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C lvllx
Izl
by (2.33). The proof is complete.
Proof of (3.~0): We take v C X, and set .f(t) = A ( t )  l v E V. The existence of df(t)/dt follows similarly. Noting At (.f (t), X) = (v, X) , we define j~(t) by
where ~ E V. Then it follows that lim
t~8
.f(t)  . f ( s )  r ts
=0. V
3.1. Generation Theories
107
Inequality (3.40) is a consequence of the stronger one:
Ilvll~,.
(3.46)
In fact we have
IIx ~ < A<(r  r x) = Jlt(f(t), x )  ( A t  A,)(j'(s), x ) + Jl,(f(s), x) = ( f %  Jlt)(f(t), x ) + J l ~ ( f ( s )  f(t), x)  ( A t  As)(r <
x)
C Ilxllv (it  sl ~ IIf(t)llv + Ilf(s)  f(t)llv + It  sl IIr
or
iii(t) ./(~)llv _< c (it ~l ~ lif(,)il~ + ilf(~) f(t)ll~ + it ~l
Its( )ll ).
(3.47)
Remember that
IIf(t)llv <_ C IIv v, follows from
llf(t)ll~
_ .At(f(t), f(t))  (v, f(t)) < c
IlVllw, IIf(t)llw
.
Similarly,
~llJ(s)ll~ ~ A,(r
r
 J~(f(s), r
< c }lf(s)llv IIf(s)llv
gives
Finally, we have
(5lif(s) f(t)ll~
< .At(f(s) f(t), f ( s )  f(t)) = ( A t  As)(f(s), f ( s )  f(t)) <
C It  ~1 IIf(~)ll. IIf(~)  f(t)ll.
and hence
] i f ( s )  f(t)iiv <_ C I t  s I tlf(s)llv <_ C i t  s I livllv, follows. Plugging those inequalities into (3.47), we obtain (3.46). Subject to the above conditions on the differentiability of resolvents, the family of evolution operators {g(t,S)}o<s
u(t, ~) = ~(~~)~(~) + w ( t , ~)
3. Evolution Equations and FEM
108 with
w(t, ~) =
~(*~(*)R(~, ~) d~,
(8.48)
where R = R(t, s) is the solution of
R(t, ~) = R~(t, ~) +
R~(t, T)~(,, ~) d~
(3.49)
for
0 + N0 R l ( t , s ) = (~t
)r
1 O~F e (t s)z 07(zI0  27r,. A(t))ldz,
(3.50)
F being positively oriented boundary of E0+~ for e > 0. Inequality (3.20) and IIg(t, s)ll < c are also derived from this scheme. More precisely, we have
IR(t, s)][ _< C
(0 _< s _< t _< T)
(3.51)
and
(0 _< s < ~ < t <_ T),
(3.52)
for 7 C (0, 1) and 5 C (0, a). They are consequences of those with R replaced by R1,
IIR~(t, s)ll <_ c
(3.5a)
and
II/~l(t, 8 )  /l~l(r, 8)11 ~ C7,5((t T)7(T 8) 7 [ ( t  r ) a ( r  s)~a1), respectively. This theory of generation is particularly remarkable as any assumptions on the domains of A(t) are not made. Consequently, it is no wonder that a little stronger assumption on the smoothness in t of A(t) is imposed.
3.2
A Priori
Estimates
In this section, we show a uniform estimate concerning the evolution operator associated with the discretized problem, which plays a key role in the error analysis. In the following lemma, V C H C V' denotes a triple of Hilbert spaces, A(t) an msectorial operator in X associated with the bilinear form At(, ) on V x V, smooth in t E [0, T] in the sense of the preceding section, and {U(t, S)}o<8
3.2. A Priori Estimates
109
L e m m a 3.2. Each/3 e (0, 1/2) admits the equality
d(t)U(t, s)  A(r)U(r, s) = A(t) [e (t~)A(t)  e (~s)A(t)] + A(t)ZZ~(t, r, s)
(3.54)
with Zz(t, r, s) subject to the estimate IIZ,(t, r, s)ll <
C,(t

r)(r

(3.55)
s) ~',
where O < s < r < t < T. Proof: We shall take the reduction process first. In fact, from the construction we have A(t)U(t, s)  A(r)U(r, s) = A(t) [e  ( t  s ) A ( t ) 
e (rs)A(t)]
+A(t)" [A(t)l'~ (~s)~(')  A(~)''~ (~s)~(~)] +A(t)" [ I  A(t)ZA(r)'] A ( r ) l  ' e (~s)A(~) +A(t) a [ A ( t ) l  ' w ( t , s)  A ( r ) l  ' W ( r , s)] +A(t) ~ [ I  A(t)~A(r) ~] A(r)'~W(r, s), so that 4
/=1
holds with
Z~(t, r, s) = A(t)lZe (rs)A(t)  A(r)lZe (rs)A(~), Z~(t, r, s) =
[I  A(t)ZA(r) ~] A(r)X'e (~s)A(r),
z~(t, ~., s)
A(O'~w(t,
:
Z~(t, r, s) :
s)  A(~.)~~WO ., ~),
[I  A(t)ZA(r) ~] A(r)'~W(r, s).
The Dunford integral gives the expression
d(t)Ze_Tm(t ) _ A(s)e_~d(~ ) = 12m ofr z'Zerz
[ ( z I  d ( t ) )  '  (zI  m(s))'] dz,
while
]l(z~
A ( t ) ) 1  ( z I 
A(s))'ll < c~ It sI 
Izl
follows from (3.41). We have
IlA(t)~e ~A(t)  A(s)~~(s)ll <_ c~ It  sl r  ~ < c
d~~lts l 
~  ~ e ~"~~176 At
G I t  sl r ~~
(3.56)
110
3. Evolution Equations and FEM
for/3 > 1. This implies IIz~(t, ~, s)ll < c~(t  ~)(~  ~)~~
Inequality
III
A(t)ZA(s);311 < C ~ l t  sl
follows from (3.26). We have
~
C / 3 ( t  r ) ( l "  8) ~1
Inequality (3.51) gives IIA(t)lZW(t,s)lt
: <_ c~
A(t)lze(tr)A(t)R(r,s)dr ( t  ~)~'d~ = C ~ ( t 
~)~
and hence llz~( t, ~, ~)11  c , ( t  ~)(~  ~),
follows. The proof of Lemma 3.2 has been reduced to
Ilz~(t, ~, s)l t = IlA(t)l~w(t, s)  A ( r ) '  ~ W ( r , s)l I <_ C,(t  r ) ( r  s ) '  ' Equality (3.48) now gives 7
Z~(t,r,s) = A ( t ) l  Z w ( t , s )  A ( r ) l  Z W ( r , s ) = ~
Z[3(t,r,s )
l=5
with Z~(t, r, s)
=
f
t
A(t)l~e (t~)A(t) [/~(z, s) /~(t, s)] dz,
z~(t, ~, ~) =
[ A ( t ) '  ~ ('z~('~  A ( ~ ) l  ~  ( ~  ~ ( ~ ]
Z~(t, ~, s)
A(t)~~(~z~(~)dz
=
+
9[P(~, .~)  ~(~, ~)1 d~,
. ~(t, s),
[ A ( t ) l  ~  ( ~z~(~  A ( ~ ) I  ~ (~z~(~]
d z  ~ ( ~ , s).
111
3.2. A Priori Estimates In use of (3.52) we have
[IZ~(t,r,s)l[
<
IlA(t)'Ze(tz)A(t)ll .
<_ c~.,
IIR(z,s) R(t,s)ll dz
( t  ~)~~(t ~)~(~ ~)~d~
+G,~
( t  z)~~+~(z s) ~+~~ 9( z  s)~'dz,
< G .
( t  z)~l+~(z ~)~+l~dz. ( ~  s) ~~
=
~)(~ ~)~1
G(t
taking "7 e ( 1  f l , 1) and 5 e (c~  fl, c~). From (3.96) and (3.56)we have
_ G(t
~)(~ z) ~~.
Combining this with
}lA(t),z~I,zl~r
A(~,)'Z~Irz/Ar II
< G(~" ~)~', we get [IA(t)~ze
r

A(r)lZe(rz)m(r)[I < C ~ ( t  r)'~(r z)  ~ + z  '
for r~ E [0, 1]. This implies
IIG( t, ~, ~)ll _<
I[A(t) ~~(~~(~  A(~)'~(~z)~(~)l[ I1~(~, ~ )  ~(z, ~)11 d~
+c~,~
= G(t
( t  ~)~(~  ~)~+~1. (~  ~)~(~ ~)~
,~)~(,~ s)~',
3. Evolution Equations and FEM
112 by taking 7 e ( a  / 3 , 1) and (5 E ( a  / 3 , a) similarly. We have Z~(t, r, $)
=
[A(t)fle(tz)A(t)]~:t r 9R(t, 8) Jr[A(t)fle (tz)A(t)  A ( r )  f l e  ( r  z ) A ( r ) ] : : :
9I~(t, 8)
11
/=8
with
Z~(t, r, s) =
[I  e (t~)m(t)] [A(t)zR(t, s)  A(r)ZR(r, s)],
Z~(t, r, s) =  e (tr)A(t) [A(r) z  A(t) z] R(r, s), zl~~ r, s) =  [A(t)Ze (t~)A(t)  A(t)f~e (~s)a(t)] R(r, s), z ~ l ( t , 1", S)
:
 [A(t)~e (rs)A(t)  A(r)~e (Ts)A(r)] R(r, s).
The inequality
IIA(t)  ~  A(~)~II < G It ~1 follows from (3.26) so that we have IIz~(t,
~, ~)11 <
c~(t 
~)(~  s)/2i' 1 .
On the other hand the inequalities
Ilzy(t, r, ~)ll _< Cz(t ,~)(,~ ~)~' and
IIz~'(t, r, s)ll _< G ( t  r)(~ s)~' follow from (3.96) and (3.56), respectively. In this way, Lemma 3.2 has been reduced to
[[A(t)zR(t, s )  A(r)ZR(r, s)[ I _< Cf3(t r ) ( r  s) zI
(3.57)
Integral equation (3.49) reduces this inequality furthermore to IIA(t)ZR,(t, s )  A(r)ZR,(,', s)l [ <_ C ~ ( t  r ) ( r  s) zl,
(3.58)
R~  Rl(t, s) being the righttrend side of (3.50). We show this fact first. In fact, in use of (3.49) we have m(t)~R(t,
s)  d ( r )  r R ( r ,
s)
= [ A ( t )  O R l ( t , s )  A(r)aR~(r,s)] + +
/'
A(t);3R,(t,z)R(z,s)
dz
[ A ( t )  O R l ( t , z )  A(r)aR~(r,z)] R(z,s) dz.
3.2. A Priori Estimates
113
The second term of the righthand side is estimated by (3.51) and (3.53):
/*
liA(t)Zll
9IIRl(t,
z) lIIR(z,
~)11 dz <
c(t

~).
Similarly, the third term is estimated as
fs r IlA(t)ZRl(t,z)  A(r)~t~l (r, z) C
dz
9 IR(z,s)
( t  r ) a ( r  z)~lds = C~3(t r ) a ( r  s) l~.
Thus, (3.58) implies (3.57). We now turn to the proof of (3.58). Recall the symmetric part Ao(t) of A(t) associated with Ao. We have the following. L e m m a 3.3. The equality
[ 0_~( z l _ A(t))_ 1  ~sO(zl  A(s))l] = A(t)lo/2 ( z I  A0(t)) 1Bz(t, s)Ao(s) 1/2 ( z I  Ao(s)) 1 holds with
Bz(t, s)II ~
c I~ 
(3.59)
sl ~ ,
where z E EOl. Proof: Define A'(t)" V , V ' b y J~t(u, v) = (A' (t)u, v) for u, v E V. It holds that
0 (zI  A(t)) 1 =  ( z I  A(t)) 07
1 d'
(t) (zI  A(t))  1
We have
0 =
(zI(zI q(zI
A(t)) 1
0
A(s))I 1
(zI
A(~)) 1A'(t)(zI 
+ (zI  A(t)) 1
A(t)) 1 [A(/~)  A(s)]

d(f)) 1 [m'(t)

(zI
[A(t)  A(s)]
9d ' ( s )
The equality
m'(s)]

(zI 
A(s)) 1
d(s)) 1
(zI 
A(s)) 1
(zI  A(s)) 1 d'(s)(zI

d(s)) 1
114
3. Evolution Equations and F E M
arises with the following terms" Bl(t, 8)
=
Ao(t) 1/2 ( z I  Ao(t)) ( z I  A(t)) 1Ao(t) 1/2 9A o ( t )  l / 2 A ' ( t ) d o ( t ) 1/2. do(t) 1/2 ( z I  A ( t ) ) ~ do(t) 1/2 9Ao(t) 1/2 [A(t)  A(s)] Ao(t) ~/9
9Ao(8) 1/2 ( z I  A(8)) 1 ( z I  A o ( s ) ) d o ( s ) ~/2, B2(t, s)
=
mo(t) 1/2 ( z I  Ao(t)) ( z I  A ( t ) ) 1 do(t) ~/2 9Ao($) 1/2 [A'(t)  A'(s)] do(s) '/2
A(8)) 1 ( z I 
9Ao(s) ~/2 ( z I Ba(t, s)
=
A o ( s ) ) A o ( s ) 1/2,
Ao(t) 1/2 ( z I  Ao(t)) ( z I  A(t)) 1Ao(t) 1/2 9Ao(t) 1/2 [A(t)  A(s)] Ao(s) 1/2. Ao(s) 1/2 ( z I  A(s)) 1Ao(s) 1/2 9d o ( s )  ~ / e A ' ( s ) d o ( s )  l / 2 ,
m o ( s ) ( z I  A ( s ) ) ~ ( z I  do(s))Ao(s) Vg
Thus the lemma has been reduced to the following inequalities:
IlAo(t)'/~ (zI
 Ao(t)) ( z I  A ( t ) ) ~ Ao(t)l/2ll <_ C,
IlAo(t)l/2A'(t)Ao(t)l/21l
<_ C,
< C It  ~1,
IlAo(t) 1/~ [n(e)  A(~)] no(~)'/=ll
IIAo(t)'/2 (zI  n ( t ) )  ' (~I  Ao(e)) no(e)'/=ll < C,
IlAo(t)
A'(~)] nol/=ll _< C l t  ~ l
~
They are direct consequences of (3.23), (3.25), and the following inequalities:
IA(t)l v,v, ~ C,
C 1 ~
C 1 ~ IIA0(t)Iv, v, < C,
I A'(t)llv, v, <_ C, A(t)  A(s)llv, v, < C I t  sl,
IIA'(t)  A'(.s)l v,v, < C I t  ,s ~. Details are left to tile reader. Now we show the following inequalities as consequences of Lemmas 3.3 and 3.1, where
9 ~ (0,1/2). A ( t ) O~ A ( t ) ~ (9
(zI  A(t))
1
0
 ~
(3.60)
<_ C o ,
( z I  A(s)) 1
A ( t ) o f f   T A ( t )  '  A ( s ) o "ff~A(s) ~
< C It  s] ~ , < CI~ It ,sl "
(3.61) (3.62)
3.2. A Priori Estimates Proof of (3.60):
115
Inequality (3.26) (with a = 1) gives
l]A(t) ~ [A(t + e) ~  A(t) z]
tl <
and hence (3.60) follows because
ate3A(t)_ ~  27rzl/r z~~c3 (zI 
A(t)) 1
dz
exists. P~oof of (J.61): If H is a positive definite selfadjoint operator in X, its spectral decomposition assures
IIH1/~ (zI  H)~I _< c for z E E01. Inequality (3.61)follows from (3.59).
Proof of (3.62):
Noting
cOA(t)Z . A(t)_Z ' A(t)Z at~A(t)~ = cot we have
A(t)~ ffiA(t)~ A(s)~ ff~A(s)~ =  ata a(t)~. A(t)_ ~ + ff__~A(s)~.A(s)_ ~
[~A(t)Z osOA(s)Z] A(t)z + CgA(s)3" A(s)Z [IInequality (3.62) is reduced to
[ff~A(t)~ ~sA(S)~J A(s)~[[ < C~ ts[ ~.
(3.63)
We have assumed the boundedness of A(t) so that (2.18) holds for any v E X. It follows
that
8 A(t)Zv =
sin 7r/3 ]i~176Zc3
7r
1
tt ~ (#I + A(t)) v dl~.
If we apply Lemma 3.3, we get
sin7cflTr/o~176
(Bu(t's)A~
(pI + Ao(s))lv, Ao(t) 1/2 (pI + Ao(t))lx).
3. Evolution Equations and FEM
116 Inequality (3.63) follows from
( OA(t)~~
~sOA(s)~l C
(fOOO#Z3 [Ao(8)1/2
( # I t
A0(8))1 '/)112)1/2
IlAo(t) ~/~ ( . I + Ao(t)) ~ ~[l~ d.
:
It 
sl ~
c~ IIA0(~)~llx. I~llx" t  ~1~
and (3.25). We are able to complete the proof of Lemma 3.2.
Proof of (3.58): We have Oq e_(t_s)A(t) A(t)ZR~ (t, s) =  A ( t )  z  ~0qe (t~)A(t)  A(t)Z~s
=
O ~
_
(A(t)_ze_(t_s)a(t) ) + A ( t ) l  . e_(t_~)a( t)
0__ A + ot ( t )  "
=
e
(ts)A(t)
1 Ot 0 f_ 2m r zZe1
27r~
(ts)z
(zI  A(t)) 1 dz
Jfr Z 1~e(t~)z (zI

A(t)) 1 dz + ~ 0 A(t)_ ~ 9e_(t_s)A(t)
1 f~ zZe (t~)zO (zI  A(t))  l dz + o__ Ot A (t)z . e(t~)A(t) 2m Ot Therefore, the equality =
16 A(t)~Rl(t's)  A(r)~Rl(r's) = E Ze(t'r's)
g=12
holds with
Z12(t, r, 8) =
1 ofr z~e (~~)z [e 
zla(t, r, z) 
1 Jfr 27rz
~e(~~)z [0oi (z:  A(t)) 1  ~0 (ZI A(t))lJ d~,
Z14(t,r,z)
=
Z15(t, r, Z)
:
Z'6(t, r, s)
=
0t0A(t)_ ~ " A(t);~ ~r0 A(r)_ ~ . A(r)f3] A(t)_~e_(t_s)A(t) ~O
A(r)_ ~ A(r);' [A(t)Ze (ts)A(t) A(t)Zc("s)A(t)],
O A ( r )  " . A(r) ~ [A(t)/~e (rs)A(t)  A(r)~ (rs)A(r)] Or
3.2. A Priori Estimates
117
Here, we have ~
,•'0
<
C
#  z e (rs)'c~176 9( t  r)# d# #

C/3(t
1")(/
<
C
p  Z e (rs)ac~176 . (t  r ) a d #
8)/51
and
IJZ'a(t,
r, s)
j~0~176 c~(t ,)~(r s)~'
:
Taking the adjoint operators in (3.62), we have
cot (t)Z " A ( t ) z
A ( s )  Z " A(s)Z
< Cz
It
 sl
This implies
]lz'4(t, ~, s)JI ~
A ( t )  ~ . A ( t ) ~  ~rA(r) ~ 9A ( r ) ~
IIA(t)~ll. 9 II~('~)A(~)ll s) ~.
_< G(tSimilarly, we have
}I~ at
(t)~ " A ( t ) z

