate asymptotic expan sion 44, 62, 71, 109, 140, 159 multivariate Newton's interpola tion formula 48, 66
order of error 14,112,134,249, 287 orthogonal projection 144, 202
P piecewise continuously differen t i a t e 166 piecewise linear continuous func tion 144 piecewise linear interpolation 161, 243, 269 piecewise strongly-regular parti tion 227 piecewise uniform cuboid parti tion 225 piecewise uniform rectangular 223 polynomial extrapolation 6 point wise convergence 102 product integration method 143 projection method 105
N Neville's interpolation formula 4 Nitsche technique 212, 225 nodal basis 262, 271 nonsmooth kernel 132 nonstationary problem 160 nonuniform mesh 290 number-theoretical method 69 Nystrom's approximation 235
Q quasilinear ordinary differential 160
R rectangular partition 203 rectangular rule 26, 70, 109, 259
Index
316
recurrence algorithm 4, 44, 63, 92 refinement 13 reflexive Banach space 148 Richardson's extrapolation 2 Riemann Zeta function 30, 30 Ritz-Galerkin method 201 Ritz projection 202 Romberg's extrapolation 5
s seminorm 205, 266, 269 simplex 90 Simpson's rule 28, 124, 162, 236, 245 singular boundary value problem 169 single layer potential theory 256 sparse grid 260 sparse grid space 267 sparse grid stiffness matrix 272 stability factor 9, 52 stationary problem 160 stiffness matrix 261 Sturm-Liouville type eigenvalue problem 163 subdivision 201 successive elimination 61 super convergence 43, 200 support 261 symmetric property 58 symmetric quadrature 31
T Toeplitz theorem 13 trapezoidal rule 20 triangular polynomial 24 truncation error extrapolation 290 r-extrapolation 290
u uniformly convergent 101 uniformly bounded 104, 251 uniformly positive definite 201
v Vandemonde determinant 3 F-ellipticity 201
w weak form 201 weak solution 201 weakly-singular kernel 61, 101, 143, 153, 235 weakly-singular integral equation 143, 153, 239