and hence
9
[[A(t)l~e(ts)A< t) _ A(t)Ze(rs)Att)[[
C/~(t r)(? ~  8) ~1 hdds by (2.19). Finally, inequality (3.56) is valid for/3 >  1 and therefore
[IA(t)Ze(rs)A(t) _ A(r)~e(~sla(r)ll <
c~(t
~)(,~
~)~'
follows. The proof is complete. Modifying the proof of Lemma 3.2 slightly, we have (3.54) with/3 = 0 and Zo(t, r, s) satisfying IIZ0(t, ~,
s)ll
<_
c ~ ( t  r)~(r  s) ~
for 0 < ~ < a. Combining this with (2.20), we get IIA(t)U(t, s)  A ( r ) U ( r , s)ll < G ( t
 r)~(r  s)  ~  1 + C~(t  r)~(r  s) ~
(3.64)
for 0 _< s < r < t _< T, where/3 E [0, 1] and ~c E [0, a). The proof is simpler and left to the reader.
3. Evolution Equations and F E M
118 3.3
Semidiscretization
As is described in w parabolic equation (3.1) with (3.3) (or (3.4)) and (3.2) is reduced to evolution equation (3.5) with (3.6). In the same way as in the preceding chapter, this equation is discretized with respect to the space variables x = (xl,x2). We triangulate f~ into small elements with the size parameter h > 0 and denote by Vh C V the space of piecewise linear trial functions. As before, Xh denotes Vh equipped with the L 2 topology. The operator norm II " Ilxh,x, is written as I1" II for simplicity. The msectorial operator in Xh associated with r h is denoted by Ah(t). Finally, Ph : X ~ Xh is the orthogonal projection. Then, the semidiscrete finite element approximation of (3.5) with (3.6) is given by duh + dt
Ah(t)Uh = 0
(0 < t < T)
(3.65)
with Uh(0) =
Phu0
in Xh. Because dim Xh < +oo and Ah(t)'s are smooth in t, they generate a family of evolution operators {Uh(t,S)}o<s
] Ah(t)Uh(t, S)ll + IlUh(t, S)Ah(S)ll <_ C ( t  s) 1 and
(3.66)
IIuh(t,~) l < c
hold uniformly in h for 0 < s < t < T. Here, the theories FujieTanabe and KatoSobolevskii are applicable. The theory of KatoTanabe works also from the argument in w We turn to the error estimate for scheme (3.65). Employing the method of Helfrich, we show
Ileh(t)l x < Ch'2t' '~011x
(0 < t < T)
for eh(t) = u ( t )  uh(t), and extend the similar result of w introduced the error operator Eh = Eh(t, s) by
(3.67) For this purpose we
G ( t , ~) = u(t, s)  Uh(t, s)P,~. Obviously,
~h(t) = E,.(t. 0)..0 holds and inequality (3.67) is reduced to
IIG(t, s)ll < c h 2 ( t  s)'
(o < s < t <_ T ) .
(3.68)
3.3. Semidiscretization Calculations of w equality
119 are valid even for temporally inhomogeneous generators. From the
O [Uh(t, r)PhEh(r, s)] = Uh(t r)[Ah(r)Ph  PhA(r)] U(r, s) Or follows
PhEh(t, s) =
~st Uh(t, r)[Ah(r)Ph
 PhA(r)] U(r, s) dr.
Introduce the Ritz operator Rh(t)" V ~ Vh through the relation (~r
x;, ~ v.).
(3.69)
The equality
Ah(t)Rh(t)v = PhA(t)v
(v E D (A(t)) C Vh)
holds similarly to (1.24). We have
Eh(t, s)
= =
[I  Ph] Eh(t, s) + PhEh(t, s) E~(t, ~) + EX(t, ~) + e~(t, ~)
with
E~(t, s)
=
E~(t, s)
=
E3(t, s)
=
/t
[I  Uh(t, S)Ph] [I  Rh(t)] U(t, s), Uh(t, r)A~(r) [Rh(t)  Rh(r)] U(t, s) dr, Uh(t, r)Ah(r)Ph [I  Rh(r)] [U(r, s)  U(t, s)] dr.
It sumces to show that
IIE~( t, ~)ll < c h ~ ( E s t i m a t e of
t 
8)1
( ( = 1,2,3).
E~(t, s)
We have shown inequalities (1.38) and (1.42) for v E V N H2(~):
II[Rh(t)  I] ~llv
<
II[Rh(t)  I] ~llx
<
Ch Ilvl ~.(~),
Ch~ll'~ll_,,~(~).
Therefore, from the elliptic estimate follows that
[IE~(t,s)ll
~_ (I + l Uh(t,s)l I . IIPhll) . l[[I Rh(t)]A(t)lll . llA(t)U(t,s)l I ~_ C h 2 ( t  s) 1
(3.70) (3.71)
3. Evolution Equations and FEM
120
Estimate of E~(t, s) We shall show the inequality
(3.72)
ll[Rh(t)/~h(S)]Vllx <Ch 2 It sl ~ IvllH~(a) for v E V N H2(f~). Then it follows that IIE~(t,s)ll
_
IIUh(t,r)Ah(r)ll
tl[Rh(t)  R~(~)] A(t)' l[" IIA(t)U(t, s)ll d~ <
c
( t  s)l+~h~d~. ( t 
= Ch2(t
~)~
Ch2(t s) 1. To show (3.72), we recall the adjoint form .A~('u, v) = .At(v, u), and denote by/~h(t) the 8)1+ c~ ~
Ritz operator associated with it:
.,4; (,Oh(t)~, ~:h) : A;(v, ~:h)
(,, ~ v, ~,, c v,,).
The inequalities
ll[/~h(t)/]vlIv~ ChlvllH~(~)
and
I][~h(t) /]vllx ~ Ch21v IH~(nl hold similarly to (3.70) and (3.71), respectively, where v C V N H2(~2). Setting z = [Rh(t) Rh(s)] v E Vh, we have
llzll
= A, (z,
A, (z,
=
(At  As) ([1  Rh(s)] v, [~h(t)A(t)*lz)
=
( A t  Ms)([1 Rh(S)] V, [/~h(t) I] A(t)*lz) + ( A ~  A s ) ( [ I  R/~(s)] v, A(t)*lz)
= ( A t  A s ) ( [ I  Rh(s)]v, [/~h(s) I] A(t)*lz) + As ( [ I  Rh(s)] v, [A(s) *  1  A(t) *1] z)
:
(,At ,As)([I Rh(8)] "u, [il~h(8)  I]
A(t)'lz)
+ .As ([I  Rh(s)] v, [I  Rh(s)] [A(8) *1  A(t) *1] z)
_<
+ c I I [ /  Rh(s)] vllv I[[1  ~h(S)] [A(s) *  ]  A(t) *]] 6' It  sl ~ h 2 Ilvl g=(~) IA(t) *'zl ,,=(.) + Ch 2 Ilvlln=(~) l] [A(S)*I  A(t)*l] ~11.~(~)
zll~
3.3. Semidiscretization
121
The elliptic estimate implies
IIA(t)*x~llH~(~) ~ c Ilzll~ and
II [A(s) *1 
A(t) *1] ZlIH2(a) < C It  sl ~ Ilzllx.
Inequality (3.72) has been proven.
E s t i m a t e of E3(t, s)
First Case
If V = H~(f~), the duality argument of Helfrich is applicable and inequality (3.22) follows. For Uo E D ( A ( s ) ) =_ D we have
jfst IIUh(t, r)Ah(r)ll.
SO ollx
I11I /~h(r)] A(t)l[[
9IlA(t)[u(t, s)  U(r, s)] A(s)lll 9IlA(s)uollx ds <_ Co
=
(t  s)lhg(t  r)~
 s)~
IIA(s)u01 x
Ch 2 IIA(s)uollx
with 0 E (0, c~). We can show
for g = 1, 2 similarly, from the second estimate of (3.21). (Details are left to the reader.) We have
IIEh(t, s)A(s)~ll < Ch 2.
(3.73)
Now, the semigroup property of evolution operators implies
Uh(t, r)Uh(r, s) = Uh(t, s)
and
U(t, r)U(r, s) = U(t, s)
for 0 _< s _< r < t _< T. The identity (3.74)
Eh(t, s) = Uh(t, so)PhEh(So, s) + Eh(t, so)U(so, s) follows with so = (t + s)/2. The second term of the righthand side of (3.74) is estimated as
IIE~(t,~o)U(so, s)ll
<_ IJE~(t, so)A(~o)~ll'llA(so)U(so, <
C h 2 ( t  s) 1
s)ll
3. Evolution Equations and F E M
122 by (3.73). On the other hand, (3.69) gives [[Uh(t,S)PhEh(So, S)II =
[IU~(t, so)Ah(So)Rh(So)A(so)XEh(So, s)ll
< [Iuh(t, so)Ah(So)Ph [Rh(S0)  I]A(so)lEh(So, S)] + ]lUh(t, so)Ah(so)PhA(So)IEh(So, s)ll C(t s) 1 {] [/~h(SO) I] A(8o)1 I 9IIE,~(~o, s)ll +llA(so)lEh(so, s)[I}. Because IIEh(So, s)ll ~ c follows from (3.66), inequality (3.68) is a consequence of
]A(t)lEh(t,s)l
<_ Ch 2.
(3.75)
To prove this inequality, let
U(t, s) = U ( T 
s, T 
t)*
(Ih(t, s) = Uh(T  s, T  t)*.
and
Then,
{U(t,S)}o<s
{U~(t,S)}o<s
and
are nothing but the families of evolution operators generated by A(t)  A ( T t)* in X and Xh respectively. The relation
Ah(t) = A h ( T 
l~h(t, s) = l~l(t, s)  [Ih(t, S)Ph = E h ( T  s, T  t)* follows. Because
I1E,,/,sIAl,/ 11 is proven similarly to (3.73), we get (3.75) as
I[A(t)lEh(t,s)[I E s t i m a t e of
Eta(t, s)
= ]lEh(t,s)*A(s)*l][
<_ Ch 2.
Second Case
If V = H I ( ~ ) , D ( A ( t ) ) varies as t changes. We cannot expect the estimate
IIA(t)U( t, s)A(s) 1]1 < C or IIA(t)[U(t, s)  U(r, s)]A(s) 1 II < Co(t  r)~ In this case, we apply the method of telescoping as
5
(t ~)E,~(t, ~) : ~ F~(t, s) g=l
 s) ~
t)* and
123
3.3. Semidiscretization
with the following terms:
fs t (r  s)Uh(t, r ) A h ( r ) . Ph [I  Rh(r)] [U(r, s)  U(t, s)] dr,
F~(t, s)
=
F~(t, s)
=
(t  r)Uh(t, r ) A h ( r ) " [Rh(s)  R~(r)] [U(r, s)  U(t, s)] dr,
F~(t, s)
=
(t  r)[Uh(t, s ) A h ( s )  Uh(t, r)Ah(r)]
9Ph [I  Rh(s)] [U(r, s)  U(t, s)] dr,
f'
(r  s)[U(r, s)  U(t, s)] dr,
F2(t, s)
=
Uh(t, s ) A h ( S ) " Ph [I  Rh(S)]
F2(t, s)
=
Uh(t, s ) A h ( s ) . Ph [I  Rh(S)] (t  s)
[U(r, s)  U(t, s)] dr.
We have to show that [IFeh(t, s)l I _< C h 2
(g = 1, 2 , . . . , 5).
(3.76)
Recall that KatoTanabe's generation theory is applicable to (3.5). Combined this with the elliptic estimate we can show the following for 0 _< s < r < t _< T, where 0 _
IlU(t, 8)  U(r, 8)IlL2(FI),H2(~2) ~_ C k { ( t  T)~(T  8) ~1 1 ( t  T)~(T  8)1} , IlUh(t,r)Ah(r)  U , ( t , s ) A ~ ( s ) l l
<_ Ck { ( t  r )  ~  Z ( r  s) ~ + ( t 
/s'
[U(t, s)  U(r, s)] dr
I[
L2(f~),H2(f~)
(3.77)
r )  l ( r  8)a},
(3.78) (3.79)
< C.
Those inequalities are proven in the following way. Proof of (3. 77):
Take u0 E L2(f~) and set u(, t) = U(t, S)Uo. We have B(x,t,D)u(x,t)
=0
on 0f~ where 13 = O/Ovc + ~. This implies B ( x , t, D) (u(x, t)  u(x, r)) : [B(x, t, D)  B ( x , r, D)] u(x, r)
there. Similarly, we have
[A(t)U(t, s)  A(r)U(r,
s)] uo(x)
+ [s
r, D)  s
t,
D)] u(x,
r)
124
3. Evolution Equations and F E M
in f~. The elliptic estimate gives
II[U(t, ~)  u(~, ~)1 ~olIH~(~)  I1~(', t)  ~(., ~)11,~(~) <_ C I~(', t, D) (u(., t)  u(, ~))11~(~) +C ILL(', t, D) (u(., t)  u(., ~))11,,,~(o~) <_ C I I [ A ( t ) U ( t , ~)  A(,~)U(r, ~)] ~o11~(~) +C I [L(', ~, D)  s t, D)] u(., ~)11~(~) +C lilt3(', ~, D)  B(, t, D)] u(., ~) I1,,,,~(o~) <_ C I I A ( t ) U ( t , ~)  A ( r ) U ( , ' , ~)11" luoll~(~) + C ( t  ,~) I1~(', ~)11~(~) <_ C IIA(t)U(t, ~)  A(,~)U(, , ~)11 I1'~o11~(~) + C ( t  ,~)(~ ~)' I1~o11~(~) Inequality (3.64)implies (3.77).
[]
Proof of (3. 78)" This is nothing but the adjoint form of (3.64) applied to A(t) = Ah(t) and U(t, s) = Uh(t, s). [] Proof of (3. 79): Letting u(., t) = U(t, S)Uo, we have s
s, D) (u(x, t)  u(r, x)) = [s
+ [s
x, D)  E(s, x,
s, D)  s
t, D)] u(z, t)
D)] u(z, r) + [A(t)U(t, s)  A(r)U(r, s)] to(Z)
in t2 and B(x, s, D) (u(x, t)  u(x, r)) : [B(x, s, D)  B(x, t, D)] u(x, t) + [B(r r, D)  B(x, s, D)] u(r r)
on cgf~. The elliptic estimate now gives [U(t, s)  U(r, s)] no dr
=
('u(, t)  u(., r ) ) d r
H2(U)
<_ C(t  s)II[E(., s, D)  s
+C
f'
I[s
r, D)  s
H2(~)
t, D)] u(.,
t)JlL~(~)
s, D)] u(, r)llL2(U ) dr
+C(t  s)I [~(., s, D)  B(., t, D)] u(, t)ll,,/~(on) +C
[A(t)U(t, s)  A(r)U(r, s)] uo dr
II
L'~(n)
<
C ( t  s) 2
Ji'u(, t)llu~(n) + C
(r s)tl'u(,
r)liu~(~)
dr
~)11 ~o11~(~) + C IIU(t, ,~)  I I. I1~,o11~(~) <_ C ( t  ~) I uoll~,(~) + C I uoll~(~) ___C I1,,,o I~(,) 9 +c(t
 ~)IIA(t)U(t,
3.3. Semidiscretization
125
The proof is complete. Now the proof of (3.76) is given as follows. First, with/3 E (0, 1) we have
IIF ( t, )ll < <
/s
 s)Ilgh(t,
9I I I  R~(~)ll.~(~).~(~)"
C~h 2
IIU(~. ~)  u ( t . ~)l ~ ( ~ ) , ~ ( ~ ) dr
8 ) ( t  T) 1
(r9{ ( t 
<
r)&(r)l
T)~(T  8) ~1 + ( t 
T)~(r  s) 1 } dr
Ch 2
and
jfs t (t  r)IIUh(t, r)Ah(r)ll
~
IIFZ(t, s)ll
U(t, ~)llL=(~),H~(~)dr
9IIRh(~)  Rh(~)l H=(~),L=(~)" IIU0", ~0 
_< c ~ h ~ fS t (~  ~) {(t  ~)~(~  ~)~~ + (t  ~)~(~  ~)1} a~ <
C h 2.
Next, with 0 < "i' < ~ _
_<
r) l U h ( t , r ) A h ( r )  Uh(t,s)Ah(s)l]
(t
9III 
R"(S) IIH=(~),L=(~)"IIU(t,
~)  U(~', ~) IIL=(~),H=(~) d,"
( t  T) { ( t  r )  l  ~ ( r  8) ~ Jr ( t  r )  l ( r 
C~h 2
8) ~}
{ (t  T)7(r  8) 13' Jr (t  T)m(r  S) 1 } dr <
C h 2.
W i t h / 3 c (0, 1), we have
I[F2(t, )ll
Ilun(t,
s)&(s)ll.
9
<_
<
III
Rh(S)IIH=(~),L~(~)
IIg(t, s)  U(r, S)[IL2(a),H:(a)" (r  s) dr
C~(t s)~h 2 9
{ (t
~)~(~  ~ )  ~  1 + ( t 
_<
IUh(t,s)&(s)
~)~(~
~ )  1 } (~ _ ~) d~
C h 2.
Finally, we have [IF2( t, s)ll
9 <
C h 2.
9liRa(s)  IllH~(~),L~(~)
[u(~, ~) 
u(t,
~)] d~
L2(~"/),H2(n)
3. Evolution Equations and F E M
126 Summing up those estimates, we obtain
IlE2(t,~)ll <_ ch~(t_ ~), All the proof for inequality (3.68) is complete.
3.4
Fulldiscretization
Fully discrete approximations are obtained by discretizing the semidiscrete equation
duh dt
+ Ah(t)Uh = 0
(0 < t <_ T)
with
Uh(O) = Phuo
(3.80)
in the time variable t. In the present section, we adopt the backward difference method with the mesh length r > 0 satisfying T = N r . T h a t is,
U~h(t + r)  U~h(t)
+ Ah(t + r)u,~(t + r) = 0
(t = n r )
(3.81)
in Xh with
~,~(o) = P~,o, where n = 0, 1,. , N. Thanks to (3.14), tile scheme is uniquely solvable and
e~h(t) = u h ( t )  U,;(t) denotes the error, where t = nr. we shall derive
Id,(t)llx _< c~t' luol x and extend a result in w
(3.82)
Combining (3.82) with (3.67), we obtain
u(t)  u;(t)l x < C (h? + r) t~ I 'uo Ix
(t = n r ) .
(3.83)
Let t,~ = nT and set
U[~ (tn, tj) =
{ ( / h + TAh(tn)) 1 (Ih + TAh(tn1)) 1"'" (Ih + TAh(tj+l)) 1 Ih.
We have ~,;(t) = ui:(t, o)&~o
(t = t,,)
and
,,,,, ( t ) = u,, ( t, 0)P,,,~,o.
(~ > j) ( ~ = j).
3.4. Fulldiscretization
127
Drop the suffix h for simplicity of writing. Because of (3.80) and (3.81), we have for t = tn that
e ~(t + r)  e r(t)
tntr
f f
=
[A(t + r)u r(t + r)  A(r)u(r)] dr
at
=
t+r
[A(t + r)u(t + r ) 
A(r)u(r)] d r  r A ( t + r)e" (t + r).
,It
Hence
er(t + 7) = (1 + r A ( t + "r) )l er (t) + (1 + r A ( t
+
T)) 1
f
t+r
[A(t + r)u(t + r ) 
A(r)u(r)] dr.
Jt
Because of e ~ (0) = 0 we obtain
e ~(tn)
 E ~(tn)Puo =
(I q 7A(tn)) 1 (I + "rA(tn_l))  1 . . . (I + TA(tk)) 1 k1
1
9[A(tk)U(tk, O)  A(r)U(r, 0)] drPuo,
(a.s4)
Er(t) being the error operator: E ,(t) = u(t, o)P  u ~(t, o ) Inequality (3.82) is reduced to those on the stability of the approximate operator U r (tn, tj+l) and the smoothness of the original one A(t)U(t, s)  A(r)U(r, s). We have proven Lemma 3.2 for the latter. As for the former, we have the following lemma, proven similarly to the continuous case: L e m m a 3.4. For each/3 E [0, 4/3), the inequality
II
tj)A(tj+l)Zll <__Cz(tn  tj) ~
(a.85)
holds true. Proof." Taking adjoint form reduces inequality (3.85) to I A(tn)ZU~(tn, tj)l] <__Cz(tn

tj) ~,
of which continuous version is proven in w
IIA(t)ZU(t, s)ll<_ Cz(t s) z All we have to do is to trace computations
in the context of discreteness.
(3.86)
3. Evolution Equations and F E M
128
In fact, inequality (3.26) holds for A(t) = Ah(t) uniformly in h as is described there. Identity (3.30) holds for A(t) = Ah(t) so that we obtain
U'(tn, tj)  (1 + rA(tj)) (nj) =
~
[(1 + ~A(t~))(~k)U'(tk, t j )  (1 + 7A(t~))(nk+l)U'(tk_x,tj)]
k=j+l
=
~
(1 +7A(tn))(~k+l) [(1 +~A(tn)) (1 +~A(tk))]U'(tk, ty)
k:j+l
= T ~ (1 + rA(tn)) ('~k+~) [A(tn)  A(tk)] U~(tk,tj) k=j+l = ~p=l
(1 + 7A(tn))(nk+UA(tn)lPPD(tn, tk)A(tk)PPVr(tk,tj).
~
(3.87)
k=j+l
Given operatorvalued functions fit = Ke(tn, tj) with t~ = 1, 2 on
D r={(t~,tj) I N>_n>_j>_O}, we define K = K1 .r /42 by n1
(K1 *~ K2) (tn, tj) = ~ Z
Kl(tn, tk)K2(tk, tj).
k=j+l
Furthermore, we set
WT(t~,tj) Y~(rn, tj)
= Ur  (1 + TA(t~)) (~j) = A(t~)qPWT(t~,tj).
(3.88) (3.89)
Then, equality (3.87) reads: m
Yq = E
H q~p , ~ U + Yq,~O,
(3.90)
p=l
where
Hq~p(t~, tj)
= A(tn)IpP+qP(1 + ~A(t~))(~J+l)D(t~, tj),
(3.91)
m ,0
=
,p
, 1
(3.92)
p=l
with
Yp~_l(t,,,tj) = A(tn)PP(1 + TA(tn)) (''j)
(3.93)
Note the elementary inequality (3.94). The desired inequality (3.86) can be derived in a similar way to that in w
3.4. Fulldiscretization
129
Details are left to the reader, but the following inequality is worth mentioning, where
O < a <wb a n d a < l :
B N(a, b)  ~1 k=l
1
 ~
< S(a, b) =
/o (1


x)alxbldx.
(3.94)
In fact, f(x) = (1  x)alx b1 is monotonically increasing in [0, 1) when b _> 1 _> a, while f(x) is convex in (0, 1) when a, b _< 1. Those facts imply (3.94). [] We are able to give the following.
Proof of (3.82):
The operator E ~ in the righthand side of (3.84) splits as
n~k E~(tn) =
(1 + ~A(tn)) 1... (1 + TA(tk))lA(tk)[e tkA(tk) e rA(tk)] dr
E k=l
1
(1 + rA(tn)) 1... (1 + rA(tk))lA(tk)~Z~(tk, r, O) dr
+ k=l
=
1
(1 + TA(tn))lA(tn)
[etnA(t~)  e rA(t~)] dr 1
+
Ur(tn, tk_l)A(tk)[e tkA(tk) e ra(tk)] dr k=l
1
~
+~
k=l
U'(tn, tk_l)A(tk)ZZz(tk, r, O) dr.
(3.95)
1
Inequality (2.19) is applied to A = Ah(r) uniformly in h. We have for n >  1 that
[IA(r)~[e tA(~) ~~A<~>]II C (t s)s nl,
(3.96)
where 0 < s _< t < oc. Supposing n > 2, we can estimate the first term of the righthand side of (3.95) as
[1(1 + TA(t~))lA(tn)ll
~
n
lie t~A(t~)  e~A(t~)ll dr 1
CT
( t n  r)r~dr <_ CT1T2(tn1
1 1
~ Crt
3. Evolution Equations and F E M
130
Next, by Lemma 3.2, the third term of the righthand side of (3.95) is estimated as
k~l
Ilg~(t~,t~l)A(t~)z[I 9IIZz(tk, r,O)ll dr 1
< C a k1= ~
, (tn  tkl)Z(tk  r ) r  l + Z d r
n
___ c~ ~ ( ~ 
k + 1 )  ~  ~ ( k ~ ) ~'
k=l n = c~
Z(~
 k + 1)~k ~1
k=l Inequality (3.94) now gives that
~~ IIg'(t~,
tk_l)A(tk)Zll 9]lZz(tk, r, O)l I dr <_ CT. k=l The desired inequality (3.82) has been reduced to
fi'
,11
To see this, take/3 C (0, 1/3). We get
n En1 o~tl k U~(tn, tk_l)A(tk) k=l
[e  t k a ( t k )
 e ~A(tk)] dr
' n1( n = E k + 1)UT(t~, tk_l)A(tk) 1+'. ~tlk A ( t k )  ' [e tkA(tk) e rA(tk)] dr k=l 1 n1
+ E(k
 1)UT(tn, tk_l)A(tk) '~
k=2 9
A(tk) ~ [e tkA(tk)  e rA(tk)] dr.
(3.98)
1
The first term of the righthand side of (3.98) is estimated by Lemma 3.4 and (3.96) as
n1 ~~.(nk+ 1) llUT(t,~,tk_~)A(t~)'+Zll. ~t.lk IlA(tk) ~ [et~A(t~)e~A(t~)]l I dr k=l
1
_< Cr Z ( r t  k + 1)'T  1  ' ~t.lk (tk  r)r'idr k=l 1 n1
rt1
< C/3 Z ( ~ 9 , k=l
]~:J I)~qT 1/~ 9T2(~T)/91
rt
:
Ca E ( n k=l
If + 1 )  ' k '1 _< C.
131
3.5. A l t e r n a t i v e A p p r o a c h
Similarly, the second term of the righthand side of (3.98) is estimated as
n1 E(kk=2
1)IIU~(t~,tk_l)A(tk)l~ll.
/i k
IIA(t,)~
[~~(~) ~~(t~)] II d~
1 n1
)1+/~
_< C 0
( k  1)(tn  t k  ,
k=2 rz1 _< c , ~ ( k k=2 n1 < Cr E ( k k=2
/~1 9
(tk  r ) r
dr
1
 1 ) ( ~  k + 1) 1+~ 9~1+~. ~ . ((k  1)~) ~~
 1)r
k + 1) r
< C.
The proof is complete.
3.5
Alternative
Approach
Inequality (3.83) is proven in a different way by the energy method. In this section, we describe the arguments of M. Luskin and R. Rannacher for the semidiscrete approximation and M.Y. Huang and V. Thom~e for the fulldiscrete approximation, respectively. A Priori Estimates
First, we show some estimates concerning the solution of the equation du d7 + A ( t ) u = f (t)
with
(0 < t < T )
u(0) = ,to
(3.99)
in a Hilbert space X over R. Here, A ( t ) is an msectorial operator associated with a bilinear form At( , ) on V x V through a triple of Hilbert spaces V C X C V*. We suppose
I.,4t(,< ~)1
<
C1 Ilu Iv IIv Iv
and
.a~(~, ~) > 5 II~ll~
(3.100)
for u, v E V, where C 1 > 0 and 5 > 0 are constants. It is also supposed to be smooth so that oq A (u, v)
and
02
exist and satisfy
IAt(,
and
A~(,~, ~)1 ~
c
I1~11~Ilvllv
132
3. Evolution Equations and F E M
for u, v C V. Problem (3.99) admits the weak form (Ut, ?J) + .At(u , v) ~ (f(t), v)
(3.101)
for v C V, where ( , ) denotes tile inner product in X. This implies (Utt , 72) nt .At(ttt, v)   A t ( u ,
v) nt (It(t), v).
(3.102)
/0
(3.103)
We have the following. P r o p o s i t i o n 3.5. The inequality
Ilu(t)ll 2x +
/0
Ilu(s)llv~ ds _< C I1~oll ~x + c
IIf(s)ll~.ds
hold, fo~ th~ ,ol~t~o~ ~ = ~(t) of (S.99).
Proof: Putting v = u(t) in (3.101), we h~ve ld
2 dt Ilu(t)ll~ + 5 Ilu(t)ll~ _< Ilf(t)llv. Ilu(t)llv by (3.100). Then, Schwarz's inequality gives

and
d dt Ilu(t)ll2x + l u(t)llv2 < C Ilf(t)l 2v. 
_
inequality (3.103) follows.
[]
Letting .A;(u, v) = .At(v, u) and A ~ = (.At + .At)/2, we suppose that
v) + (B(t)u, v)
.At(u, v) = A~
(3.104)
holds with a bounded linear operator B(t) : V ~ X. This assumption is satisfied if .At(, ) is associated with a secondorder elliptic operator with real coefficients. We have the following. Proposition
3.6. The inequality
ilu(t)ll 2v +
/0
Ilu~(s)llx2 ds _< C Iluoll 2v + C
/0
f(s)l
x
ds
holds for the solution u = u(t) of (3.99). Proof: Putting v = ut in (3.101), we have
Ilu~ I~ + A~(u, u~)
= (f,
Ut)
< 
1 2 1 ~ II/11X + 2 Ilu~ I~
"
On the other hand, equality (3.104) gives
1d
1
.At(U, Ut) = ~~.At(U, U)  = A t ( U , U) nrZ
1.B.t.u,( ( )
Ut)
Q
(3.105)
133
3.5. Alternative Approach
Those relations imply lutll x~ + ~dA , (u, u ) <_ C I/ll ~x + c I1~ Iv. Then, inequality (3.1o5) follows from (3.100) and (3.103). P r o p o s i t i o n 3.7. The inequality Ilu,(t)l x~ +
/o
I~,(s)llv~ ds _< C Ilu~(0)ll x2 + C l'~o I.~
(If(s)I~,. + II/~(~)11~v*) ds
+ C
(3.106)
holds for the solution u = u(t) of (3.99). Proof." Putting v = ut in (3.102), we have
ld 2 dt I1~'11~ + 61~,11~, < <
c II~llv I1~ Iv + f~llv. I~1 v 1
~11~ ~v + C' I1~ Iv~ + C IIf~llv* 9 2
This implies
Ilu~(t)ll ~x +
/o
Ilu~(sDllv~ds_< II~,(o)ll~ + c'
/o
II~(~)ll ~vds +
c
/o
II/t(s)ll~.
ds.
Then, inequality (3.106) follows from (3.103). P r o p o s i t i o n 3.8. The inequality
/0
s ~ Ilu,(s)ll~, ds _< C Iluoll.~ + c
ji
(ll/(s)[Iv. + II/~(s)ll 2v.) ds
holds for the solution u = u(t) of (3.99). Proof: Putting v = t2ut in (3.102), we have
ld 2 dt (t2 I]utl]~.) + 5t 2 Ilu, ll ~v <  ? A , ( ~ , u,) + t I1~11x~ + t2(ft, ut) 

v + c II~llv + t II~ll~ + ~t ~ II~,ll~, + c II/~ll 2V. 9
This implies d~ (t2 I[ut[ ~) +
I[ut][y _
g + t II~,llx + c IIf~ll~.
(3.107)
3. Evolution Equations and FEM
134 and the inequality
/o
s ~ Ilut(s)ll*v ds _< C
/o t (ll~(s)ll~
+ s lut(s)12x + lift(s)Iv*) ds
(3.108)
follows. Next, putting v = tut in (3.101), we have
t i1~,11~x+_did (tAt(u,
u))

At(u, u) t .At(u, u) t At(tut, u) t (f, tut)
<
~t ~ll~t ~v + c~ II~llv + c~ IIf I~.
for c > 0. The inequality
jot
II~t(~)ll ~ d ~ <_~
/~t
~ II~t(~)llv~ d ~ + C ~
jot (11~(~)11'v +
If(~)ll~.)d~
follows. Taking e > 0 small enough and combining this with (3.108), we have
/o s21lut(s)ll~ds c /o { Ilu(s)ll~ + II/(s)I~. + IIf,(s)I~. } ds. _<
Then, inequality (3.107) follows from (3.103). Proposition t llu,(t)ll~ +
3.9. The inequality
i
X
t
II~.(s)ll ~ ds <
c ilu,(o)ll x~ + c II~ollx~ + c
/o'
(llf~(~)llx~ + IIf(~)llv.) d~ (3.109)
holds for the solution u = u(t) of (3.99). Proof." Putting v
~ Utt
in (3.102), we have
IlUttll 2X Jr,At (ut, utt) : At (u, utt) + (ft, tttt).
(3.110)
Here, note that
12ddt.At(ut, ut) = <  At (.tz, tttt)
.At (ut, uu) + ~1 At (ut, ut) + 1 (But,
r
?ztt) + C I1..,11~
1
Utt)
2
d 9 ~  diAt (11., ?zt) ~ At (.~z,"~l.t) + At (ut, ut) <
d dt
A,(,~, .,~,) + c I1,,, } + c
ul ~v ,
135
3.5. Alternative Approach and 1
We obtain
d d &(~, ~,,) < C I~,1 ~ + CII~I ~ + CIIf, ll~ u.ll~: + aTA,(~,, u,)+ ~7 Multiplying t now gives that
'(
)
t lluttll2x + ~t A~(u~, u~)+ At(u, ut) < c(t +
1)(llut]l
2v + I~1 ~ +
Ift I~)
This implies
tA~(~,~) +
/o
~ll~,(411~ <_ c
/o
(ll~,(~)llt + 11,,(~)115 + IIf~(~)ll~) d~.
Inequality (3.109)follows from (3.103)and (3.106). Semidiscrete Approximation Let fZ C R 2 be a convex polygon, and A t ( , ) a bilinear form on V x V with V = H~(f~) (for simplicity) induced from a second order elliptic differential operator. The associated msectorial operator in X  L2(f2) is denoted by A(t). We take the evolution equation
du + A ( t ) u = O (O < t < T) dt
with
u(0)=u0
with
uh(0) = PhUo,
and its semidiscrete finite element approximation
duh dT + Ah(t)Uh = 0
(0 < t < T)
described in w and w respectively. Now we develop the error analysis by the energy method. Recall t h a t Rh = Rh(t)" V +Vh denotes the Ritz operator associated with At( , ):
At(Rh(t)v, Xh) = At(v, Xh)
(v e V, Xh r Vh)
(3.111)
We have the following. L e m m a 3.10. Any v = v(t) c C 1 ([0, T] + V) satisfies the inequality
Ila~(v nhV)llH,(~) <_ Ch ~j (llv g~(~)+ IIV~IIH~,(~>) fork
1,2 and j = 0 , 1 .
(3.112)
3. Evolution Equations and F E M
136
Proof." Equality (3.111) implies
.A~(a,(v  R,,v), x~) = A,(,,  R,,~. x~)
(3.113)
for Xh E Vh, and hence
a l a~(v  R~v)ll~
~
.At (oqt(y  R h V ) , Ot(v  f~h'U))

.At (Ot(V  R h V ) , Otv  Xh)  A t ( v  R h V , Xh)
___ c (lla,(v follows. T h e results in w
Rhv)liv Ila~v

 xhll,,,. + IIv  Rhvllv
Ilxhllv)
are summarized as
II(Rh(t)
1)v g,(n) <
ChkJll~llH~(n)
(3.114)
for k = 1,2 and j = 0, 1. Taking Xh = Rh(t)Otv(t), we get (3.112) for j = 1 and k = 1,2. The remaining case j = 0 is obtained by the duality a r g u m e n t similarly, in use of (3.113) and the elliptic estimate of A(t). It is left to the reader. [] Suppose u0 E V, and take the solution Uh = gh(t) E Xh of duh
dt
+ Ah(t)gh = O (O <_ t <_ T)
~(0) = R~(0)~0.
with
We have the following. Lemma
3.11.
The inequality
Z
~ II~h(~)ll~d~ ~
Ch~l ~o11~"
(3.115)
holds for eh = u  Uh. Pro@
We have
(9~,,, x,,) + A~(~,,, xh) = 0 for Xh E Vh. Taking Xh = Rh(t)eh(t), we get ld
2dtll~hll~c+ A,(~h,~h)
=
(0,~h, (1  Rh)~h)+ At (eh, (1  Rh)~h)
=
(ateh, (1  Rh)'U) + .At (eh, (1  Rh)'U)
(
<_ c~ h~ I1,~11~(~) + Ila,~,~llx
) +cllehllv
for c > O. This implies
d dt
1~,,112 x + ~hl ~ _<
Ch ~
( I1~11~,,~(~)
+ Ila~,,llx
)
137
3.5. A l t e r n a t i v e Approach
by (3.100), and hence
/0
II~h(~)ll ~V dx
<
Ch ~
<_ c h ~
I1~(~) .:(~) + IIO~h(~)ll
ds + II~h(o)llx
II~(s)
+ ch ~ ~ol follows. The elliptic estimate gives
Ilu(s)ll/_z~(r~) < c IIA(~)~(s)Ix : c Ilu,(~)l x, while inequality (3.105) of Proposition 3.6 is valid for u = Uh uniformly in h. The righthand side is further estimated from above by ch ~
(11~ol ~ + IIRh(0)~oll~) <
Ch ~
I1~o
and the proof is complete.
[]
If one assumes the inverse assumption, L 2 orthogonal projection Ph " X ~ X h satisfies
llPhllv_v <_ c
(3.116)
by Proposition 1.5. Then, the inequality
fo ~ II~h(~)tl ~V d~ < is proven in a similar manner, where eh = u assumed in the following lemma.
Ch ~
I1~o11~,
Uh. However, inequality (3.116) is not
L e m m a 3.12. I f uo E V A H k  J ( ~ ) we have
~0 t II~h(~)ll x~ d~ <
c t l  J h 2k
Iluo
2
,
(3.117)
where k = l , 2 and j = 0 , 1 . Proof:
Given t > 0, we take the backward problems (v, O~w)  A~(v, w) = (v, e~(s))
(v 9 V)
with w(t) = 0 in X and
(x~. 0 ~ , )
 As(X,. ~ , ) = (x~. ~,.(~))
(x~ e v~)
with wh(t) = 0 in Xn, respectively. Those problems have unique solutions w = w(t) 9 C O ([0, t), D (A(t)*)) A C O([0, t ] , X ) n C ~ ([0, t ) , X )
(3.118)
3. Evolution Equations and F E M
138 and
~
= ~(t)
~ c ' ([o.t]. x h ) .
respectively. In spite of the inhomogeneous terms in the righthand side, the argument in the proof of Lemma 3.11 is applicable with the time variable t reversed. A quick overview reveals that
/o'
w(~)  w~(~)l ~. d~ _< C h ~
/o'
I~,(~)1 ~ d~.
Combining this with inequality (3.105) implies that
L
' (llo~(~)<
/o' 1'
(21a, w,,(~) ~x + 2 IIO, w(~)llx~ + h ~ IIw,.(~)  w(~)ll~) d~
< c 
h2 II~(~)  ~ , ( ~ ) l l ~ ) ds
o.~,.(~)+
Ile,(s)ll ~ ds.
(3.119)
X
Taking v = eh(s) in (3.118), we have
II~,,(s)ll%
=
(e~(s),O~w(~))  A~(~,,(s),w(s)) {(~,,. o ~ ( ~  ~ , , ) )  .4s(~. ~  w,,)} + ( ~ , , . O s ~ , , ) 
..4~(~,,.w,,).
Here, eh = ( u  RhU)+ (Rhu uh) and
(~,,. o~(w,, ~))  .4~(x,,. ~ , ,  ~) = o holds for Xh E Vh. It follows that
I~,(~)11~
=
{(~
~,,~. o ~ ( ~  ~ , ) )  A s ( ~  R.,,~. ~ d + ~ (~,,. ~ )  ( o ~ . w~)  .4~(~,,. ~ ) .
~,,)}
Furthermore, the equality
(O~eh, Xh) + A~(Ch, Xh) = 0 holds for Xh E Vh. W e have
I1~(~)11 ~x = ( ~  R,,,~, o ~ ( w  w , , ) )  A ~ ( ~  R ~ , ~ 
d
w,,)+ y~ (~,,, ~,,).
In use of e,,(0) = wh(t) = 0, we h a v e
/0'
II~,,(~)ll X~ ds = _
/o' ]o'
( ( ~  ~,,~,, a~(w  w , , ) ) 
~
A ~ ( u  R,,~, w  w,,))d~
(llOs~  a~,,,,,, ~ + t,, ~ w  w , ,
+ c~
/0'
I~)d.~
(I ~  ~,,,~11~ + h ~ I1,~ ~,,,~11~)d~
(3.120)
139
3.5. Alternative Approach
for c > 0. We obtain
/o
I I ~ ( ~ ) l l ~ d ~ < C
/o
(ll**Rh~ll ~~ + h ~ I1~ R ~ I I ~ ) d ~
by (3.119)and (3.120). Inequality (3.117)follows from (3.114). As is described in w
the following theorem implies Ileh(t) x < Ch2t1 Iluollx
by Helfrich's duality method. T h e o r e m 3.13. If Uo E V r~ H 2 (f~), the estimate le.(t)llx < CA ~
(3.121)
luo H~(~)
holds true. Proof." Putting = [(t)= Rh(t)eh(t)= Rh(t)u(t)
Uh(t) E Vh,
we have ({t, ~) 2t r
~)

(Ot(t~hU), ~)  (OtUh, ~) Jr r
=
(at(Rhu), ~)  (atu, ~) (a~(~  R~), ~).
:
~)  At(Uh, ~)
Multiplying t implies
1 d (t I1{11~)
2 dt
1 t~t,(~, {) = ~ I1{11~ 
+
t(at(u

Rhu)
~)
and hence
t ll{(t)ll~: < c 
follows. We obtain
J/o'
I1{(~)11~ d ~ + C
t Ileh(t)ll 2X < t l l ( u  Rhu)(t)ll~: + C + c
/o
f'
~ Ila~(u R~u)(~)lt ~ d~
Ile~(s)ll~ d~
I I ( u  R~u)(~)ll~ d~ + C
Here, Lemma 3.10 applies. We get
X
jo
~ Ila~(u R ~ ) I I ~ d~.
t lleh(t)ll 2X < C foot Ileh(s)llx2 ds + Cth 4
{
max [[U(S)IIH2(~) + O<s
)}
8 [[U(S)IIH2(~) + [[Ut(8)[[ u~(n) 2 ds
140
3. Evolution Equations and F E M
For the righthand side we have
fo' I1~(~)11~ d~ X
< 
C t h 4 I1~o11~(~)
by (3.117), and also
I~(~) IH~(~) _< c II~oll.~(a) and
Jo"(
u(~)ll ~
)
~
d~ < C I ~oll ~
by Propositions 3.7 and 3.9, respectively. Inequality (3.121) now follows.
Fulldiscrete
Approximation
The energy method is also applicable to the error analysis for fully discretized problems. Here we study the backward difference finite element method,
~;(t + r  ~ ( t )
+ Aj~(t + T)U "(t h + ~) = 0
T
with
~;~(o)
=
Phi0,
w h e r e t = t n f o r 0 _ < n < N. We drop the suffix h, and the error is denoted by e(t) = u ' ( t )  u ( t ) .
The inequality
L e m m a 3.14.
n1
t 2 Ile(t)ll x2 < c~ 2 I~011x~ + Cr ~
IIe" (tj)llv.2
(3.122)
j=l
holds for t = tn. Proof:
Given v = v(t), we put
Otv(t) = v(tj)  v ( t j _ l ) T
for t = tj with j _> 1. We have (ut, v) + At(u, v) = 0
and
(O~u~,v) + A~(u ~, v) = 0
for v E V and t  tj. It follows that
(Ote(tj), v) + At3 (e(tj), v) = (Tj, v) ,
(3.123)
3.5. Alternative Approach
141
where
1 f~~, 1 (~_ "~j = Otu(tj)  ut(tj) = ~
tj_,)~.(~) d~
(3.124)
Putting ~(ty) = tje(tj) and ~j = ty3'j, we get
(Ote(tj), v) 4 At3 (e(tj), v) = (~j 4 e(tj1), v) . Letting v = ~(tj) now gives that
1 (~(tj)  e(tj1), ~(tj)) 4 Atj (~(tj), ~(tj)) = (~j 4 e(tj_l), ~(tj)). T In other words, the inequality 1
:2 (ll~(tj)ll~II~(tj

12) 4 TAt, (~(tj), ~(tj)) = T (X/j 4
 1)112 x + r 2 IIO~(tj)
e ( t j _ l ) , e(tj))
holds and we obtain ila(tj)ll x~  I I ~ ( t j  1 ) l l
~ + 2 w ~ l ~ ( t j ) l l ~V X
<_ 27 (Sj 4 e(tj_l), ~(tj)) < 2~ { c (llgjll ~v . + I l e ( t j  1 ) l l v ~. )
+ ~

Ila(tj)llv}
9
We have II~(tj)ll X ~  I l e ( t j  1 ) l l x< 2Cw ([l@jl[~. + Ile(tj1)[2v.)
and hence
t 2 Ile(t,)llx = II~(t,)lI2x <__2 c T
tj 1l'TjIlv. + j=l
Ile(tj)lIv 9
{3.125)
j=l
follows. Equality (3.124) implies ii,),jll2 _~ 1 T
/t;
' (8  t j _ l ) 2 Ilutt(8)ll 2. d8 1
and hence tj II~jll..
follows because t y ( s with f = 0 implies
_
..
ftj 'j1
tj1) <_ Ts for s E [tj1, tj]. On the other hand, equality (3.110) IIuttllv. < Cllutllv + CIlullv.
(3.126)
142
3. Evolution Equations and F E M
Therefore, we get
n
2
2
< ~~
j~0tn
~ I1~.(~)111 V* d~
j=l
< c wi
/o ' (ll~(~)ll~+s ~ll~r
_ cw 2 Ilu011~ by Propositions 3.5 and 3.8. Inequality (3.122) is now a consequence of (3.125). For the operator T(t) =_ A(t) 1, the relation Ilfllv.
~
IIZ(t)fllv
(f ~ Vh)
holds uniformly in h. In use of T'(t) =  A ( t )  l A ' ( t ) A ( t )  l ,
I Z'(t)fllv <_ C
Ilfllv.
we have
9
In terms of T(t), equality (3.123) is written as
T(tj)Ote(tj) + e(tj) = T(tj)3,j. Setting Fj = T(tj)~,j, we get
Ot [T(tj)e(tj)] + e(tj) = Fj + [OtT(tj)] e(tj_l). Here we take L 2 inner product with T(tj)e(tj).
We have
(T(tj)e(tj), e(tj)) ~ I e(tj)l
~.
and
(Ot [T(tj)e(ty)] , T(tj)e(ty)) 1 = 2T (llT(tJ)e(tJ) 12X  I l T ( t j  l e ( t j  1 ) I x
+ 7IIOt [T(tj)e(tj)]llx),
as in the proof of L e m m a 3.14. It follows that
_1 2
(llZ(ty)e(tj)ll 2x [IT(tj 1 )e(tj_ 1 )1 x2 + ~ II0, (T(tj)e(tj))ll 2 ~) 2 x / + ~ I~(tj)llv. 7<_ Ilrjllv. IIT(tj)e(tj)llv + II[O~T(tj)] e(t~_l) x IIT(tj)e(tj)l x
with a constant # > 0. The righthand side is estimated from above with some t, 6_ (tj_l, tj)"
I FjlIv. IIT(tj)e(tj)llv § IIT'(t,)e(tj1)llv T ( t j ) e ( t j ) I x _< c (IIFjlIv. e(tj)llv. + Ile(tj,)l v. IIZ(tj)e(tj)l x) 1
2
2
_< ~. (lle(tj)llv. + I e(tj_~)llv.) + C (llT(tj)e(tj)ll 2X + IIFj I~*).
(3.127)
Commentary to Chapter 3
143
We obtain 1
9
3
1
2
2
2
+ ~  , (If~(tj)lf v. II~(t~ ~)11~.) _< c~ (llT(tj)~(t~)fl~ + Ilrjlt~.) and hence
]]T(tn)e(tn)]] x2 + ~ s
II~(tj)ll~. < c ~ s
j=l
Ilrjll ~v.
n + c~ ~lf(tj)e(tj)ll2x
j=l
j=l
follows. Then the discrete Gronwall's inequality implies that n
IIT(t~)e(t~)llx + f
Ile(tj)ll 2V* <  CT ~ j=l
Ilrjll V* 2
"
(3.128)
j=l
Here, Schwarz's inequality gives JIFjll2v.
IIZ(tj)'~jll~. < 1__

T
~ (s  tj_~) 2 ]lZ(tj)u~(s)l[~. ds 1
similarly to the proof of Lemma 3.14. On the other hand, T(t)ut + u = 0 implies
T(t)utt =  u t  T'(t)ut. Therefore, given s E [tj1, tj] there exists s. E (tj1, tj) satisfying
<_ Ily(s)utt(s)llv. + f IlY'(s,)u,(s)lIv. _< Ilu~(s)llg. + IIT'(s)~,(s)ltv. + w IlT'(s,)utt(s)llv.
IIT(tj)u,(s)llv.
<__ c (ll~(~)llv. + II~,(~)llv + ~ II~,(~)llv.). In use of [lutllv. = I A(t)ullv.
< C I1~11~, w~ h~v~
IIT(t~)~.(~)llv. <
c
(ll~(*)llv
+ ~
II~,(*)llv)
by (3.126). Consequently, the inequality
~  ~ Ilrjll ~. _< C j1
j=l
1
(~ tj_~ (11~(~)11~+
<
c~ ~
<
c w 2 IIPu0112

(ll~(s)ll =v +
I1~(~)11~)d~
~ Ilu,(~)ll~)ds (3.129)
X
follows from Propositions 3.5 and 3.8. Combining (3.122) with (3.128) and (3.129) gives inequality (3.82):
II~(t)lt~
< CTt1
II~ollx.
3. Evolution Equations and FEM
144 Commentary
to C h a p t e r 3
3.1. The generation theory of TanabeSobolevskii was given by Tanabe [377] and Sobolevskii [353], independently. Inequality (3.17) is a consequence of the elliptic estimate of Agmon, Douglis, and Nirenberg [3]. The generation theory of FujieTanabe was given by Fujie and Tanabe [132]. Inequality (3.23) follows from the coerciveness of At. The righthand side may be replaced by C~ Izl, but this form simplifies the treatment of fractional powers of A(t). The proof of (3.23) is also given in Tanabe [378]. Generation theory of KatoSobolevskii was given by Kato [201] when p1 is an integer and Sobolevskii [354] for the general case. Inequality (3.27) was made use of by Heinz [169] and Kato [202], [200]. For the theories of fractional powers of msectorial operators, see Kato [199]. Inequality (3.26) was also proven there. Equality (3.3o) is due to P.E. Sobolevskii. The theory of KatoTanabe was given by Kato and Tanabe [209]. See also Suzuki [368] and ~3.3 for the fact that smoothness of bilinear forms confirm the assumptions of the last theory. Most of those generation theories are described in Tanabe [378] in details. 3.2. Inequality (3.55) was proven by Suzuki [368]. Proof of (3.64) was also given there. Note that necessary assumptions for A(t) are those on the theory of KatoTanabe and inequalities (3.60)(3.62). Once they are provided, the Banach space structure induces the conclusion. 3.3. Inequality (3.67) was proven by Fujita and Suzuki [147] for boundary condition (3.a) and Suzuki [3661, [36s] for the general case. Related works were done by Helfrich [171], Fujita [140], Suzuki [365], and Sammon [335]. 3.4. For the detailed proof of (3.86), see Suzuki [368]. Inequality (3.82) was proven by Suzuki [366], [3681 Related works were done by Sammon [335]. 3.5. Error analysis on the fully discrete approximation of temporally inhomogeneous parabolic equations has been done independently by Baiocchi and Brezzi [23], Huang and Thom~e [184], [185], Luskin and Rannacher [256], [257], and Sammon [336], [337] Contrarily to Suzuki [366], [368], the Hilbert space structure of the problem were used systematically there. The works described in this section were done by Luskin and Rannacher [256] and Huang and ThomSe [185]. The error analysis for schemes with higher accuracy were done by Baiocchi and Brezzi [23], Luskin and Rannacher [257], and Sammon [336], [337].
Chapter 4 O t h e r M e t h o d s in T i m e D i s c r e t i z a t i o n We have studied fully discrete finite element approximations for parabolic equations with the forward, the backward, and the CrankNicolson schemes for time discretization. However, many other methods have been examined in numerical computations. The present chapter is devoted to systematic studies for them.
4.1 Let
r(z)
Rational Approximation of Semigroups {etA}t>_o be a uniformly bounded (Co) semigroup on a Banach space X. We take
= (1 + z) 1 and describe the idea. It is a rational function provided with the properties
r(z) ~ r
_.it_0 (]Zl p + I )
as z ~
0
for p = 1, and
Ir(z)l _< 1
for Rez >_ 0.
They are referred to as being of order p and Aacceptable, respectively. As we have seen, the operator r'~(TA) = (1 + TA) n is regarded as an approximation of e tA for t = nT, where ~ > 0 denotes the time mesh size. Based on those facts, we can argue as follows. First, the relation
d~d (r~(sA)e_n(T_s)A)
=
nr n  l ( s A ) (r'(sA) 4 r ( s A ) ) Ae n(rs)A
 nrn+l(sA)sA2en(~s)A holds, and therefore we have [r~(TA)  enrA]A 2  n
rn+l(sA)se n(~s)A ds.
Next, from the semigroup theory, the uniform boundedness of the semigroup represented as
lietAll _< C
(0 < t < oc)
(4.1)
is equivalent to the stability of its backward difference approximation:
]rn('rA)ll <_ C 145
(4.2)
4. Other Methods in Time Discretization
146
Those relations imply
II[rn(~n)  en=A]APlll <_ CtT p
(4.3)
for p = 1, where t = n7. Taking this observation in mind, we shall give error estimates for the approximate o p e r a t o r rn(rA) when the semigroup {etA}t>_o is holomorphic and r(z) is a general rational function of order p and Aacceptable. For this purpose, we recall the notion of Aacceptability of G. Dahlquist and generalize it in connection with a sector in the complex plane, where 0 E (0, 7r):
z0 = {z e C l 0 _< larg zl _< 0}. Namely, we introduce the following notations: 1~ T h e rational function
r(z) satisfies (i)o if Ir(z)l_ 1 for any z E E0 and Ir(oc)l < 1.
2 ~ T h e rational function
r(z) satisfies (ii)o if Ir(z)l < 1 for any z C E0 \ {0}.
3 ~ T h e rational function some 5 > 0.
r(z) satisfies (iii)o if Ir(z)l < 1 if 0 < Izl < ~ and z E E0 with
If I~(z)l _ 1 for any z E E0, then r(z) is said to be A0acceptable. Therefore, Aacceptability means A~/2acceptability in this terminology. Any rational function r(z) of order p (>_ 1) has p r o p e r t y (iii)o for 0 E (0, 7c/2). In fact, for p E [0,0] we have 0
Op where ~ = x/~l. Because of r/
r(O) = eZlz=o = 1
and
r'(0) =
~2eZ dz z=O
" 1,
it holds t h a t c)fi r(Ps
=  2 R e e~ =  2 c o s ~
_<  2 c o s 0 < 0
(4.4)
p=0
and hence the conclusion follows. If r(z) is the Pad~ a p p r o x i m a t i o n of e z with degrees n and m of the numerator and the d o m i n a t o r , respectively, then it is of order p = n + m. Furthermore, according to n < m, n = m, and n > m, it has properties (i)o, (ii)o, and (iii)o for some 0 E (0, 7r/2), respectively. In fact, in this case we have r(z) = R,,,m(z) = P,.m(z)/Q ......(z) with (n + m  j)!n!
P.,m(~)
=
(~ + .~)!j!(~ _ j)! (~Y, j=O
m
Qnm(Z)
(n+mj)!m!
x'z_.,(,, + ,~,)b~(,~,  .J)~J j=O
4.1. Rational Approximation of Smigroups
147
The relation
]Rnm(Z)

I <_ c zl n+m+l

(z ~ 0)
is known. Let
(n + r e  j ) ! n ! anm(j) = (n + re)!j!(n j)! for 0 < j _< n. If n <_ re, we have
anm(j) > 0
and
a,~m(j) <_ anm(j).
j =0
T h e inequality
j =0
m
< E anm(J)PJ cosjqp j=o
= Re
Qnm(pe +zv) 5r 0
follows if m~p E [0, 7r/2]. Under the same assumption we also have n
IIm
Pnm(pe+'~)[ =
anm(j)(p) j sin(tjqp)
_< E j =0
j =0 m
anm(J)PJ sinjqp
m
< E a~m(J)PJ s i n j ~ = j =0
E anm(J)PJ sin(+jqD) = IIm j =o
Q,~m(pe+~)l.
Therefore, I~(p~)l ~ 1 holds for qp E [0, 7r/(2m)]. If n < re, r ( o c ) = 0 follows and (i)o holds with 0 = 7r/(2m). If n = re, we have Ir(oc)l = 1 < + o c . T h e n the inequality r(z)l < 1 for z E E~/2m is a consequence of the classical P h r a g m 6 n  L i n d e l 6 f theorem. The m a x i m u m principle now gives Ir(z)l < 1 for z E E0 \ {0} with 0 C (0, 7r/(2re)). This means t h a t r = Rnm satisfies (ii)o in this case. Finally, r(z) satisfies (iii)o always for some 0 E (0, 7r/2) as is described. It is known t h a t the Pad6 a p p r o x i m a t i o n Rnm of e z is An/2acceptable if and only if r e  2 _< n _< re. If n = re and n = r e  1, Rum is subject to a recursive formula based on the continued fraction expansion. Namely, we have the expansion e z = Z
1Z
1+ Z 2~


3+.
expressed by Z
C z 
Z
Z
Z
Z
 i+~5+~
Z
Z
g+..+~
2j_1
This implies eZ=l+
2Z
2~
+
Z2
6
+
Z2
~
Z2
+    + ~
2(2j1)
+ 9
....
4. Other Methods in Time Discretization
148
Accordingly, a rational approximation H2k+l(Z)= G2k+l(Z)/F2k+l(Z)of e z is introduced inductively with F1 = 1, Fa = 2  z, G1 = 1, G3 = 2 + z, F2j+I
=
2 ( 2 j  1)F~j_I + z2F2j_3,
and
Ggj+I = 2(2j  1)G2j1 + z2G2j3. Similarly, H2k(z)= G2k(z)/F2k(z)is defined by Fo = 1, F2 = 1  z, Go = 0, G2 = 1,
F2j =
{
2(2j1)+
2 } 2j3 z
2jlz2F2j_4, F2j2+2j 3
and
G2j = {2(2j  1 ) +
2
2j 3
"[
2j
z f G2j2 + 2j
1
3
G2j4.
Then, the relations H 2 k + l (  z ) = Rkk(z) and H 2 k (  z ) = Rkl,k(z) can be verified. Let A be an operator of type (00, M0) on a Banach space X, where 0o C (0, 7r/2) and Mo >_ 1. The holomorphic semigroup generated by  A is denoted by {e tA }t>o. Suppose that A is bounded for simplicity. We can show the following. T h e o r e m 4.1. The estimate T
en'AII < C ( t )
P
(4.5)
holds for t = nT under one of the following assumptions, where fJ1 > 0 is a constant and 0>00" 1~ r(z) has property (i)o. 2 ~ r(z) has property (ii)o and 7 IIAI < M~ < +oo. 3 ~ r(z) has property (iii)o and T [JAIl < /l}I~ < ~.
We need several lemmas. constants, respectively.
Henceforth r > 0 and C > 0 stand for small and large
L e m m a 4.2. If a rational function r = r(z) is of order p and Aoacceptable with p > 1 and 0 E (0, 7r/2), then there exist constants cy > 0 and fl > 0 such that [rn(z)  e nz ] _< C n zl p+I e ''~'ee(z)
for z C Eo with
Izl
_ ~.
(4.6)
149
4.1. R a t i o n a l A p p r o x i m a t i o n o f Smigroups Proof." We have
rn(z)

e nz ~ s
r J  l ( z ) ( e  z  r ( z ) )
e (njT1)z
j=l
and
It(z)

eZ[ <
C
Iz[ p+I
(Izl < ~o)
for some Oo > 0. This implies [r(z)[ _< e Re(z) + C (Re(z)) p+I for z E E0 and Izl _< a0. T h e function f ( t ) = e  t + C t p+I satisfies f ( 0 ) = 1 and f'(0) < 0, and hence f ( t ) <_ e ~t holds on t E (0, al) with some /71 > 0 and ~ E (0, cos0). We get
for [zl <_ min(crl, a0) and z e E0. Therefore, if Izl _< (7  m i n ( a l , a0) and z e E0, we have
Irn(z)  e nz ] ~
s
e(J1)~Re(z)c [zip +1 e(nj+l)cosO'Re(z)
j=l C n ]Z[p+I e /~nRetz) .
The proof is complete. Lemma
4.3.
Under the a s s u m p t i o n of Theorem ~.1, the estimate
II[<(~A)  e ~] A~II ~ C < holds f o r t = h r . Proof: Take M > 0 satisfying r [IAII < M, ]~/'1 < M, and s < M < 5 according to the cases in T h e o r e m 4.1, respectively. We may suppose t h a t a < M in Proposition 4.2. We take a p a t h of integration F0 divided into three parts, Fo = F1 U ['2 U ['3 with {90<01 <{9"
Pl

{)~=pg+ZOl [ O ~ p ~ g T  1 } ,
r 2
=
{A =
r~ =
pc+Z~ I oT 1 < p < M T  1 } ,
{M~'~'~l
I~J < 0 1 }
Then, we have 
2m
+
1 =
I+II+III.
+
2
( r ~ ( r z )  etz) z  p ( z I  A) 1 dz
3
4. Other Methods in Time Discretization
150 In use of Proposition 4.2, we have
IlZll <_ c
n.Tp )P+aeZnwc~176 P
=
c
.~p?+'~~(n'o)c~176 ( )
/0
)  ~ d ~ ~ = C~~. P
From the a s s u m p t i o n follows t h a t
I11:
__
c ]r
+
[ l ( z l  A)'II dzl
2
C
fl_p~dp = CTP. T 1
P
Finally, the inequality
II(zI
a)l]l ~
7
1
I z l  {IAII
M
7IIAll
holds on F3 and hence
IXIII ~_ c Jr ( ~(~z)l +
I  zl)Izl
~' I ( A  A)']I Idzl
3
< c 
f
o,
0,
(M~') ~
Md~
M~ItAII
 CM Tp
follows. In the first two cases of the theorem, the constants C in the estimates of III]] and IIIIi] are independent of M, while CM goes to zero as M , +co. Therefore, the t h e o r e m follows. [] Proposition
4.4. I f r = r(z) satisfies (iii)o, then each 5' E (0, 5) admits/3 > 0 satisfying
Ir(z)l _< e ~'zj. If ~ = ~(z) satisfics (ii)o, thcn ~' > 0 can be taken arbitrarily large. Proof: As is noted, inequMity (4.4) holds. Any/30 E (0, cos0) admits a0 > 0 satisfying
I~(S~)I _< oxp (/3op)
for p c (o, ~o) ~nd I~I< 0. Next, let max {Ir(peW)llp E [c~0,~'], ~ C [0,0]} = 1  c < 1, and take /~1 > 0 satisfying exp (/31~') = 1  e .
Then, wc have
lr(pe'~)] _< exp (/~,p) for p c (00, (Y) and [~p] < 0. S e t t i n g / 3 = min(/30, ~1), we get the conclusion.
[]
4.1. Rational Approximation of Smigroups
151
L e m m a 4.5. In the last two cases of Theorem ~.1, the estimate
IIA~<(wA)II < C~(n~) ~
(4.7)
holds for c~ > O. Proof: Taking a path of integration as in the proof of L e m m a 4.3, we have 
+ 27"~'~
=
1 '~F2
(nTz)arn(Tz) (Z[  A) 1 dz 3
I+II.
From the assumption, we can take ~' of Proposition 4.4 in M < ~'. We have the following:
I111 IIHII
<
< 
Then
c
c
Z
(n"rz)C~e ~nrpc~
dD = Ca,
P
fie ( n T M ) ~ e  Z M n ~ d ~M< C ~ . ol
M/1)1

the proof is complete.
[]
Helfrich's duality argument now gives the following.
Proof of Theorem ~.1 for the last two cases: 6, m E N a n d O _ < ~  m _ < 1. We have
Divide N E N into n = t~ + m with
rn(TA)  etA = (re(TA)  ee~A) APrm(~A) + ee~dAp (r'~(TA)  em~d) . Operator norm of the first term of the righthand side is estimated as II[rl(TA)  eI~A]APlI . IIAPrm(TA)I < CTP(mT) p < C / n p = C(~/t) p by Lemmas 4.3 and 4.5. T h a t of the second term is estimated similarly by the adjoint form of those lemmas. The proof is complete. [] Generally, r(c~) = 0 does not arise even in (i)o. We cannot take d' = +cx~ in Proposition 4.4. For the first case of Theorem 4.1 to prove, therefore, we require some more considerations. L e m m a 4.6. Let f l, f2 " R+ ~ R+ be continuous functions satisfying
n f l ( r ) dr < +oo r
and
dr < +oo 7"
for R > O, and let ~ be a meromorphic function provided with the properties IF(z)l < f~ (Jz[) and l~(z)  ~ ( o o ) l < f2(Izl) for lz[ <_ R and largz I = 0 ~ , where O~ > 0o. Then, the inequality
{ fl(r)~ dr + ~00Rf2(r) d~. + I~(oo)1
II~(A) < CR Jo(.R holds true.
152
4. Other Methods in Time Discretization z
Proof." Letting h(z) = qp(z) 1 + z qp(~176we have ~(A) = h ( A ) + ~(oo)A(1 + A) 1 Here, lid(1 + A )  l l _ C holds and we have
[I~(A)II <_ IIh(A)ll + Cl~(c~)l. To estimate IIh(A)ll, we recall the path of integration F given in w oriented boundary of E01, which produce the equality
h(A) = ~
the positively
h(z) (zI  A) 1 dz.
We have z
lh(z)l = ~ ( z ) for Izl _< R and larg z I =
01,
I + ~ (oo)
f l ( [ Z [ )  4  [ z [ [~(Cx:))l
and 1
Ih(z)l= ~(z)~(~)+
1
l+z ~(~)
A(Iz[)+~l~(~)l
for Iz[ >__R and larg z I = 01, respectively. Those inequalities imply IIh(A)ll
_< c
=
C
{/0"
( k ( r ) + rl~(oc)l) dr +
{/0" 'r/0" fl(r)
r +
) } } dr
A ( r ) + 1I ~ ( r ) l
r
r
'" ( R +
f 2 ( r ) r +
~(oc)l
The proof is complete.
r
.
[]
Now, we complete the proof of Theorem 4.1.
Proof of Theorem 4.1 for the first case: Take 01 E (0o, 0) and a > 0 as in Proposition 4.2. The function ~(z) = e . . . . . r"(z) satisfies (4.6) if z E E0, and Iz <_ a. On the other hand, making ~ > 0 smaller if necessary, we may suppose that sup{Ir(z)ll
z~E0,,
zl _ ~ } = e
z
from the assumption. Furthermore,
~(z) r(o~)l_
cI ~I
(Izl >_ ~)
holds because r = r(z) is rational. Those relation imply rt1 j=O
4.2. Multistep Method
153
and hence ~(~)  ~(oo)1 _< ~~,zlcos0, +
c~nZ/Izl
(4.8)
if z C E01 and Izl >_ a. Finally, we have
I~(oo)1 ~ e n~.
(4.9)
Inequalities (4.6), (4.9), and (4.9) give
II~(~A)II = II<(~A)by L e m m a 4.6. The proof is complete. Theorem 4.1 is applicable to the semidiscrete finite element approximation of the parabolic equation, dub + Ahuh = 0 dt
(0 <_ t <_ T)
a~d
uh(O) = Phuo
in Xh with Ah, Xh, and Ph as before. We recall that the spectrum of Ah lies in a parabolic region in the complex plane uniformly in h. Each 0o E (0, rr/2) admits constants M0 _> 1 and ~ independent of h such that Ah  .X is of type (00, Mo). Taking Vh = e)'tUh instead of Uh, we can make the exponent 0 > 0 as small as we like in applying T h e o r e m 4.1. On the other hand, the inverse assumption guarantees the inequality IIAh[I <_ 7h 2 with some "y > 0. We have the following. T h e o r e m 4.7. Let r = r(z) be a rational function of order p (> 1) satisfying one of the following conditions for some 0 > O: 1~
(~)o.
2 ~ (ii)o and r h 2 < M1 < +c<~. 3 ~ (iii)o and r h 2 <_ M1 < "~15. Then, the estimate (~.5) holds for t = n r with a constant C > 0 independent of h. Applied to the semidiscrete finite element approximation with higher accuracy, Theorem 4.1 gives natural results from the viewpoint of the correspondence of the rate of convergence with respect to the time discretization and the space discretization. Then, theorems on backward, forward and CrankNicolson schemes in w167 and 2.6 arise as special cases.
4.2
Multistep
Method
Time discretization schemes studied in the preceding section may be called the singlestep m e t h o d totally, as the value un = u(tn) is determined by t h a t of un1 = u(tn_l)
154
4. Other Methods in Time Discretization
for tn = nT. Multistep m e t h o d of order q(>_ 2) determines Un from the values at the preceding qsteps, Un1, U~2,''" , U~_q, after determining u l , . . . , Uq1 suitably. In this section we study this kind of schemes adopted to the evolution equation
du
dt
+Au=O
(O<_t<_T)
with
u(0)=uo
(4.10)
on a Banach space X, where A is of type (0, M) with 0 E (0, 7r/2) and M >_ 1, so that  A generates a holomorphic semigroup {e tA}t_o. Suppose that A is bounded for simplicity. In the numerical scheme which we are studying, one determines U l , ' " , Uq1 from u0 by a singlestep approximation in use of a rational function first, and then computes Un+q through the relation q
E(ai
t TbiA)Itn+i : 0
(4.11)
i=0
for n = 0, 1 ,   . , where ai, bi E R. W i t h o u t loss of generality, we take aq = 1 in (4.11). Setting q
P(r
q
a~r
= E
and
S(() = E
i=0
bir
i=0
we call scheme (4.11) the multistep m e t h o d (P, S). It is said to be of order p(_> 1) if the following relations hold: q
q
q
q
q
Ea~=0, Eia,= Eb,,
EiJa~=iEiJlb~
i=0
i=1
i=0
i=0
(2
/=1
Letting 0 ~ = 1 and 0 . 0 I = 0, we can write them simply as q
q
E ijai = j E iJlbi i=0
(4.12)
i=0
for j = 0, 1 , . . . ,p. To understand tile meaning of those equalities, take A = d/dx and compute tile Taylor expansion around r = 0 of the finite difference operator q
L~[y] = E
[aiy(x + iT)+ fbiy'(x + iT)].
i=0
Then, equalities (4.12) follows if tile coefficients of T up to pth powers are let to be 0. If q = 1, scheme (4.11) is either one of the backward, the forward, the CrankNicolson, and the modified CrankNicolson (of order 1). Letting w((, z) = P ( ( ) + zS(() and 0 E (0, rr/2), we introduce the following notations: 1) The m e t h o d (P,S) satisfies (III)o if any root (j (1 _< j _< q) of P ( ( ) = 0 is simple and lies on the closed unit disk t(I _ 1, and moreover, if (j is in (j = 1 the inequality Re Aj/[Ajl > sin0 holds for Aj = CjS(Q)/P'(Q).
4.2. Multistep Method
155
2) The m e t h o d (P, S) satisfies (II)o if it has property (III)o, and any root Cj(z) (1 _< j _< q) of w(r = 0 is simple and lies in the open unit disk [r < 1 for
z e x0 \ {0}. 3) The m e t h o d (P, S) satisfies (I)o if it has property (II)o, any root of S(~) = 0 is simple and lies in the open unit disk [C[ < 1, and bq > O. If (P, S) has property (III)o, then the requirements for Cy(Z) (1 < j < q) stated in (II)o holds if 0 < Iz] < ~ and z E E0, where ~ > 0 is a constant. In fact, Cj(z) is continuous in z and we may suppose t h a t Cj(0)  Cj by reordering the number if necessary. We have only to show that [r
< 1 holds if 0 < [z I < ~ and
z e E0 for some n > 0, assuming I(jl = 1. Because ~j is simple, g(p, r differentiable in p at p = 0, where [r < 0: 09
8p
= [(j(pe~r
is
(o, r = 2Re (r162
The relation
r
 s(r162
follows from P(Cj(z)) + zS(Cj(z)) = 0. We have ~p 0g (0 , Oh) =  2 R e Aje ~r and hence
sup
{ ~~1 6 2
r
holds. This gives the assertion. We make use of the rational function r(e) of order p  1 to construct the approximate solution of (4.10) for the value of ui = u(ti) with i = 1 ,   . , q 1 and then take the multistep m e t h o d (P, S) of order p for the values un = u(t,~) with n > q to determine. We have the following. Theorem
4.8. The inequality T
P
holds for t = nr ~ (0, T] with n > q, if one of the following condition is satisfied with 0>0o" 1) ( P , S ) and r have properties (I)o and (ii)o, respectively. 2) (P, S) and r have properties (III)o and (iii)o, respectively, together with the condi
tion
IIAII _< Me <
min
6, ~, ~q[
.
Here, 6 > 0 is so taken as for r(z) to have no poles in z[ < ~ and z E Eo. Furthermore, ~ > 0 is the constant described above as (II)o holds for 0 < z I < ~ and z C Eo.
4. Other Methods in Time Discretization
156
In the second case, min(5, n, 1/lbql ) can be replaced by m i n ( ~ , l / b q [ ) and min((~,n), respectively, if (P, S) satisfies (II)o and bq > O, respectively. Setting
5~(z) = a~ + b~z aq + bqz
(1 _< i _< q),
we can write relation (4.11) as q
5,(TA)un+i = 0
(n = O, 1, 2 , . . . ) .
i=0
First, we study the functions un = u,~(z) of z E C satisfying q
E
5~(z)un+i(z) = 0
(4.13)
i=0
for n = 0, 1, 2 , . . . . We have the following. Lemma
4.9. If ~i(z) (1 <_ j <_ q) are distinct, then un(z) satisfies q
I~n(z)l <_ C ~ ICj(z)l" j=l
fo, ~ ,~ >_ q, ~h~,~ c
>
o ~ a ~on, tant d ~ t ~ , ~ n ~ d
by
"~o,"" ,~q,, ~Plr
inf I C j ( z )  r ~#j
Proof." From (4.13) follows q1 i=0
while the relation q
p(r
+ zS(r
= (a~ + ~qZ) ~
(r  r
j=l
implies q1
q
~ =  Z ~ ( z ) ~ ~ + 1I (~  ~ ( z ) ) i=O
j=l
with ~i = cri(z) (i = 0 , . . . , q  1) determined by
{P(r162 (aq + bqz)
~' i=0
J
and
4.2. Multistep Method
157
Now we write ai and un for ~i(z) and un(z), respectively. If ui = (Q)i holds for i = 0, 1 , . . . , q  1 with some j, then we have un = (Q)n for n = q, q + 1 ,  . . In fact, we can show inductively that Un+q
 
~i'U'n+i
 
(~i ( ~ j ) i
. (O)n

(o)q+n
i=0
Therefore, in the case that q
(/=O,.,q1),
j:l
(4.14)
for some 0/j E C, we have q
(n=q,q+ 1,..). j=l
This implies
<_ l<_j<_q m a x I jl
q j=l
The linear transformation S"
(o/1,
..
. , 0/q)T~.+
(UO,
..
.
, U q _ 1 )T
is expressed by the matrix 9 9 9
~'1
"'"
1
~'q
9
~.q1
.
...
,
~'q1
whose determinant is that of Vandermonde. Since Q's are distinct, UO,
" " " , ~q1}
of (4.14) can represent an arbitrary element in C q. Then the l e m m a follows immediately. []
If some of Q(z) is not simple, the inequality
c(14'/l)rn(z)i
q
E j=l
ICj( )l
holds with m = re(z) > 2 standing for the m a x i m u m of their multiplicities.
158
4. Other Methods in Time Discretization
T u r n to the scheme in consideration. T h e a p p r o x i m a t i o n o p e r a t o r is denoted by T~(A):
un = T~ (A)uo Define the rational function sn(z) inductively as sn(z) = r(z) n for n = O, 1 , .  . , q  1 and q
E
Si(Z)Sn+i(z) = 0
(n = O, 1, 2 , . . . ) .
j=o We have T , ~ ( A ) = s~(TA) and hence q 5
(z)
_
=
i=0 holds with q
Fj(z) = E 5~(z)e(i+i)z"
(4.15)
i=o In the following lemma, the error o p e r a t o r T , : ( A )  e tA is represented by Fj('rA). Define the rational functions 7i(z) (j = 0, 41, 4 . 2 , . . . ) inductively as "~i(z) = 0 for j < 0, "),j(z) = 1 for j = 0, and q
=o k=O for j > 0. Lemma
4.10.
rite identity q
e t"+~A  T~+q(A) = E 7 .... j('rA)Fj('rA) j=o q1 j (4.16)
j=O k=0
holds for n = O, 1 , . . . Proof." E q u a l i t y (4.15) gives
j=o
j=o z=o n q j=o i=o n+q j=O
4.2. Multistep Method
159
where J
Bj(z)  ~
?,k(z)(~jk(Z)
(j : O, 1 , . . . , q)
k=O
and q
J~q+j(Z)  E
"~/rtJk(Z)(~qk(Z)
(j : O, 1, 2 , ' ' " ).
k=0
We have Bj(z) = 0 for j = q , .  . , n + q  1 from the definition of 7j(z). We also have q
]~n+q(Z) 
E ~/k(Z)(~qk(Z)
 (~q(Z) = 1,
k=O
so that qi
e(n+q)z  Sn+q(Z) 
~/nJ(Z)PJ(Z)  E j=o
j
E
~/nk(Z)(~Jk(Z) [eJz  3j(Z)]
j=0 k=o
follows. This implies the lemma.
[]
Estimate of
II~xp (t~+~m)  T~+q(A)[[ is reduced to those of 7j(z) and Fj(z) in this way. For the former we have the following. 4.11. Let (P, S) have property (III)o, and suppose (II)o for 0 < Izl < and z c Eo with t~ > O. Then, each ~' e (O, min(~, 1/Ibql)) admits constants C > 0 and /3 > 0 satisfying Proposition
for Izl < ~' and z E Eo. Proof." As we have seen, the roots Q(z) (1 _< j _< q) of P ( ( ) + inequalities
ICj(~)l _< ~.,zj
z S ( ( ) = 0 satisfy the
(1~1 _< ~', z e r~0)
for some ~ > O. This implies
IO,n(Z)l < Ce znlzl
(Izl < ~', z e Eo)
by L e m m a 4.9. Here, the constant C > 0 depends on % ,  . . , ~[q1, which are polynomials o f / ) 0 , " " , (~q1. The latters are bounded on Izl _< ~' < 1/Ibql, hence C > 0 also is bounded there. Recall t h a t aq 1. The proof is complete. [] :
As for Fj(z) we have the following.
4. Other Methods in Time Discretization
160
P r o p o s i t i o n 4.12. /f (P,S) is of order p(>_ 1) and ec' < 1/Ibq] , then the inequality
IFj(z)l ~ C [zlp+I ejRez holds for Re z >_ 0 and Iz <_ ~'. Proof: Letting v(t) = e tz, we have F J ( z ) : ;~"Si(z)e(J+i)Z:(aq+bqz)l
{ ~~:o  ~ " a i v ( j + i )  ~  ~i:o "biv'(j+i)}
Here, (P, S) is of order p, and tile righthand side is equal to
{~~ fjj+l(J"~it)Pv(P+l )(t) i=, a~ p!
(aq t bqz)i
~~lJ+i(J~it)Plv(p+l) ( p  1)!
dt  i=, bi,y
In fact, we have
Aai 
fj+~
Jj
(j + p! i  t)Pv(P+l)(t ) dt
t=j+i
=
,,
 
(t)
t=j
g=O
~.v(e)(j) + v(j + i) e=O
and
B~ =
jj
j+i (j + i  t) p1 )! v(V+l)(t) dt (p 1
[ ~(j+it)elv(e)(t)]e=l ( f  1),
t=j+i _ 
ie1
_ ~
~v(e)(j)
+ v'(j + i).
e=l ( e  i)!
t=j This implies q
q
i=1
i=0

e=l
C!
i ea~ 
i=0 q
= E i=0
by {4.12}.
{aiv(j + i )  bv'(j + i)}
g
b~
i=O
dt
}
.
4.2. Multistep Method In
use
161
of
_Fj(Z) = (aq F bqz) 1
ai i=1
p! JJ
•

(t) dt bi
i=1
with
v(p+l)(t) = (z)p+le tz,
we
)!
JJ
v(P+l)(t) dt
(p 1
}
get the desired inequality.
[]
Note that if bq > 0, we can take ec' < rc and a = oo in Propositions 4.11 and 4.12, respectively. Now, we can give the following.
Proof of Theorem ~.8 for the second case: Take A C
/1//1, rain
, n,
The rational function r(z) satisfies
I~j(z)  ~Jz I < c I~" for 1 < j < q  l , z I < a, a n d z E Ee with s o m e a > 0. Recall (4.16). In the second term of the righthand side, we have
I')'nk(z)hjk(z) [e jk  rJ(z)] l <_ C Izl p e z(~k)lzt
(4.17)
for 0 < k _< j < q  1, Izl < a0 = m i n ( a , ~ ' ) , and z E Ee. On the other hand, given M E (M1, ~'), we have
I'Yn~(z)hj~(z) [~jz
~j(z)]l_<
C~,(~~)lzl
forao <__ ] z ] _ < M a n d z E E o . We take a path of integration Fo = F1 U F2 U F3 with 01 E (0o, 0): pl

{ZpetzO,
I
0 ~ p ~ O0T1} ,
r~ =
{z = ~•176
F3 =
{A4T l e zq~ I I~1 _< 01 }"
~o~1< p_< M~~},
It holds that
"/nk(TA)(~Jk'(zA) [etjA rJ('rA)]
=
27c~
=
I+II+III.
1 (]; 1
+
fr 2
+
d / r3)
"Ynk('rz)(~jk.('rz)
[e j~z rJ(Tz)] ( z I 
A) 1 dz
4. Other Methods in Time Discretization
162
Each term admits of the following estimates for n = q, q + 1,."
II111
<
c fo ~ ~Z(nk~'o(W)~2 = C(ndp
k) p < Cn p
I
IZZll
<_ c
IlzIzll
< c
__ _ eZ(nk)wdP  c
o~~
_pdp <_Cn off(nk) P e
P
p,
eZ(nklMd~< Cn p. 9
01
The second term of the righthand side of (4.16) is estimated as
[q1 j
Z Z ~,,_k(~A),~j_,~(',A)[~,jA _ ,.j(.,_A)]
~ CTt p.
j=0 k=0
To estimate the first term, we make use of Propositions 4.11 and 4.12. The inequality
Is j=o
<_Cne~nlz',zP+l
follows for z I _ < M a n d z E E 0 .
In use of
j :0
1ItI"2 =
~OfF ) s
(zZA) dz 1
3 j=0
IV+V,
we get
II/Vll
<_ c
ne_~nrP(7.p)p+l do = C r _ p P
~0 ~176
and
IlVll
< c 
_01 .
0,
~.~~""
M 
~
M1
dw < o~~. 
The first term of the righthand side of (4.16) is also estimated as
[["
~_ CTz p.
j=O
The proof is complete. W h e n (P,S) has property (III)o, we can replace min(,~., 1/Ibql) by 1 / b q in Proposition 4.11, and accordingly, min(5, 1/[bql)by min(5, 1/bql ). Similarly, min(~, K, 1/Ibql)is replaced by min(~, K) in tile case of bq > 0 from tile reason described after the proof of Proposition 4.12.
4.2. Multistep Method
163
Now we give the following.
Proof of theorem 4.8 for the first case:
Because (P,
ICj(z)l _< ~z~zf
S) has
property
(I)o, the
inequality
( z l _ ~', = e zo)
holds for arbitrarily large ec' > 0 with s o m e / 3 > 0. T h e conclusion of Proposition 4.11 still holds for any ec' >> 1. M a k i n g / 3 > 0 smaller, we have
~j(z)l _< ~~ < ,
( =1_ ~', = ~ x0),
I~'~(~)1 _< c~ n~
(z >~', z~Eo)
(4.18)
so t h a t
follows as in the proof of Proposition 4.11. To estimate the first t e r m of the righthand side of (4.16), we take M > r IIAII and represent n
1
+ ~r ) s %_j(rz)Fj(rz)(zIA)
E %_j(rA)Fy(rA) = j=0
2 7"i~
i nt_l,2
3
1
dz
j=0
= I+II. Similarly to the case (b), we have
IZll _< o~p with the constant C > 0 independent of M. If j _> 1, we get
Vn_j(~z)Fj(~z) (~Z A) 1 & <_C
~("J)eM~+'~jMc~176
3
01
of which the righthand side goes to zero as M + oo. To estimate the term for j = 0, we make use of the a r g u m e n t s in the preceding section. First, we note lim Fo(z)= Izl~oo zEEo
bo bq
and b0
M a k i n g / 3 > 0 smaller if necessary, we have e/3n
ITn(z) "Tn(oo)l _< C ~
Iz
( el >_ R, z e r,o)
4. Other Methods in Time Discretization
164
for R > 0. Actually, from the proof of Lemma 4.9 this is reduced to e~n
I~j(z)" ~j(oo)nl ~ c ~
where ~j(z) (1 _< j _< q) denotes the root of P(r ~(z) o  ~(oo).l
:
(1=1 _> R, z ~ ~o),
Izl
+ zS(r
,~3(z)  ~ ( o o ) ,
(4.19)
= 0. Inequality (4.18) implies
n1 . }2 ~;~~(z)~y(oo) k:0
<_ ~"~ I~j(~)  ~ j ( ~ ) t , where
arises because (P, S) has property (I)o. Inequality (4.19) has been proven. On the other hand, If0(z)l _< c and
IF0(z) F0(oo)l
ao  bo/bq 1 +bqz
+ ~q
C t Ce Izl cos01 77, Izl
~(z)~ ~
i=1
are obvious. We obtain b0 <_ C e  ~
%~(z)ro(~) ~.(oO)Vqq
(1
)
~
+ etzl c~176 .
The first term of the righthand side of (4.16) is estimated as
irj=0a by Lemma 4.6. We proceed to the second term. Recall (4.17) with
js ~ e_~,~Opp dp = Cn_ p P We note
+ I~._k(oc)~j_.(o~)] ]~J(z) ~J(o~)].
(4.20)
Here, 0 _< k _< q  1 in this term and inequality (4.19) assures e~n
17,~k(z) 7n~(oo)l < C ~
tzl
165
4.3. Product Formula for Izl ~ ~0 and z C E0. Furthermore, 3' and (~jk satisfy C (~jk(z) (~jk(oo)l _< 757 Izl
and
I t ( z )  r(oc)l _<
C
Izl
for Izl _> a0, respectively, because they are rational functions. We also have 17n(CC)l < Ce zn
and
I~nk(z)l _< C.
Summing up those relations, we see that the righthand side of (4.20) is dominated above by CeZn/Izl. The second term of the righthand side of (4.16) is estimated as q1
j
E E 3`nk(TA)(~Jk(TA)(et'A rJ(TA)) <_C/n".
j=0 k=0
The proof is complete.
4.3
Product
Formula
From the classical theory of Lie groups, it holds that one paraeter families of subgroups {exp (  t A ) } t > o and {exp (  t B ) } t > o satisfy
)in lim [r ( t ) ( o r n,+c~ nt m
=exp(t(A+B))
where r = exp (  t A ) and r = exp (  t B ) . H.F. Trotter extended this property to (Co) semigroups on Banach spaces. It has been called Trotter's product formula. Error estimates in the operator norm were known for bounded operators, but only strong convergence has been discussed for the other cases. However, D.L. Rogava has succeeded in giving them for analytic semigroups. We shall describe the simplest case. T h e o r e m 4.13. Let A1 and A2 be positive selfadjoint operators on a Hilbert space H satisfying IIAll/2etA2All/21 < C
(4.21)
for t e [0, J], and suppose that A = A1 + A2 is selfadjoint with D(A) = D(A1) N D(A2) and furthermore, D(Aal/2) C D(A~/2) N D ( A 3/2) holds. Then, T ( t ) = etA2e tA1 satisfies the estimate
sup
tC[O,nJ]
IIe tA  T ( t ) n l l
with a constant C > 0 independent n = 1, 2, 3 ,   . .
< C~ lx/'n
(422)
4. Other M e t h o d s in T i m e Discretization
166 This t h e o r e m bounded domain "10a = 0, and A2 We take a few
is applicable, for example, if A1 is the differential operator  A on a ft C R n with s m o o t h b o u n d a r y 0f~ provided with the b o u n d a r y condition = V ( z ) >_ 0 with V ( z ) sufficiently smooth. preliminaries. Let
A =
fO ~
AdE(A)
be the spectral decomposition of a positive selfadjoint A operator in H. t >_ 0 and c~ E [0, 2], we have
First, given
OG
[(I + t A ) 1  e tA] A ~ =
[(1 + tA) 1  e t~] (At) ~ d E ( A ) , t ~.
Because sups>0 I[(1 + s) 1  e ~] s~l < +oc, this implies
II [(I +
t A ) 1  etA]
A II
ct~.
(4.23)
(s > 0).
(4.24)
We also have tile elementary inequality (1  e  S ) '/2 <_ s 1/2 + 1
To see this, let f ( s ) = s  S / 2  ( 1  e  S ) '/2. We have limst0 f ( s ) = + o c and l i m ~ + ~ f ( s ) =  1 . To show i f ( s ) < 0 for s > 0, suppose the contrary. The case i f ( s ) = 0 is equivalent to
g(t)  t 2  t a  31ogt = 0
fort=e
s/3 > 1.
However, this is impossible because g(1) = 0 and
g'(t) = t  2 ( t 
1)(2t 2 + 2 t 
1) > 0
for t > 1.
Inequality (4.24) implies (1  e  ~ ) 1 d lE(A)'u 2
=
<_
/0
<_
((tA) 1/2 + 1) 2d E ( A ) u 2 =
IIA / ul
(t1/2A1/2 + I ) u l 2
+ I1~11)2
We obtain (4.25) forttCHandt>0. Now we show the following. w
167
4.3. Product Formula
L e m m a 4.14. The operator W ( t ) = e tA  etA2e tA1 satisfies
IIAT1/=W(t)A~/=]I< Ct=
(4.26)
for t c=_[0, J]. Proof: We have
W ( t )  [e tA  (I it tA) 1] Jr[(/t
t A ) 1  ( I
+ tA2) 1 (I t tA1) 1]
+ [(I + tA2) 1  e tA2] (I + tAl) 1 q e tA2 [(I + tA1) 1


etAil.
The first term of the righthand side is estimated by (4.23) as
IIATX/2[etA(I+tA)l]Aa/2[[ IlA71/2AI/2[I II [e tA  (I nt ~A) 1] A2I[ ~_~C~;2. Here, the inequality [IA1/UAT~/211 < C follows from Heinz' inequality and D(A~/2) C D(Aa/2). Similarly, the third and the fourth terms of the righthand side are estimated as IlAT1/2[(I+tA2)letA2](I+tA1)lAa/2[I
_<
IIAI~Ag'=[I II[(, + ~A~)I  e~=] A;=ll
and
IIAT1/2etA2[(Ilt;A1)letA1]A3/211 IIAll/2etA2All/21t [[[(/ttA1) 1  e tA1] A1211 llAal/2A3/211 ~ Ct 2, respectively. Finally, writing (I + tA) 1  (I + tA2) 1 (I + tA1) 1 = (I + tA) 1. tA2 (1 + tA2) 1 tA1 (I + tA1)  t , we have
<_ I{ATI/=AI/]I I[(tA) The proof is complete.
(Z + tA)lll [leA= (Z + tA=)*ll lllllAT' A []
168
4. O t h e r M e t h o d s in T i m e Discretization
We are r e a d y to give the following. P r o o f of T h e o r e m 4.13: we p u t E(t)

e tA
L e t t i n g t = n r , U ( n ) = e nrA, U1 = e  r A ' , and U2 = e rA2,

(
e tnlA2
)~  
. e tnlA]
g(n)
(82gl)

n
and W = W(7) = e rA  e  r A 2 . e rA' T h e n , it holds t h a t
E(t)

8281. W . U(?z  2) Jv ...Ji (U2U1)n1. W . U(O) 1) 4 U2U~/2. (I 81) 1/2" V. U(Tt 2) nt  ' ' " "Jr U 2 U ~ / 2 . S n  2 . ( I  81) 1/2 V. U(0),

U1) 1/2
=
W U ( n  1) +
=
WU(n



where V = (I
U~/2W
and
We o b t a i n
n1 E(t)
=
W U ( n  1) I U28~/2 ~
S k1
(I

81) 1/9" VU(Tt  1~ + 1).
k=l Now we shall show
li~ 1(, ~,)1,~ I < ~
1~,,~
~4~/
for k = 1 , 2 , . .  . In fact, t a k i n g A / + 6 for A/ and m a k i n g 6 ~ 0, we may suppose t h a t A/ k 6, where i : 1, 2. In this case, we have
I1~' ~1)1'~1l< (' ~'~/1'~
'~,J <~"
~
'~'J ~ ~'~
For s = e ra C (0, 1), we o b t a i n
I1~~' (, ~l)"'l < II~,ll~1 Ill, f,~' I1(1 ~')"~11 _~ 82(k1)(1 Here, f ( O ) = / ( 1 )


8) 1/2 ~ 8k1(1
= O, and m a x f (s) = f (So)
sE[O,1]
holds for So = (2k  2 ) / ( 2 k  1) C (0, 1). We get f ( s ) <_ (1  So) 1/2 = ( 2 k 
1) 1/2


8) 1/2 ~ f(s).
169
C o m m e n t a r y to Chapter 4
and inequality (4.27) follows. On the other hand, we have U 1/2 (I  U1) 1/2 (TA1) 1/2 ~ ~o~176z~ (1  eTA) 1/2 (TA)I/2dE(A)
with SUps>o e s/2 (1

eS)1/2 s 1/2 < +oc. This implies [IuI/2(IU1)I/2(TA1)I/2[]
<_C
and hence [IV. g ( n 
1)[[
k
[IU1/2 ( Z  Ul) 1/2
(TA1)I/2II.T1/2IAll/2wA3/2II [[A3/2U(TLk1)[[
G C (n  k) 3/2
(4.28)
follows for k = 1, 2, , n  1 if t E [0, nJ]. Inequalities (4.27) and (4.28)imply
I
2u 1
.
k=l <
.
.
. n1
g2ll Igll1172 c . ~
i,i[ k  l / 2 ( n  k) 3/2
k=l 1
9n 1 = C n 1/2
ft k=l because
1 ~l (k)1/2 (
k) 3/2 ..~
n k=l
n
1
J0 1
x1/2( 1 _ x)_a/2d z ,.0
()n
1/2
_1
The proof is complete.
Commentary
to C h a p t e r 4
4.1. This section follows the description of [148]. For later developments, see Crouzeix, Larsson, Piskarev, and Thom6e [100], Palencia [312], and Sava% [338]. See also the monograph Thom6e [383]. The notion of Aacceptability was introduced by Dahlquist [104], and studied by Ehle [118], axelsson [16], Cryer [103], Dahlquist [106], and others. For the equivalence between (4.1) and (4.2), see Yosida [410], Kato [205], and so forth. Recall that i f  A is a generator of a contraction semigroup, then constant C in those inequalities is taken to be 1.
4. Other Methods in Time Discretization
170
General rational approximations of (Co) semigroups were studied by Hersh and Kato [174] and Brenner and Thome~ [61]. Inequality (4.3) holds if r(z) is of order p and Aacceptable. In this section, we applied the method of Baker, Bramble, and Thom~e [25] for the approximation of holomorphic semigroups. A related work was done by Bramble and Thom~e [58]. The notions (i)o, (ii)o, and (iii)o were introduced by [148]. The RungeKutta method is regarded as a rational approximation of semigroups. Stability and error analysis were done by Crouzeix [97], Keeling [215], Ostermann and Roche [311], Lubich and Ostermann [254], [255], and Nakaguchi and Yagi [275] including the case applied for quasilinear equations. The fact that the Pad~ approximation r = R,.m of e ~ is An/2acceptable if and only if m  2 < n < rn was conjectured by Ehle and proven by Wanner, Hairer, and Norsett [402]. Related works are Birkoff and Varga [36], Ehle [119], and Saff and Varga [329]. Pad~ approximation Rnn(Z) can cast the implicit RungeKutta method of Butcher [69] applied to du/dt + Au = 0. The use of continued fraction expansion of the exponential function was proposed by Mori [270], where Aacceptability of Hn(z) is proven. It has also the practical use. Baker, Bramble, and Thom~e [25] showed Theorem 4.1 for the case that A is selfadjoint under weaker assumptions on r(z). The first case of this theorem is due to LeRoux [234]. Large constants in stability or error estimates can cause unreliabilities to the numerical computation. For this topic, see Lenferink and Spijker [232] and the references therein. 4.2. The first case of Theorem 4.8 was proven by LeRoux [234]. We have followed [148] for the notions (I)o, (If)o, and (IlI)o. Multistep methods for ordinary and partial differential equations were also studied by Dahlquist [105], Zl~imal [421], Nevanlinnca [293], Raviart [324], Crouzeix [98], and Palencia [313]. 4.3. Product formula for (Co) semigroups was introduced by Trotter [388]. Nelson [292] gave a different proof making use of the Feyman path integral. Chernoff [78] extended the formula to a general principle as follows: Let {F(t)}t> 0 be a family of nonexpansive operators and {etC}t>_ 0 a (Co) semigroup on a Banach space X. Suppose that )~ > 0 admits slira [I + At 1 ( I tl0
F(t,))] 1 :r = (I + )~C)  1.'/;
for each :r E X. Then, it follows that slimn_.~ F(t/n)'~z = etCz locally uniformly in t E [0, oo). Later it was generalized to nonlinear semigroups by Brezis and Pazy [64]. See w for details. On the other hand, Kato [206] studied unbounded perturbations, and Feit, Fleck, and Steiger [123], [124] applied the formula to find eigenvalues numerically. Related works were done by Chernoff [79], Kato and Masuda [208], and Kuroda and Toshio Suzuki
[228]. Concerning the case of bounded operators, see M. S~zuki [362], [363], [364] and the references therein. R.D. Rogava's paper is [327]. There, O ( l o g n / v ~ ) is asserted under the assumption of D(A1) C D(A2), instead of assuming (4.21) and D(A3~/2) C D(A32/2) A D(A3/2). Related works were done by Ichinose and Tamura [186], [187], [188].
Chapter 5 Other M e t h o d s in Space D i s c r e t i z a t i o n
Continuous problems have several fine structures and it is desirable to realize them in the discretizing process. In this chapter, we describe three techniques of FEM; lumping of mass, upwind difference, and mixed elements. The first two are concerned with the dissipative structure, and the last one is related to the variational structure. Those methods are efficient particularly in nonlinear problems as we shall describe in the following chapter. In the last two sections, we describe the method of boundary elements and that of charge simulations. Both of them are associated with fundamental solutions for specific operators and reduce the quantities of computation in some cases.
5.1
L u m p i n g of M a s s
We suppose that f~ C R 2 is a polygon. In Chapter 2, we studied the finite element approximation of the initial boundary value problem
utAu=O
i n f ~ x ( 0 , T)
with
Ulo~ = 0
ult= o = Uo(X).
and
This problem, however, is provided with the dissipative structure such as the maximum principle and the L 1 contraction. Those properties remain valid even in nonlinearly perturbed problems and control basic features of the solutions. Numerical schemes kept with those properties, therefore, are desirable from both theoretical and practical points of view. Method of lumping of mass realizes them.
S c h e m e of
Lumping
Recall that {%} is a family of regular triangulations of gt, and Zh the totality of vertices subject to Th locating in Ft. The set of continuous functions, linear on each T E Th and taking the value 0 on the boundary, is denoted by Vh. For a E Zh, the function Wa E Vh is given by 171
5. Other Methods in Space Discretization
172
10 (x = a)
Wa(X) =
(X e Zh \ {a}).
The family {Wa J a C 27h} forms a basis of Vh and the interpolation operator rrh" W + Vh is defined by
7ChV= E
v(a)Wa,
aEZh
where W = {v e C(U) Jv = 0 on Of/}. Each a C Zh takes the barycentric domain denoted by D~. It is nothing but the closed convex hull of barycenters of the triangles in rh containing a as their vertices. Let 1 0
~,(x) =
(xeDa) (xe~\D~).
and denote by Vh the vector space generated by {~a I a C 2"h}. The linear transformation Mh " Vh +Vh, sometimes referred to as the lumping operator, is defined through Wa ~+ Wa. Let M;: 9Vh + Vh be the dual operator associated with the L 2 inner product, and
I ( h = M;: Mh" V~ ~ Vh. Given u0 E W, we can show that the scheme d ~KhUh + AhUh =
0
with
Uh(0) = 7rhuo
(5.1)
in Vh has the desired properties, where Ah indicates the finite element approximation of  A . In the weak form, it is written as d
dt (gh, wa) + (VUh, Vwa) = 0
with
(Uh.(0), Wa)
=
(7rhUO,Wa)
for a E Zh, where 'uh = MhUh. The operator Mh is an isomorphism and Kt71 = M,71 (M,7:)* is welldefined. Problem (5.1) is equivalent to d'It h
d7 + K~'A,,~,,,
:
0
with
u,,(O) :
~o.
There are works by T. Ushijima concerning L 2 and L ~176 convergences of approximate solutions. Here, we study L 1 and L ~ structures of (5.1) in details. In this section, Xh and Xh denote the spaces Vh, and Vh equipped with the topology induced by Ll(f~), respectively.
5.1. L u m p i n g of Mass
173
/2stability Before doing so, we show L p stability and convergence. First, we have a constant C >_ 1 independent of h satisfying
c'
IIM~x,~ll,.,,,(~) <
xh
I~,(~)_< c IIMhxhll.,(r)
(s.2)
for Xh E X h , T C rh, and p C [1, co]. This implies
(s.a) In fact, take the canonical reference element T of which vertices are P0(0, 0), PI(1, 0), and /:'2(0, 1). Inequality (5.2) is then reduced to the case T = T from the regularity of {%}. Linear functions on T, provided with suitable boundary conditions, form a finite (up to three) dimensional vector space and the desired estimate follows because any two norms are equivalent there. Furthermore, because the extremal cases p = 1 and p = oc are valid, that inequality is uniform in p E [1, oc] by RieszThorin's interpolation theorem. This gives (5.2). It is possible to evaluate C _> 1 explicitly by elementary calculations in use of the least angle of the vertices of T for instance, and is left to the reader. As a consequence, lumping operator is shown to have the following property, where p E [1, cxD]: Uh e X h +U
in LP(t2)
=:::a
To see this, take g > 0 and w E W satisfying
M,~
Jlu
MhUh + U
(5.4)
in LP(t2)
wJls,(a) < e. In use of
 ~ = M~ [~,  ~ , ~ ] + ( M , ~ , ~  ~) + (~  ~).
we have
IIMhUh  UlIL~(~) < c (llu,~  ~ I,..,,(~) + c + I1'~  ",,~'~ll~oo(~)) + IIMu,,,,.~  wll,r.,o~(~)
9
Here, w is uniformly continuous on f~ and we obtain lim sup I l M h u h  Ullr,(a ) _< Ca. hi0
Because e > 0 is arbitrary, we get the conclusion. Now we study the uniform boundedness of Kh " X h + X h with respect to t h e / 2 norm. Taking Xh E X h and v E W, we have
_< II~hllLp'(~)" IIMh~vlILp(~)<II~IILp'(~)IIP, vlIL~(~)
5. Other Methods in Space Discretization
174
with 1 / p ' + 1/p = 1, where Ph'L2(f~) ~ Xh denotes the L 2 projection. If {7h} satisfies the inverse assumption, {Ph} is regarded as a family of uniformly bounded operators on LP(gt) by Theorem 1.5. In this case we have
IIPhvll~.(~) _< c IIv
(5.5)
~,(~)
for p E [1, cx~]. The inequality
I(M;N,,..'v)I _< c ~hll~,,(r~)IIv
L,(~)
follows and hence
IIM;~,.II~p,(~) _< C I ~,.11...(~) is obtained. Combining this with (5.5), we have
I K.~.II~(~) _< C l ~ l ~(n)
(xh e x~).
(5.6)
If p = 2, the inverse assumption is not necessary for this inequality to derive, because IPh r2(n),r2(n) < 1 always holds true. The L 2 projection P h " L2(f~) ~ Xh has the same property and we have I](Kh) 1X~hIIL~(~)< C IX~hllL~(a)
(X~h C X~)
(5.7)
if {Th} satisfies the inverse assumption, and without it if p = 2.
Maximum
Principle
In the following theorem, the inverse assumption is not necessary. 5.1. If any T C Th is an acute or right triangle, scheme (5.1) is provided with the maximum principle so that
Theorem
0 <_ fh E Xh
==~
etg':'n"fh >_ 0
(5.8)
holds for the associated semigroup {etK;i ' Ah }t>_O on Xh. To see this, let Bh be the set of vertices subject to rh locating on Cgf~. Even for a E Bh, we can define a continuous function Wa on f~, linear on each T E ~t~by I
Wa(X) '~
0
(x = a) (:1;, E •h U ~h \ {a}).
Then we have
E aEZhUBh
wa(a)= 1
(z E~).
(5.9)
5.1. L u m p i n g of Mass
175
We can also take the barycentric domain Da even for a 9 •h" Taking Wa as
Wa(X) =
(xeDa) (X 9 \ Da)
1 0
similarly, we have
Z Wa(X)1
(x 9 ~).
(5.10)
aE:ThUI3h Let a, b E Zh t2 Bh with a =fl b. It is obvious t h a t
(Wa, Wb) "0.
(6.11)
(V~a, VWb) <_0
(5.12)
The inequality
is a consequence of the following l e m m a proven similarly to L e m m a 1.9. Lemma
5.2.
We have VWa" VWb ~ 0
(5.13)
on each T E ~h if a, b C ffh U Bh and a sC b. Proof: I f a E Z h U B h is not a v e r t e x o f T E ~h, then Vwa = 0 o n T and (5.13) is obvious. Let a be a vertex of T and Ta be the face of T not containing a. Because the lefthand side of (5.13) is invariant under the r o t a t i o n of the variable x, we may assume t h a t T~ is located on the line x2 = 0 and a lies in {x2 > 0}. In this case OWa/OXl = 0 and hence 0w~ 0Wb V w ~ . Vwb = Ox2 " Ox2 follows. Let {Pj [ 0 < _ i < _ 2 } be the vertices o f T with P0 = a. Because T i s a r i g h t or acute triangle, the vector e = ( 0 ,  1 ) is expressed as 2
e = ~_s aj eoj j=l
with some aj >_ 0, where e0j = P 0 ~ . j = 1, 2. It holds t h a t 0W a
Ox2
Here e 0 j . Vw~ is a negative c o n s t a n t on T for
2
= e . VWa = ~ a j e o j j=l
9VWa <_ O.
5. Other Methods in Space Discretization
176
On the other hand we have e0j 9Vwb _> 0 (j = 1, 2) if b r a, and
Owb > 0 O3g 2

holds similarly on T. Therefore, inequality (5.13) follows and the proof is complete. We are now ready to give the following.
Proof of Theorem 5.1: Note that (5.8) is reduced to A > 0, fh E Xh
===> m_ax(Ih + AK,71Ah) 1 fh _< max{0, m_ax fh}. fl
(5.14)
f2
Here, the welldefinedness of (Ih + AKs
1 is also included. In fact, (5.14)implies
fh <_ o
~
(I,, + ~ K s
f. <_ 0
fh > 0
==~
(Ih + AKs
1 fh >_ 0.
or equivalently,
Then, tile fundamental relation
etKs
Ah
=
lim (Ih + tn 1Ks lAb) n 71,'+00
assures (5.8). L e t [ . ]+ = m a x { . , 0 } .
(5.1,5)
Property (5.14)follows if max_ Uh _< max_ rrh [fh]+
(.5.16)
is proven for Uh C= Xh and fh = (Ih + AI(s In fact, then fl + = 0 implies u h+ = 0 and hence fh = 0 gives 'uh = 0 is obtained. This means the welldefinedness of (Ih + AI(,71Ah)I " X,, +Xh and also (5.14). For (5.16) to prove we may suppose that the maximum of the lefthand side is attained at some a E :2h with a nonnegative value, because the righthand side is nonnegative. Suppose those situations. It holds that
(a~, ~h) + A (v,,,h, Vx~) = (L, ~,~) for Xh E Xh. Let Xh = IDa"
(a~, tea)+ A (v,,,,,,, W*,a) = (/,,, r&) We have (gZh, g&) = uh(a)Isupp Wal
and
(fs ~ )
= fh(a) supp ~ l .
(5.17)
5.1. Lumping of Mass
177
Furthermore, the equality (VUh' VWa)

E Uh(b) (VWb, VWa) bEZh
=
~
(~(b)  ~ ( a ) ) ( W ~ , W a ) + ~.(a) ~
bEZh

(W~, W a )
bEZh
~
(~th(b)  ,/th(a))(VWb, VWa)  "/th(a) ~
bEZh\{a}
(V~Ub, Y e a )
bEI3h
holds by (5.9). Because a E/h attains a nonnegative maximum of uh we have (Vuh, VWa) ~ 0 by (5.12). Those relations imply
m~x_ ~ = ~(~) <_A(a) <_m_~xfZ gt
or equivalently (5.16). The proof is complete.
Llcontraction Now, we proceed to the following. T h e o r e m 5.3. Under the assumption of Theorem 5.1, scheme (5.1) has the property of
L 1 contraction expressed as O <_ fh E Xh
=:~
f MhetK[lAhfhdx
(5.18)
where {etK: 1mh}t>_O denotes the associated semigroup on Xh. Proof: Similarly, property (5.18) is reduced to ~ M~ (Zh + )~KhlAh)l fh dx < faM~fh dx
(5.19)
for 0 <_ fi~ E Xh and )~ > 0. We have only show it for fi~ = Wa with a E Ih. Writing
Uh = (In + AKhlLh) 1 Wa = E
~bWb~
bEIh
we have ~b ~ 0 by (5.14). Relation (5.17) with A  Wa and )(~h  wc is expressed as E
~b { (Wb, We) ~ /~ (VWb, VWc)} : (We, We),
(5.20)
bEIh
where c E :Zh. We shall show that
E = ~ M h ( I h + ) ~ K h l L h ) 1 Wa d x 
= ~
~ bEIh,CEBh
c~ (w~, w ~ ) .
MhWa dx
(5.2~)
178
5. Other Methods in Space Discretization
This implies (5.19) by (5.12) and ~b _> 0. In fact, we have
b6 Ih
b6 Ih ,c6 Ih U Bh
b61 h ,c6 Bh
c6 Ih U Bh
c6 B h
b,cE I h
by (5.10) and (5.20). Applying (5.9) for the last term, we get
bE Ih
cE Bh
cE Bh
This means (5.21)by (5.11).Property (5.19) has been proven. Llestimate
In connection with the L 1 structure, elliptic estimate (1.69) is also worth noting. We shall study its discretization here. We emphasize that the convexity of f~ is not necessary in the following theorem. Furthermore, not all T E rh must be acute or right triangles. Remember, however, that {rh} is supposed to be regular. T h e o r e m 5.4. If {7h} satisfies the inverse assumption, then q E [1,2) admits C > 0 satisfying
IIA;~K,,PhlI~I(.),..,,~(~) <
(s.22)
c.
If any corner point of [2 does not have the angle 27r furthermore, the convergence lira hi0
IlAhlI(,~Pj~v 
Alv[Iwl,q(n)
= 0
(5.23)
holds for v E LI(f~), where Ph denotes the L 2 projection. Proof: Because {Th} satisfies the inverse assumption, {Ph} is regarded as a family of uniformly bounded operators on L2(f~) and we have (5.5). If we apply (5.6) for p = 1, inequality (5.22) is a consequence of (1.69). Taking v E Ll(f~), we get it as
Turning to the proof of (5.23), we note that it is reduced to limhlo IAh:I(hPhv  A lvllH:(a )  0
(v E L2(~)) .
(5.24)
5.1. Lumping of Mass
179
To see this, let us admit (5.24) for the moment. Given v E Ll(f~) and e > 0, we can take w E L2(f~) satisfying IIw  VlILI<~) < c. In use of (5.22), (1.102), and (1.69), we have
IIA;lKhPh~ Aiv
wl,q(f~)
]As
_<
(v  w)[lWl,~(~) +
IIA~ ( w  ~)1 w~,~<~)
<_ C (c + IlA~'KhPh~  A  i ~ l l / r ( a ) ) . Therefore, letting h I 0 and e ; 0, we get (5.23). We now prove (5.24). Taking v E L~(a), we have [ A h l K h P h  A 1] v
=
[ A h l p h  A  I ] W + A h l p h ( I ( h  y) Phv
=
( R h  I ) A  i v
+ R h A 1 (Kh  I ) Phv,
where Rh" V ~ Xh denotes the Ritz operator and relation (1.24) is m a d e use of:
Rh A1 = Ah 1Ph Because v E L2(f~), it holds t h a t A  i v E V = H~(f~). We have lim hlO
II(R
 i) A  i v
V
= 0
by (1.39). On the other hand we have IIRhllv, v ~ c , and therefore the relation lim IIR~A 1 (K~  I ) P , ~ l l v hi0
= 0
follows from lim IIA 1 (Kh  I ) P ~ l l ~
= 0
hi0
(5.25)
Thus, we have only to derive (5.25) for ( 5 . 2 4 ) t o prove. We shall show wlim (Kh  I) Pt~v = 0 hlO
in L2 (f~).
(5.26)
T h a n k s to the assumption to f~, the operator L 1 9 L2(f~) + U~(f~) = V is compact. Relation (5.26) implies (5.25), and in this way, (5.24) is reduced to (5.26). For the proof of (5.26), we note t h a t (5.6) with p = 2 implies
II(Kh  I)PhlIL=<~),L=<~) ~ C. Therefore, (5.26) is reduced to the case of v E W similarly. In fact, assume (5.26) for v = w E W. Given v E L2(t2) and c > 0, we take w C W satisfying
5. Other M e t h o d s in Space Discretization
180 We put Jh =
(Kh 
I) Ph for simplicity and take u E
L2(f~).
T h e n we have
I(&v,~,)l < II& ( v  ~)IIL=(~)II~llL=(a) + I(&w,,~)l _< g e Ilull~=(~) + I(Jhw,~)l. Letting h + 0 and g + 0, we get (5.26) for v E L2(f2). a similar argument now reduces (5.26) (for v E W ) to
(5.27)
lim ((Kh  I) Phv, w) = 0 h,t O
for w C W. We have only to verify (5.27) for v, w E W. For this purpose we note t h a t
((ir
 ~) e~,,, ~)
=
(M;M,,P,,v, ~)  (e~v, w)
=
(MsPhv, MhPhw)  (Phv, w)
=
(M,,Phv, MhP,, (I  re,,)w) + (Mhre,,v, M h r e , , w )  (Phv, w) + (MhPh (I  re,,)v, Mhrehw).
For the first term of the righthand side, we make use of (5.3) and (5.6) as
IIM,~PhvlIL,(~) < c IIP,,v gl(~'2) ~ C II,, L1(~2) and
IIMhPh (I

reh)w
IL=(a) < IIPh (I  ~h) WlIL=<~> ~ C I1(1  ~h)W L=(~),
respectively. It follows that
I(MhPhV, MhPh (I  reh) W)[ < C l v l l z x ( a ) [ ( I  reh) ~IIL~(~) ~ 0 from the uniform continuity of w c W . Similarly, we have
I(MhPh (I  re,,)v, Mhrehw)l
,0.
For the rest terms we make use of (1.40): lim liP,,,, vl e=(~) = o hlO
and also lim []Mhrehv V[[f~o(a) = lim [[MhrehW  W [L~(a) = 0, h~0
h~O
which is a consequence of the uniform continuity of v and w. We get that
I(A6,ro, v, MhrehW)  (Phv, w)l   ~ 0 as h i 0, and relation (5.27) has been proven.
[]
An analogy of the above theorem is verified for schemes without lumping more easily.
5.2. Upwind Finite Elements
5.2
181
U p w i n d Finite E l e m e n t s
Continue to suppose that Ft C R 2 is a polygon. Let b(x) be a Lipschitz continuous vector field, and f, g scalar fields on t2, respectively. The problem Au+b.
Vu=f
inf,,
u=g
oncgf~
(5.28)
describes the phenomenon of convectiondiffusion, so that u(x) is subject to the flow generated by b(x) and diffuses by itself. In this sense, the system keeps basic features of the problems in fluid dynamics. If the convection term is stronger than that of diffusion, then the standardized m e t h o d of w is not efficient. Here, one sees t h a t the value of u at x is under the influence of its values upwind with respect to b(x). Upwind finite element method is based on this observation and we describe the simplest case in this section. We make use of the following notations of the previous section. Let {Th} be a family of regular triangulations of acute type so that any T E Th is an acute or right triangle. The set of piecewise linear continuous functions on f~ is denoted by Wh, and Vh = Wh M V, where V = H0X(f~). The sets of interior and boundary vertices are denoted by 2h and Bh, respectively. Each a E :ZhU Bh admits We E Wh with the value 1 at a and 0 at each b C :ZhU Bh \ {a}. Interpolation operator 7rh'C(gt) ~ Wh is defined by
rrhf= E
f(a)Wa"
aCZhUBh
We also make use of the lumping operator Mh " Wh + Wh and set MhXh = Xh in short. Given a C 2h, we take T(a) E rh which meets the vector b(a) with the origin a, and set
Bauh = ~
(b(a).
VUhlT(a) ) Wa.
aEIh
We study the scheme to find Uh E Wh satisfying Uh  r~hg E Vh and ( v u , . Vx ) +
=
(5.29)
for Xh E ~ . We can show the following Theorem
5.5. Suppose one of the following conditions:
1) Each a E Ih admits a sequence a l , . . . , am C Ih, am+t E 13h with ai+l is a vertex of T(ai) for 1 <_ i < m. 2) Each T E Th is an acute triangle.
Then, (5.29) is uniquely solvable and the discrete maximum principle f, g < 0
follows. (f, Xh).
~
uh < 0
(5.30)
The righthand side of (5.29) may be replaced by the term without lumping,
5. Other Methods in Space Discretization
182
Discrete maximum principle (5.30) can imply the L ~176 bound of the Ritz operator (1.59) and also IIu uhliL~(a) = O(h), although details are not described here. If p (~h, Xh) is added to the lefthand side of (5.29), the conclusion of Theorem 5.5 holds without the assumption, where p > 0.
Proof of Theorem 5.5: From the Fredholm alternative, the solvability follows from the uniqueness in (5.29). The latter is a consequence of the discrete maximum principle. We have only to derive (5.30), supposing that 'uh E Wh is a solution of (5.29). It is proven similarly to (5.14) and we shall describe the sketch. Because uh = 7rhg on oqf~, we have only to show urn(a) 5 O, supposing that m a x g u h is attained at a E Zh. Let Xh = w~ in (5.29). We have
(w,,,, v x , ) > 0
(5.31)
(BhUh, Xh) = supp Wa] (b(a). Vuh)r(a) > 0.
(5.32)
and
Because the righthand side is nonnegative, the lefthand sides of (5.31) and (5.32) are equal to 0. If the first assumption holds, then a~ attains maxg'uh for i = 1, 2 , . . . , m + 1. This implies m a x ~ u h _< 0 by 'uh = rrhg _< 0 on oqf~. If the second assumption holds, we have (VWa, Vwb) < 0 for a : / b . Then, the inequality (5.31) implies that 'uh is a constant on f~. Then m a x g u h _< 0 follows similarly. To prove the latter part, we take A > 0 and u~, E Wh satisfying u h  rrh9 E Vh and
(~h, ~,~) + a (v,,,,h, vxh) + (~h,,,,, ~,~) = (~Tf, ~,~)
(5.aa)
for )C~ E Vh. Without those assumptions, we can show m_ax a uh _< Inax { m_ax a rrhf, I ~ X rrh } .
(5.34)
In fact, if a E 5~ attains max~u,, we have (B,,u,,, ~,,) >_ O. Then, relation (5.33) implies (5.34). The proof is complete. []
5.3
Mixed Finite Elements
The m e t h o d in consideration is concerned with the variational problem with constraints. To describe the idea, we take the Dirichlct problem for the Poisson equation Ap = f
in [2
with
p = 0
on cgf~,
(5.35)
where f~ C R 2 is a polygon. Putting 'u = Vp, we llav(; u
Vp = 0
and
V . ',, = .f.
(5.36)
183
5.3. Mixed Finite Elements
Let V = L2(f~) 2, W = H~(~), and B ' = V : W ~ V. The dual operator B = ( B ' ) ' : V "~ V' , W _~ W' is defined through the representation theorem of Riesz. Similarly, f 6 L2(ft) determines 9 e W', regarded as an element of W, by (g, q)w = (f, q)L2(fl) for q 6 W. Under those preparations, (5.35) has an abstract form, to find (u,p) 6 V x W satisfying
u + B'p = 0
B u = 9.
and
(5.37)
As' we shall see, this problem is realized as an Euler equation on u for a variational problem with a constraint, where p acts as a Lagrangian multiplier. Then, it is transformed into another variational problem without constraints, where the solution (u, p) is to be a saddle point. Method of mixed (hybrid) finite elements arises naturally as a discretization of those structures. Although it had been applied to many problems in engineering related to solid mechanics, hydrodynamics, electromagnetic theory, and so forth, the discovery of such a fine structure made it much more reliable practically. This section describe the fundamental ideas.
Abstract
Theory
We take the abstract theory first. Let V and W be real Hilbert spaces, and ,4 and B be bounded bilinear forms on V x V and V x W, respectively. We have
1"t4(u, ~)1 ~ C1 for u , v E V, where C1 > 0 is a constant. variational problem with constraints,
II~lIv IIvll~
Given g c W and F E V', consider the
(5.39)
inf u ,7, where U = {v E V l B ( v , q ) =
(5.38)
(9, q)w for any q E W} and
y(~)
= ~
A(v, ~)  F(~)
The bounded linear operator B : V ~ W is defined by
(B~, q)~ = z(~, q)
(q ~ w ) .
We have U = {v E V I B y = g} and hence U # 0 if and only if g 6 Ran B, which means the existence of Uo E V satisfying Buo = g. Then it holds that U = {u0} + Ker B. It is a closed atiine space in V and the s t a n d a r d argument described in w the condition
A(v, ~ ) ~ ~ I vll~
(v
e
Ker B)
applies. If
(5.40)
5. Other M e t h o d s in Space Discretization
184
holds with a constant a > 0, variational problem (5.39) is uniquely solvable. The solution u C V is characterized by u
u0 E U
and
.A(u, v) = F(v)
(v C Ker B ) .
(5.41)
Let c~ : V' + V be the canonical isomorphism defined through Riesz' theorem, and A : V + V the bounded linear operator associated with A:
A ( ~ , ~) = (Au, ~ ) ~
(~, ~ 9 v ) .
Tile second equality of (5.41) is expressed as
(oF
Au, v)v = 0
(v 9 Ker B ) ,
or equivalently, a F  A u E (Ker B) • in V. Here, the dual operator of B is realized as B' : W + V through Riesz' theorem: V' " V, W' _~ W, and then (Ker B) • = Ran B' follows. Under the assumption that Ran B' : closed
C V,
(5.42)
u solves (5.41) if and only if it admits p E W satisfying
A u + B'p = o F
B u = g.
and
(5.43)
Note that p acts as a Lagrangian multiplier here, and (5.37) obeys a form of (5.43). If one prefers to tile weak formulation, (5.43) may be replaced by
.A(u, v) + B(v, p) = F(v)
B(u, q) = (g, q)w
and
(5.44)
for a n y v C V a n d q C W . Problem (5.43) is reduced to another variational problem without constraints. In fact, (u, p) solves (5.43) if and only if it is a stationary point of 1
,]('u, q) = ~A(v, v)  F ( v ) + (q, B v  g)w
(5.45)
on V • W. Actually (u, p) is a saddle point,. To see this, take q E W and v C V. If (u, p) solves (5.43), then we have 2(,,,, p)  2(,,,, q) = (~,  q, B,,,  0 ) ~
= 0
and
! A ( . . , .~,) ~A(.,,,, . . ) 2
F(.v
u)+
(~, B ( v 
~))~ 1
= A(.,~, v  u)  r(,,,  ..,) + (p, B ( v  ..,))~ + ~ A ( v  .,,,, ,,,  u) 1 A ( , v  ,u, v  u) a F , v  "u)v + ~
=
(Au + B'p

_1 2 A (v  u, v _ u) > O.
185
5.3. Mixed Finite Elements
In particular, the relation 2 ( u , q) _< 2 ( u , p) _< ST(v, p)
(v e V, q E W)
(5.46)
follows as is expected. Relation (5.42) is equivalent to the existence of k > 0 satisfying
IlB'qllv =
sup ~o~
> kllq ~li~

for q C (Ker B') L. Because (Ker B') • = Ran B, this means sup
B(v, q) > k Ilq w
(q E Ran B).
(5.47)
v~.\{0~ II~ll. 
Here, the closed range theorem says that Ran B' C V is closed if and only if Ran B C W is so. Because (Ker B) c = Ran B', similarly it is equivalent to the existence of k > 0 satisfying
sup
qeW\{0}
15(v, q) > k II~llv q Iw 
(v e R a n / 3 ' ) .
(5.48)
We have the following. T h e o r e m 5.6. Let V and W be Hilbert spaces over R, and ~4 and 13 be bounded bilinear forms on V x V and V x W , respectively. Suppose (5.38), (5.~0), and (5.47) (or equivalently (5.~8)). Let A : V + V and B : V + W be the bounded linear operators associated with A and 13, respectively, and take g E Ran B. Then, the solution (u, p) r V x W of (5.~3) (or ( 5 . ~ ) ) exists, and is characterized as a saddle point of 2 defined by (5.~3). Here, u C V is unique, while p E W is unique up to an additive element in K e r B ' . Actually, p is taken uniquely in Ran B, and then we have Ilullv + IIPlIw < C (IIFIIv, + Ilgllf),
(5.49)
where C = C (5, C1, k) > 0 is a constant. Proof: From the above arguments follow the equivalence between (5.46) and (5.43), the existence of the solution (u, p), and the uniqueness of u. The orthogonal decomposition u = Ul + u2 with Ul C Ker B and u2 C Ran B' gives Bu2 = g, and assumption (5.48) assures
I1~11~ _< k'I Ilgll~. Furthermore, A u l = ~ F 
Au2  B'p and hence
A(ul, v) = f(?3)  A(u2, v) follows for v E Ker B. Putting v = ul, we obtain
II~lll. _< ~_1( FII., + 61 lu~ll.) _< ~1 (llrl ., + c~k' I gll~).
186
5. O t h e r M e t h o d s in Space Discretization
Similarly, the orthogonal decomposition p = Pl + P2 with Pl E Ker B ~ and p2 E Ran B gives B~p2 = o F  A u and Ilp211w <_ k  i (IiFlIv , }Ca IlulIv) by (5.47). Inequality (5.49) follows and p is taken uniquely in Ran B except for an additive element in Ker B'. T h e proof is complete. []
Poisson
Equation
Above t h e o r e m justifies ( 5 . 3 6 ) i n the following way. Let V = L2(f~) 2, W = H~(f~), A(u, v) = (u, v)c2(n)2, and B(v, q) =  (v, Vq)r2(n)2. The bounded linear operator t9' : I/V ~ V is realized as B ' = V. From Poincar6's inequality, IIB'qllv IIVqlIL~(n)~ provides a n o r m on W. Hence d a n B = w , Ran B ' = v , (5.47), and (5.48) follow. Given f E L2(f~), we set F = 0 and define g E W ' ~_ W by ( g , q ) w = (f,q)L2(n) for q E W. The abstract theorem assures a saddle point (u, p) E V x W of 2(v,q)=
1
~11 v 2L2(n)2  (v, Vq)c:(a)2  (f, q)L:(n)
satisfying (5.43). We have (U, V)L2(f~)2 ~ ( V p , V)L2(12)2
and 
('a, Vq)L2(fl)2 : (f, q)L2(n)
for any v E L2(gt) 2 and q E H~(f~). This implies p E H~(f~), Vp = u E H I ( ~ ) , and V u : f E L2(ft) 2. In particular, p E H2(f~)VI H~(ft) is a solution to (5.35). P r o b l e m (5.35) is formulated differently in this framework. Let V : H(div, f~):
H(div, ~) : {~ c r~(~)~ I v . ~ c r~(~)} It is a Hilbert space with the inner product
(~, V)v : (u, ~)~(~) + ( v . u , v . ,v)L~(~) 9 Taking W = L2(ft), we set A(u, v) = (u, 'V)L2(fl)2 for u, v E V, and B(v, q) = ( V . v , q)r2(a) for v E V, q E W. This time tile b o u n d e d linear operator B : V + W is realized as B = V., and we have Ker B = {v E V I V . v = 0}. Condition (5.40) holds as A(v, v ) =
IIv ~/
(v E Keg B ) .
We shall show t h a t inequality (5.47) holds for any q E W = L2(~). This implies Ran B = W in particular. In fact, take (b E Hg(f~) satisfying V  V c b = q. We have [IV011L=(a) _< C IlqlIL=(a) and hence v = Vq5 E V satisfies v 2V  I V r IL2(~) 2 _ ( C 2 + 1) Iq IL2(fl) 2 2 [ ]lV " V 4) 2g2(fi) < 9
187
5.3. M i x e d F i n i t e E l e m e n t s
T h e n (5.47) follows as
z3(~ q) ,
=
I~lIg
llqll ~
w
1
>
IlVllv 
Ilqllw.
v/C 2 + 1
Let F = 0 and g = f 9 W = L~(f~) in (5.45). There exists a saddle point (u, p) 9 V x W of d(v,q)
1
2
: ~ Ilvll L 2 (a)2
t
(q, V " v  f)L2(a)
Relation (5.43) reads;
 (~, ~ ) ~ ( ~ ) : (p, v 9~ ) ~ ( ~ ) and ( V . u, q)L2(a) = (f, g)c2(a), where v C H(div, f~) and q C L2(f~) are arbitrary. This implies f = V  u , p E H2(f~), and Ploa = 0. In particular, p E H2(ft) A HI(f~) solves (5.35).
Vp = u,
The Stokes System In the above problem, f~ may be an arbitrary b o u n d e d Lipschitz domain in I~n. We have an i m p o r t a n t example formulated as above for this case; the stationary Stokes system described as Au+Vp=f
and
in f2
V.u=0
(5.50)
with u = 0
on 0~.
(5.51)
Here, u and p are vector and scalar fields on f~, respectively. For V = HI(f~) n and W = L2(f~), its weak form is introduced; to find (u, p) E V x W satisfying
~(
V u <9 Vv) d x  (p, V . v)r2(n) : ( f , V)L2(a)n
(5.52)
and
(v.~,
q)~(~) = 0
(5.53)
for any v E V and q E W. Here, OU i z,3
OV i
5. O t h e r M e t h o d s in Space Discretization
188
for u = (u i) and v = (vi). We introduce the b o u n d e d bilinear forms
A(~, ~) = s ( w  w )
dx
and
and take F ( v ) = (f, V)L2(n). and 9 = 0. Condition (5.40)is a consequence of Poincar6's inequality. To examine the closedness of Ran B or Ran B', let q E Ker B'. This means (q, V 9V)L2(a ) = 0 for any v E V. The space Ker B' is composed of constant functions and hence Ran B = (Ker B') • = {q E L2(~) I (q, 1)L2(n) = 0} follows. If 0f~ is Lipschitz continuous, it is known t h a t any q E Ran B admits v E V satisfying Vv = q
Ilvll~ <
and
C llqlIw,
(5.54)
where C > 0 is a constant determined by f~. In this way condition (5.47) holds true. In use of
IVvll~= ~
~
g~<~)
1,3
for v = (vi), problem (5.50) with (5.51) is formulated as (5.44), or finding a saddle point of d ( v , q)  ~1
Vvll~
(f, V)L2(n), 4 (q, V ' V ) g 2 ( n )
oil H~(f2) n x L2(f~), where T h e o r e m 5.6 is applicable. The variational structure induces the discretization. Under the assumptions of Theorem 5.6, we prepare a family of finite dimensional subspaces of V x W denoted by {Vh x Wh}h>0. Assume the property that any (v, q) E V x W admits a sequence {(vh, Wh)} in (Vh, Wh) E ~ X Wh satisfying lira (ll v  vhlIv 4IIq  qhlIw) = o. td.O
The linear operator Bh : Vh* Wh is defined by
13(vh, qh) = (S,,vh, qh)w for vh E Vh and qh E Wh.
5.3. Mixed Finite Elements
189
It is reasonable to assume (5.40) and (5.47) with V, W, and B replaced by Vh, Wh, and gh, respectively, with constants ~ > 0 and k > 0 independent of h: inf ~or~\~o~
r
(5.55)
~h) ~___5,
I1~11~
inf
sup
B(Vh, qh)
> k.
(5.56)
Then, RitzGalerkin method (5.44) is realized as to find (Uh, Ph) E Vh • Wh satisfying
.A(~h, vh)+ Z(vh,ph) = P(vh)
(vh ~ Vh)
(5.57)
and
s ( ~ , q~) : (g, q ~ ) ~
(q~ 9 w ~ ) .
(5.s8)
Those relations are equivalent to
AhUh + B~hPh = PhcrF
and
BhUh = Qh9,
respectively, where Ah : Vh ~ Vh denotes the linear operator associated with Alyhxvh, B~ " Wh ~ Ws ~ l/h ~ V/~ the dual operator of Bh " Vh ~ l/Vh, and Ph" V ~ Vh and Qh : W ~ Wh the orthogonal projections. This means that Qhg E Ran Bh is necessary for this discretized problem to have a solution. Then Theorem 5.6 applies and there exists a unique (uh, Ph) E Vh x Ran Bh satisfying (5.57) and (5.58). This solution is also a saddle point of ,7 restricted to Vh x Wh, with ,~ defined by (5.45). Based on those considerations, it is natural to assign the requirement QhRan B c Ran Bh. Taking orthogonal complements in Wh, this is equivalent to assuming Ker B~ C Ker B'. Under those circumstances the following theorem holds, where (u, p) denotes the unique saddle point of J on V x W belonging to V x Ran B. T h e o r e m 5.7. We have Uh]ly + l i P  Phllw < C ( inf Ilu  Vhl]y + inf liP qh]lw~ \ VhE Vh qh E Vh / with a constant C > 0 independent of h. Proof: Equalities (5.43) and (5.57)(5.58)imply A ( u  Uh, Wh) + B(Wh, p  Ph) = 0
and
13(u  Uh, rh) = 0
for any wh E V~ and rh E Wh. Taking vt~ E Vh and qh C Wh arbitrarily, we have A(v~  uh, wh) + S(w,~, q~  p~) = A(vh  u, wh) + S(w~, q~  p) and B(~

~,. <~) : s ( ~ .  ~. <~).
(5.59)
5. Other M e t h o d s in Space Discretization
190 or equivalently,
(5.6o)
Ah(Vh  Uh) + B'h(qh  Ph) = PhA(vh  u) + PhB'(qh  p)
and s,,(v,
 u,,) = Q,S(v,,
(5.61)
u).
Theorem 5.6 is applicable to (5.60) and (5.61) with V, W, A, and B replaced by Vh, Wh, Ah, and Bh, respectively. We have I~h  ~hl v +
Iqh  p , llw
<_ c tlPhA(v,,.  ~)llv + C ( PhB'(qh  P)I v +
QhB(v,,. ~) ~) _< C ((tlmll + IIBII)Ilvh **llv + IIBII IIq,~  PlIw),
and hence II?t
~
'/t h v + I p

PhlIw 
I1~,  v,,. v + I p

qh
_< c ' (11,, vhllw + l i p 
w
+
Iv,,.  ",,,,. Iv + Ilqh

p,,.
w
qhllw)
follows. Thus, inequality (5.59) has been proven. Although it is not easy to construct such a family {Vh x Wh}h>0 with the required properties in T h e o r e m 5.7, many ways are known now.
5.4
Boundary
Element
Method
(BEM)
If one makes use of the fundamental solution, then tim boundary value problem is reduced to an integral equation on the boundary. Boundary clernertt method arises, associated with that structure. Let f~ C R n (n = 2, 3) b c a bounded domain with Lipschitz t)oundary 0f~, and consider tile boundary value problem A'~J = f
in [2,
'u g
on ~)f~.
The fundamental solution
1 F(:r) =
~
1
(rz 2)
log T~
1
47r [:,:[
(rz = 3)
satisfies  A F = ~(0), where d (lcnot(.'s Dira("s delta function. If on(: takes w(:r) =
.[~ F(:r
:,j).f(y) dy,
(5.62)
191
5.4. Boundaly Element Method problem (5.62) is reduced to the case f = 0: Av=0
int2,
v=g
on0t2
(5.63)
The fundamental solution induces also single and double layer potentials
F(x) = foa V(y)F(x  y) dS v and
o
~ r ( ~  v) ds~
respectively. For ~ E 0t2, we put A+(~)=
lira
xCgt+~
A(x),
where ~+ = ~c and ~ _ = ~. If V is continuous, we have [F] + =_ F + 
F_ = 0
on 0~.
(5.64)
Given x c 0t2, we can take the principal value for F(x) to define. On the other hand (O/Onv)F(x .) is regarded as an Ll(0t2)valued continuous function of x E 0t2. Letting
Ho(~) = foa V(Y) ~~vF(~ Y) ClSy for ~ E 0t2, we have 2H+  i V + 2H0
on 0~.
(5.65)
In the Fredholm theory, property (5.65) is applied to reduce (5.63) to an integral equation on the boundary. In fact, we have v(x)=2foa#(y)o~v['(xy)
dSy
for x E ~, with # satisfying
#(~)  2 / o # ( y ) o ~ v F ( ~  y) dS v = 9(~) for ~ C 0t2. Linear operator K :
C(OFt) ~ C(Ot2) defined by f
K,
(5.66)
=
0
jo ,(v)~ r(.  v) dS v
is compact by AscoliArz~la's theorem. Unique solvability of (5.66) then follows from the Fredholm alternative; in fact 1/2 is not an eigenvalue of K.
5. Other Methods in Space Discretization
192 However, the simple iteration scheme #nI1  g ~ 2 K p n
(n = 0, 1, 2 , ' ' " )
does not always converge because  1 / 2 is now an eigenvalue of K. In particular, }~n(2K)nr does not converge for the corresponding eigenfunction r  1. To see this, let us apply Green's formula to Av=0
inf,,
Ov On =g
on Oft.
(5.67)
We get
~(*)=fo (~(v)~ r(*y)+v(v)r(*.~J))ds~ for x C f~. Therefore, from (5.65), it follows that
/o
/o
o
for ~ E 0f~. Because v = 1 is a solution to (5.67) for g = 0, this implies the assertion that  1 / 2 is an eigenvalue of K. The boundary element method avoids the difficulty in the following way. First, one derives from (5.63) that v(x)=fo n
O~F(z  Y) + ~~v(y) 9F(x _ (g(Y) On
y))
dS~
(5.68)
for x E ~. Then, (5.65) gives 1
~v(~) + f o n v ( y ) ~ F ( ~ 
~.])dS~ = fon J~v(y) . F(~ y) dS~.
If the integral equation ~o q(~)F(.(:t j ) d S y =
~f(~)+
Jo.f(:~j)~F(~y)dSy
(5.69)
in ~ C oqf~ is solved for q = (O/On)v, then formula (5.68) gives the solution v(x). This section is devoted to the wellposedness of the scheme; we show the unique solvability of (5.69). It was founded by several authors with the efficiency of the application of RitzGalerkin's or the collocation method. Let
and
A(p,q)= fo~.s ,, v(e)q(e)r(e ,),~,s~,s,, The following theorem indicates that the bilinear form A( , ) is realized as a bounded coercive bilinear form on H1/2(0f~) x Hl/'2(Of~). Then, the unique solvability of (5.69) is a direct consequence.
5.4. Boundaw Element Method Theorem
193
5.8. There exist constants C1, C2 > 0 determined by gt such that
c, Ilqll~H1/2(0~"~) __< A(q, q) _< 02
Ilqll 2u_,,,~(o~)
(5.7o)
forqEX. Proof: Fourier transformation of a rapidly decreasing function f C S ( R ~) is given by .T" = f (~) = f
e'Z~ f (x) dx,
n
where z = v/L~. Given q c X, we take Tq E D ' ( R ~) defined by (r
Tq) 
fOa qr dS
for r E D ( R n) = C~~ Its support is contained in O~. It is also regarded as a compact measure on R ~ and Tq is taken as a C ~ function on R n. Now we show
A(v, q)= (Z~)nJ~ 1___~(~)tq(~) d~
(5.7].)
for p, q E X, with the righthand side converging absolutely. In fact, if n = 2, we have f'({) = p.v. ~
1
+ c(~({)
(5.72)
with a constant c. From the assumption, we have Tq(0) = 0. Regarding P C 8'(R2), we can define P 9Tq C S'(R2). By (5.64), it is regarded as a continuous function on R2:
r 9T~(.) = f
JO
r ( ~  y)q(y)
dSy .
We also have
f (r 9T~) = t . ~q =
p.v. ~
1
+ ~a(r
}
Tq(~)
9t~(r = ir
Let ~b E $ ( R 2) be realvalued. The inverse Fourier transformation is given as
~(~) = (2~)~ a~f~ ~'~ ~(~) d~ and hence
f~ f0r(~y)q(y)dS~r
= (r,
Tq,r (5.73)
5. Other Methods in Space Discretization
194 follows. We take p C C ~ ( R 2) satisfying 0_
supppc{lxl<_l},
and
[ JR
2
p dx = l.
Let p~(x) = e2p(x/e) for e > 0. Given p c X, we put G = P~ * Tp. The lefthand side of (5.73) for r = ~ converges to (F 9Tq, Tp) = A(p, q) as e I 0. To handle with the righthand side, we shall show that 72 1 2
B~c~us~ Ir
IT, 9I~1 IT,I, th~n the righthand sid~ of (5.73) convcr~r to
i~l= q(~)" ~~p(~) d~ as e $ 0 by the dominated convergence theorem, and hence equality (5.71) follows. It is easy to see that (5.74) is reduced to the case p = q. Then we have
=
d~ + I_<~
I~l ~
1 ,>
12
d~ = I + II.
1~
The second term of the righthand side is estimated as
II<_2
~1+1~12 d~CllYqll2_,(~=).
Here, we have
[[Tqll/_/_I(IR2)  sup { <2/2,Tq) I w ~ HI(I~2), [Iwtti/l(iR2) ~__1 } .~ s u p { ~ n q v d S =
v E H'/2(Of~), ,.wl H,/2(on) <_ 1} (5.75)
]qlln'/~(on)"
On the other hand, we have ]7~q(() ]bq(0)] < ].~a (e .... ~  1 ) q ( x ) d S x _< [e 'x~  1[ ,,,/2(oa)[IqllH1/:(on) < Cl~l Those relations imply
<
ll ll "
Ilqll.,/~(0n).
195
5.5. Charge Simulation Method
Because ] = I for n = 2, the righthand side of (5.70) is proven. The lefthand side of (5.70) follows similarly from (5.75)as
I~
dg > ~f 
i5 dg ~
Ilq
I~~/=(o~) 9
~ 1+1~
If n = 3, we have 1
f'(~) = ~~ E L~oc(IR3).
(5.76)
It also holds t h a t [Tq(O)[ 1(1, Tq)l
~ O q [H_1/2(0~).
Noting those relations, we get (5.70) and (5.71) similarly. The proof is complete.
5.5
Charge
Simulation
Method
(CSM)
The m e t h o d in consideration is also concerned with (5.63) and make use of the fundamental solution F(x). We take N points y ~ ,  . . , YN outside f~ and the solution v(x) is approximated by
N
(5.rr) j=l
Here, the coefficients Q 1 , " " , QN are determined by
v N(xj) = f(xy)
(5.78)
for j = 1 , . . . , N , where X l , ' ' " ,XN are taken on 0f~. Sometimes {yl," "" ,YN} and { x l , . .  , X N } are called the charge points and the collocation points, respectively. In this section we study the case that gt is a two dimensional disc, ft = {x E R 2 I Ixl < P} with p > 0, and the charge and the collocation points are so located as yj = Re ~(j1)~ and xj = pe ~(j1)~ respectively, where z = x/~l, R > p and 0 = 2rc/N.
Wellposedness Equation (5.78) is expressed as G Q = f , where G = (gij) with gij = F ( x i  yj), Q = T ( Q 1 , ' " , QN), and f = T ( f ( x l ) , . . . , f(XN)). In the above case, it follows t h a t 1
gij = ~~ log p  Rez(Ji)~ I ~ Lji.
5. Other Methods in Space Discretization
196 Therefore, G is a cyclic m a t r i x given as
Lo LN1 G
L1 Lo
9
.
L1
L2
999 LN1 999 LN2 .
9..
Lo
and hence :
p=O \ k=O
follows for w = e *~ L e t t i n g N1
~;(z) = ~
~r
(z P~),
k=O
we o b t a i n N1
det G = I  I ~P(P)"
(5.79)
p=0
We can show the following. 5.9. It holds that
Lemma
(p = 0 mod N)
 ~ log Iz~  ~ 1 ~p (z) = I 1 N
for
Izl
1 ( ~z~lR)
Iml
Z TM
(otherwise)
(5.80)
< R.
This implies cpp(p) r 0 for p = 1 , . . . , N holds by (5.79). Theorem
5.10.
1, in particular. Hence tile following t h e o r e m
The matrix G is nonsingular if and only if RN

pN
?s 1,
(5.81)
in which case scheme (5.78) is uniquely solvable. Proof of Lemma 5.9: In the following, the equivalence class associated with m o d u l o N is d e n o t e d by "". If p = 0, then w p = 1. In this case, we have N1
qpp(z) 
1 27r Z
k=O
1
=
N1
1 log Iz  Rwkl  ~~ log 17I Iz  Rwkl
2; log I~~  ~ 1 .
k=O
5.5. Charge Simulation M e t h o d
197
In the other cases, p = 1 , . . . , N 
r(~R~
1, we have 1
~) =
2~ log ]z  Re,,,k ]
1 log ]Ra,,k] + log 1R2~1 (logR+Re ]og(l w  k z ) )
2~ 1
 R7
log R 
Re
27r

n1 n
Writing z = rP ~ we obtain
F (z
R w k)
=
n=l n
1 27r
log R  Re
1 27r
log/7E~nn n=l
1
w  n k e mO
~
R
(~,~o~~
+
~,~0~)
'
and hence
~(~) =  ~ 1 ~ ~
~1
log R  ~
k=0
~r
(e,,Ow_nk + e_~nOwnk )
n=l
follows. Because of N1
Z J~
{ N (e  o) =
k=0
0
(5.s2)
(otherwise)
we get
1
~(~)
=
4~
_
1 Z 47r
n=l
_1
r
~
~ _1 n
n>l
n
(eznOod(p_n) k +
e_mOcd(p+n)k )
k=0
r
n P n ~N + T~ 1
1 n
r
e'~~
n>l
pn=_O
p+n=O
(5),m, ,too 4 7r n GZ
~l
e
,
mp
and the proof is complete.
[]
5. Other Methods in Space Discretization
198 Convergence
Now we study the problem of convergence. Let f~ (n C Z) be Fourier coeMcients of f so that
f(x) = E fnem~ ncZ
I,fnl
holds for x = pe ~~E O~t. We suppose ~
+cr
<
(5.83) First, we note tim following.
nEZ
L e m m a 5.11.
We have vN(x) = ~ f,~n(p)l~,~(x) n,EZ
(5.84)
.for x E f~ under the assumption of (5.81). Proof." We take 1 N~
gij = "~ ~
~)p(P) 102p(ij)
p=0
and set d = (gij). Then, the (i, j) component of G G is given as N
E
1
LkiN
k=l
~gP(P)lcup(kJ)
=
p=0
=
1 N1
N
 E ~gP(D)I~ N p=0 1
~ k=l
N1
N E cdP(iJ) : 5ij p=O
by (5.82). We have (~ = G 1 and hence
D(I QJ
=
~=
=
ff
1/
N ~=0
~?p(p)la2p(j1) p=0
~p(ki)f
(t00)k1)
k=l
follows from Q = G  i f . Plugging this into (5.77), we get
~p(p)lcoP(J1) ~"~ (.x.)P(k1)f (/9(,,0kl)
uN(x) = ~ ~ j=l
p=0
1 ~
=
N
N
N z_., ~pp(p)l ~~ ~p0,) F ( x  R J  ' ) E p=O
j=l
k=l
1 g1
" N E
p=O
r (x/~(.,o jl)
k=l
~pef (owe)~p(X)
(,pp(p)i e=O
~p(ki)f
(pcdk1)
199
5.5. Charge Simulation Method by the definition of ~p(z). This implies
(;)
:
vN (x)
p,j =0
1 N
N1
N1
E (~gP(R)I(cgP(X)E fn E CO(np)J p=0
nEZ
j =0
N1
: E ~gP(P)199p(X)E fn
(5.85)
n=p
p:0
by (5.83) and (5.82). The function Cp(Z) is harmonic and hence
I~p(p) l~p(x) l _~ 1 holds. The righthand side of (5.85) converges absolutely and we obtain (5.11) by (~gq= (~gp for q _= p. The proof is complete. [] Equality (5.83) implies r ) bl znO e nEZ
for x = rP ~ C Ft. We obtain eznO
~(x) } ~(p)
The function eN(X) is also harmonic and hence
II~N .o(~) _
max
Ixl=p
I~,(~)1 ~ ~ IAI ~
(5.86)
nEZ
follows with an=
I
sup Iezn~ oe[o,2~) I
~,~(pe ~e) ~n(p)
Now we show the following. L e m m a 5.12. We have an = a _ n < m a x { 2 ,
1+
log log
(j~N __ fiN)
(587)
for n E Z and
NllogRI
(n=0) (5.88)
an ~_ 8n ( p ) N  2 n NnR .for N >> 1, respectively.
(n= 1,
,N~)
5. Other Methods in Space Discretization
200
Proof:
T h e relation ~n(z) = ~_n(z) implies a  n = N
1
an.
If n ~ O, we have
Iml
m~n
and hence / an _< sup  1 + oe[o,2~) \
~..
(pe~~
<2
~,,(p)
follows. If n  0, we have 1 I~,~(p~'~ < ~I log lp~e'N~ RN] I < ~1 flog (pN + RN) I.
This implies log
(R N k pN)
log
(~N
an_<1+
__
pN)
and inequality (5.87) is proven. To show (5.88), we note
a~ _< I~.(p)11 sup ]e'~~
~ (pe'~
oe[o,,) If n = 1 , . . . , N, we have N
~lml
N~ ~ 47rn '
where/3 =
p/R.
We also have
l~,,,o~,,(p) _ ~,, (p~,o) [
=
N _
7;7
(~ .. ~ 
Here, the r i g h t  h a n d side is e s t i m a t e d from above by
_ ~,(~+.)0) + 9~N" (e ....o _ ~_ ,(~N,~)o} )
N
~{
47r
e=l gN + n < N k(flgN+n
S ~§
(~ ....0

2rre= 1
=
27r(Nn)
gN+n +gNn =
N/3Nn < 2 ~  ( N  n )
gN n fleNn)
g N + n t3(e~2n
(1+
oo
)Z/3(eg=l
+g.Nn
1)N __ N[3Nn  27r(Nn)
1 +/3 2n 1  / 3 N"
5.5. Charge Simulation Method Because
/~N <
201
1/2 for N large, then we get 2N
~Nn
~(N~)
"
Inequality (5.88) for the cases of n = 1, 2 , . .  , N  1 is proven in this way. The case n = 0 follows similarly in use of the first equality of (5.80). The proof is complete. [] We are ready to prove the results on convergence. In the first theorem, the b o u n d a r y value f is supposed to be realanalytic. Therefore, the solution v of (5.63) has a harmonic extension to {x C R~ I Ixl < to}, where r0 > p. 5.13. Suppose R r 1 and (5.81). Then, there exists a constant C = C~,n,ro >
Theorem
0 such that
I vN ,11~(~)
~ c,~ sup I~(~)1, Ixl =To
wh ere
~=
{
~T;;o (~o < R ~) plR
(pro > t~~)
so that r C (0, 1). If R = 1, then v N(O) = 0 always holds, regardress with the values yy and Qj. Therefore, assumption R 7~ 1 cannot be eliminated. The following theorem deals with the case of more rough data. 5.14. Under the assumptions R 5r 1 and (5.81), the following results holds, where fn (n C Z) denote the Fourier coefficients of f indicated by (5.83):
Theorem
(i) If (ii)
~~n IAI < §
If fn =
O
t h ~ limN_o~ II~  ~11~(~) = 0.
(1~1~),
then
I ~  ~11~(~) = O (N~+'),
wh~
~ > 1.
Proof of Theorem 5.13: The function f is supposed to be realanalytic, and therefore, Cauchy's inequality indicated as
p) i~i sup ~(~)1
IAI _< ~0
(5.s9)
~,=~o
holds. Letting m = [N/2], we split the righthand side of (5.86) as
I1~11,_~(~)
m
_~ ISola0 +
~( n=l
SI ~ S2~ S 3 .
oo
AI + IS~l)an
+
E
(IAI + Ifhi)an
n=rn+l
(5.90)
5. Other Methods in Space Discretization
202
T h e first a n d t h e s e c o n d t e r m s of t h e r i g h t  h a n d side are e s t i m a t e d by (5.89) a n d (5.88), w h e r e a = R ~/(pro): $1
$2
8
<
( P ) N sup Iv(x)l ,

NJlogRI
<
Z

2
R
P
Ixi=ro 9
8n
N 
n=l sup
16 ~
p N2n n
v(z)l
Ixl=ro
sup Iv(~)l Ixl=ro
ct n n=l
N
<
ol 1 \ro/ 16 1  oz
Izl=~o
(P)x sup I~(~)1 ~
(~
Izl=ro
< 1).
O n t h e o t h e r h a n d , t h e last t e r m is e s t i m a t e d by (5.87):
s3 _< 2M~ ~
n=m+l
=
TO
sup I~(*)1
Ixl=ro
2Mn <~00) P m+l 1 1  ~ sup I~(~)1 ro Izl=ro
<

1  p
sup
ro
Ixl=ro
v(x)l.
If a > 1, we have p2/R2 < p/ro a n d hence
II<~ll~o~(r~) < _
~ c
~ '
Vo
N/2 +o
s
TO
+ c
p
sup Iv(*)l Ixl=ro
sup Iv(~,)l
Izl=,'o
follows. If a < 1, on t h e o t h e r h a n d , we have p2/R2 > p/ro a n d hence o b t a i n
Tile p r o o f is c o m p l e t e .
Proof of Theorem 5.14: We have I.fI  IlfllL~o(o~)
[] T a k i n g m = [N/3], we
estimate
Commentaly to Chapter 5
203
each term of the righthand side of (5.90) as follows: S1 < 
(fl)N
8 NllogR
R
s~ _<s~v (~p)~/3 ,
Ilflc~(Oa),
(9O
S3_<(I+MR) ~
(I/~l+fAn).
n=[N/3]
The last term is o(1) and O ( N a+l) according to the first and the second assumptions on fn, respectively, and the proof is complete. []
Commentary to Chapter 5
5.1. An important feature of hyperbolic or dispersive systems is the law of conservation. M. Mori and others tried to realize them in discretized schemes. The precise definition of the barycentric domain is as follows. Let T be the twodimensional simplex with the vertices P1, P2, P3. Then, its barycentric coordinate
{A(x", T, Pi)} 3i = l of a point P = P(x) C R 2 is defined in the following way: Put Pi = (x~, x~, x~) and set
(~) 1 A(x;T,P~)=~det
1 1 x I xl
1 x~
for x = (Xl, X2) and
A = det
1 1 1 x I x21 x31 x I x~ x~
Now, given a E Zh [.J Bh, we put Tta = {T E 7 h l a ~ T} and
rer~ Then, the barycentric domain Da with respect to a is defined as
Da = U {z E T I A(x;T,b)_< A(x;T,a) TeT~
(b E I~ \ {a})}.
5. Other Methods in Space Discretization
204
T. Ushijima's works on scheme (5.1) were published in Ushijima [396] and [395]. RieszThorin's interpolation theorem is described in [127] and so forth. As is described in the commentary to Chapter 1, see Brezis and Strauss [65] for the L ~ estimate of elliptic operators. For the fact that A ~ : L2(t2) ~ H(~(t2) is compact if t2 has no corner with angle 2re, see Ne~as [286] and Grisvard [164]. Results obtained in this section keeps to hold for the general dimensional case under the following condition for the triangulation. Given T E rh, let V(T) be the set of vertices of T. Furthermore, for a E V(T), let Ta be the n  1 dimensional simplex (i.e. face) of which vertices are composed of the elements in V(T) \ {a}. Write f(a,T) for the line containing a and perpendicular to Ta. Then the condition is described as
If n  1, this assumption always holds. If n  2, it is equivalent to saying that any T E % is an acute or right triangle. Here, n denotes the space dimension of the domain in consideration. See w for related arguments. Relation (5.15) is obvious because dim Xh < +o~. But it provides a key identity for E. Hille to construct (C0) semigroups on Banach spaces. K. Yosida made use of another aI)proximation for t h a t purpose, now referred to as the Yosida approximation. Some variants are made use of in w167 and 6.3. See also Hille and Phillips [179], Yosida [410], Kato [205], and Tanabe [378]. 5.2. The properties of L ~ stability and convergence were established by Tabata [373], [374] for scheme (5.29) with # (Eh, Xh) added to the lefthand side and without lumping in the righthand side , where # > #o = C(7)Ilbll 2L ~ ( n ) " Here, C(7) > 0 is a constant determined by the parameter 7 > 0 arising in connection with the regularity of {%}. See also [375]. 5.3. The inverse operator of the canonical injection a : V' , V is sometimes referred to as the duality map. Note that the bounded linear operator A : V , V is different from that of the previous sections defined through the Gel'fand triple V C H C V'. Variational problems of saddle point type arise in many areas. See Aubin and Ekeland [15], Rabinowitz [320], and Suzuki [369]. In connection with H(div, f~), the function space H(rot, f~) is also introduced when f~ C IR3. It is composed of the vector field 'v E L2(t2, R 3) satisfying V x v E L2(t2, Ra). See Girault and Raviart [160] for those spaces and applications. The existence of v satisfying (5.54) is also proven there. See also L e m m m a 7.32 in w Instead of (5.52) with (5.53), the Stokes system may be formulated as to find (u, p) E H(~(f~) n x L2(t2) satisfying 2~
(e~j('u), e,j(v))L~(n )  (p, V 9v)L:(n ) = (.f, v),:(n),~
i,j
for ~ c H ] ( ~ ) n ~nd ( V . ~,, q)~(~) = 0 for q c L : ( ~ ) , w h ~
1 (O'u ~ Ou~) r
= ~ \o.~, + ~
Colnmelatary to Chapter 5
205
This way is more natural from the physical point of view. Then, Korn's inequality takes place for Poincar~'s one to confirm the assumptions of Theorem 5.6. The continuous Stokes system and its finite element discretization were studied by many people, including Cattabriga [70], Kellogg and Osborn [217], and Crouzeix and Raviart [101], Bercovier and Pironneau [30], Glowinski and Pironneau [163], LeTallec [237], Stenberg [357], respectively. Mixed finite elements for magnetostatic and electrostatic problems were studied by Kikuchi [219], [220], [221], [222], [223]. Concerning the scalar equation, the Stokes system, and the equation of linear elasticity, see Brezzi, Douglas, Jr., and Marini [67], Raviart and Thomas [325], Crouzeix and Falk [99], Crouzeix and Raviart [101], Falk [122], and the references therein. Inequality (5.56) is referred to as Babu~kaBrezziKikuchi's infsup condition. It was introduced by Babu~ka [18], Kikuchi [218], and Brezzi [66] independently. Concerning actual examples of Vh and Wh satisfying the assumptions of Theorem 5.7 in use of finite elements, see Giraut and Raviart [160] and Brezzi and Fortin [68]. Under this condition, the discretized nonstationary Stokes system is treated similarly by the semigroup theory. See Okamoto [307] for details. The NavierStokes system is the fundamental equation in fluid mechanics. Operator theoretical approach for nonstationary problems was initiated by Kato and Fujita [207] and Fujita and Kato [143]. For other references, see the commentary to w Its finite element discretization was studied by Okamoto [308] and Heywood and Rannacher [175], independently. Error estimates given by the former are a priori; they hold only under reasonable assumptions on the initial value. The latter's are, on the other hand, a posteriori so that are valid under some additional estimates of the solution. Up to now, fundamental theory for this system is unsatisfactory, especially for the case of three space dimensions. A posteriori approaches are intended to compensate such a situation by numerical computations. See also Bernardi and Raugel [34], Heywood and Rannacher [176], [177], [178], and Rannacher [322]. 5.4. For layer potentials, see Courant and Hilbert [94], Kellog [216], and Garabedian [155]. Theorem 5.8 was proven by LeRoux [233] for n = 2 and Nedelec and Planchard [288] for n = 3. We have followed the method of Okamoto [309] for the proof, where q E X is taken in L2(cgf~). In the actual computation, the method of collocation is more realistic than that of Galerkin's. Iso [191] and Hayakawa and Iso [168] made error analysis for this method applied to the Neumann boundary value problem. Related works were done by Arnold and Wendland [13], [14]. Several monographs and surveys were published concerning the boundary element method. See Sloan [350] and the references therein. For the theory of distributions, particularly equalities (5.72) and (5.76), see Yosida [410] and Schwartz [348]. 5.5 CSM is also called the fundamental solutions method. Since the pioneering work by Bogomolny [41], the efficiency of the scheme has been clarified rigorously. Theorems 5.10 and 5.13 were obtained by Katsurada and Okamoto [213]. Theorem 5.14 was proven by Katsurada [210]. For other charge and collocation points, it can happen that the
206
5. O t h e r M e t h o d s in Space Discretization
coefficient matrix G becomes singular. Some examples are given in [210]. Problem (5.63) is invariant under the following transformation: (1) x H c~x, yj H c~yj, where c~ is a constant (2) f (x) ~ f (x) + c and v ( x ) ~ v ( x ) + c, where c is a constant. Scheme (5.77) has lack of such properties. K. Murota proposed N
j=l
where Q o , ' " ,
Q N are determined by VN(Xj) = f ( x j )
(j=
1,...,N)
and N
k=l
This scheme is provided with such properties. In this case, tile conditions R g  oN ~ 1 and R ~ 1 are not necessary for the wellposedness and the error estimate to hold. Katsurada [211], [212] studied the case that c%2 is a Jordan curve in use of the conformal transformation. An application to the free boundary problem was made by Shoji [351]. Furthermore CSM is a powerful method to obtain numerical conformal mappings. See, for this topic, Amano [6], [7], and a m a n o et. al [8].
Chapter Nonlinear
6 Problems
Many problems in applied sciences and engineering are formulated as nonlinear partial differential equations. Fortunately, in accordance with the progress of analytical theories, reliability on numerical computations has been advanced extensively. In this area we have several monographs such as Girault and Raviart [159], [160], Kf{~ek and Neittaanmgki [227], and Zen{gek [417]. This chapter selects the following topics: (1) iterative method for solving unstable solutions for elliptic boundary value problems, (2) qualitative features of finite difference solutions for semilinear parabolic equations, (3) finite element scheme for degenerate parabolic equations based on tile L 1 structure.
6.1
Semilinear Elliptic Equations
A typical example of the nonlinear elliptic boundary value problem is
Au
lul p1 u
in ~
(6.1)
with u = 0
on 0f~
(6.2)
for p E (1, ec). In this problem, u = 0 is the trivial solution and nontriviM solutions are linearized unstable so that the simple iteration scheme U k + 1 ~ (   Z ~ )  1
I~1 ,~ ~
(6.3)
hardly converges to them. Furthermore, the existence of nontrivial solutions depends sensitively to the domain: dimension, topology, and geometry. To fix the idea, let ~ C R 2 be a convex polygon. Then, it is known that problem (6.1) with (6.2) has infinitely many solutions. The functional
J(v)
3f~(1 ~IWl ~
1 p+l
I~
,p+) '
d~
takes place of the energy, and u is the solution if and only if it is a critical point of J on
v = H0~(~). A least energy solution denotes the one which attains the minimum of J on the set of nontrivial solutions. It is positive in g2 and attains d = inf J > 0, H 207
(6.4)
6. Nonlinear Problems
208 where
c~(a) Note that J(v) =
(1 1) ~v4r

Ilvl
Lp+l(f~)
IlWllL2(n) for v 9 H .
>0
"
We call (6.4) Nehari's variational
formulation and N" the Nehari manifold. By Sobolev's imbedding theorem we have inf I WilL2 > O, yEAr (fl)
(6.5)
and this implies d > 0. It is known that d is equal to the mountain pass critical value so that d is attained by a solution with the Morse index 1 if it is not degenerate. Subject to the underlying variational structure, the iterative sequence {uk}k=0,1,2,... in consideration is constructed on iV" in use of the scalar multiplication operator Q" V\{0} . Af. Actually, it is not hard to see that any v 9 V \ {0} takes unique t = t(v) > 0 satisfying Q(v)  t(v)v 9 N'; t h a t is,
t,_l = II Ilvl ,+1 Lp+l(gt)
Then the sequence is defined as Uk+l = Tuk for T v = Q (  A ) 1 Ivlplv and u0 9 V \ {0}. More precisely, Uk+l
~
tkwk 9 Af
with
tk > 0
and
wk = (  A ) ~ ]'uk p  1
Uk.
(6.6)
Here, the positivity is preserved so that if Uo _> 0, then 'uk > 0 for k = 1 , 2 , . . . This sequence is provided with several fine properties. First observation is the following. L e m m a 6.1.
We have t(v) <_ 1 and J ( T v ) <_ J(v)
(6.7)
for v 9 N'. Here the equality holds if and only if v 9 Af is a solution to (6.1) with (6.2). Proof: Take v 9 N" and set u = T v = tw, where Aw=],vVlv
inFt
with
w=0
inO~.
We shall show that t _< 1 and J(u) _< J(v). In fact, because of v 9 A/" and (6.8) we have
iwll~ ,+1 L~(~) = I1~1 L~, l(~) = ( v w , w )
< IlVw L~<~)IlWllL~(~)
or
IlVvllL2(~) _< IIW,,llL2(~) 9 Therefore, again by v E H and (6.8) we have
IIv~ll ~L~(~) = (1~1 ~' ~, w) _<1~1 ~ + , (~) w i l L . , (~) = IIw, IIh=(~) ~+' ~11..+,<~) _<
v~
" a=(~) Ilwll~. 1(a)
(6.8)
6.1. Semilinear Elliptic Equations
209
This implies o < IVwll ~
w
p+l
and tP1 __
IlWll~(~)
Ilwl'+l I LP+
<
1 (f~)

1.
1 Isl p+l is a convex function, the inequality Because s H F(s) = K+7 1
p + 1 ( r/
p+l

[~" p+l
1
) ~ 2 (?]2 __ ~2) ~[p1
holds for r/, ~ E R. We have
J(tw)
=
t2
~IlVwll ~L2(a)
tp+ 1
p+ 1
IIw''p+l IILp+~(n)
< t 2 {ll~Twl122(f~)(W2, IvIP1)} 'JF( 12

Here we have IlVwll ~~(~) 
1 ) ivlp+l p+l i ZL~+~(a)
2
(1~1~~ ~, w) <
Qj~
v ~ Ivl ~  '
)1/2 (/~)1/2
w ~ Iv ~_1
= IVvll~(~) (w ~, I~1~') 1/~ <_ IlVwll~(~) (w ~, I~ ,,1)1/2
(6.9)
and hence
IlVwll 2L2(f~)< (W2, I'Vpl) l follows. We have (6.7) as j(u) < (1 2 
1 p+l
)
tI~i~+l Lp+I(~) = j ( ~ ) .
Assuming the equality in (6.7), we get the equality in the second inequality of (6.9). This implies w  Av with a constant A. Then A  1 follows from v C Af, and therefore, v solves (6.1) with (6.2). The proof is complete. [] To describe the dynamics of the sequence {uk} defined by (6.6), we introduce its wlimit set as
in w
for s o m e
c
},
where W = {v E C (  ~ ) l v = 0 on c9t2}. The following theorem provides the reliability on this iterative method in the numerical computation.
210
6. Nonlinear Problems
6.2. The set W(Uo) is nonempty, compact, connected, invariant under T, and contained in the set of nontrivial solutions of (6.1) with (6.2). The sequence {[ Vuk[[L2(a)} is monotone decreasing and bounded from below by a positive constant. We have tk T 1 and distw (Uk, W(Uo)) ~ 0 as k , oo. Finally, the sequence {uk} is compact in W.
Theorem
Proof." The fact t h a t Tw(uo) C W(Uo) is obvious. Because of J(uk+l) _< J(uk) and uk C N" for k >_ 1, the monotonicity of [[Vuk c2(n) follows, while inequality (6.5) shows that it is bounded from below by a positive constant. We have Wk :
(_/~)1
and
l?2k p1 Uk
uk+l = tkwk
(6.10)
IIW~IIL~<~)
with 0 < tk _< 1 and = o(1). Then, the standard bootstrap argument gu~r~nt~s II~kllH~(~/ = o(1). Therefore, {uk} c WNH~(f~) is compact and in particular, w(uo) is a nonvoid compact set provided with the property that d (uk, w(u0)) ~ 0. II Any {u;} C {uk} admits a subsequence {u~} C {u;} and u E H~(f~) satisfying u k ~ u in H~(f~). This implies u E iV" and  i Uk,, ~i u"k
zk  (  A )
~
(A)
lul ~' u 
1
w
in H 2(f~). We obtain w 7f 0 and (t~)p_~
"
IVz~ll~(~/
=
,
ilzkl p+, gp+,(n)
I Vwll ~L2(fl)
p1
= t,
> 0
(6.11)
IIw p+l Lp+,
and klim ~oo
IIt k"z k

t,w
[H (a) = O.
This means Tu~
~
Tu
in H 1(f~).
Here, we have I!
J(Uk+l) ~_ J(Tu'k) by L e m m a 6.1. Sending k ~ oc, we get
J(u) <_ J(Tu). Then the latter part of L e m m a 6.1 assures that (6.1) with (6.2). In particular, w(u0) in contained furthermore, Tu = u. The last relation gives t, = It remains to show that w(u0) is connected in W. exists because of the monotonicity of IlVukllL~(a).
u E N" is a (nontrivial) solution for in the set of nontrivial solutions, and 1 in (6.11) and hence tk * 1 follows. First, we note that l i m k _ ~ Vuk IL2(a) Next, we have tk T 1 and hence
I1~7 (Ukt1   uk)ll ~L~(~  I l V u k + ~ l l ~L2 (gt)  2 ( W k + l 
,
Vu~) + IlVu~ll ~L 2 ( f ~ )
I v u ~ + , l l ~~(~>  2 ( V w ~ , v u ~ ) +
IlW~ll =L2(a) + o(1)
211
6.2. Semilinear Parabolic E q u a t i o n s
holds. Here, we have
(Vwk, Vuk)

I uk p+~1

V?~k
~ 2(a) L
so that l i m [IX7 ( U k + 1  k,oo
Uk)llL2(f~)  0
follows. In use of the bootstrap argument for (6.10), this implies lim Iluk+l  uk w  O.
k~oG
Now, suppose that a~(u0) C W is not connected. Because it is compact, then we have nonempty disjoint compact sets A and B satisfying cJ(u0) = A U B. If dist (A, B) = 35 > 0, we have N such that I]Uk+l ukll~y < 5 for k > N. From the definition, there exists kj ~ oc and k~ > kj satisfying dist (uk~,A) < 5 and dist (uk~, B) < 5. Therefore, there exists some kj" in k i < kj" <_ k~'i satisfying distw(uk~,, A t2 B) > 5. This contradicts the assumption, w(u0) = A U B. The proof is complete.
6.2
[]
Semilinear Parabolic Equations
Vast references are devoted to the qualitative theory on semilinear parabolic equations. Among t h e m are the studies on the behavior of the solution globally in time. In this section, we take their finite difference analogues. We study the blowup of the solution, taking in f~ x (0, T)
ut = uxx + u 2
(6.12)
with Ulo a = 0
and
ult= 0 = Uo(X),
(6.13)
where f~ = (0, 1) and uo(x) is a nonnegative continuous function on f~. For this system the classical solution exists locally in time. H. Fujita showed that it cannot be continued globally in time if the initial value Uo(X) is large in a certain sense. This case is referred to as the blowingup of the solution, because then limtTTm~x I1'~(., t)llL~(a) = + o c follows, where Tm~x denotes the maximal time for the existence of the classical solution. F . B . Weissler showed that if the initial value is symmetric with one peak, then the solution blows up at onepint, x = 1/2. Finite difference analogues for those facts are studied by T. Nakagawa and Y.G. Chen. We provide the uniform mesh for f~ denoted by /~ " X 0 =
0 ~
Xl
~
X2
~
"'"
~
XN

1
6. Nonlinear Problems
212
with xj = jh for j = 0, 1, 2 , . . . , N. Then, a piecewise linear function u n is identified with its values on the nodal points, u] for j = 0, 1, 2 , . . . , N. On the other hand, the time discretization is subject to Tn=Tmin(l
,
]]un]]LP(FI) 1 )
'
where T = Ah 2 with )~ > 0 and p E [1, oc]. Here, it should be noted that all (1 _< p _< ec) are equivalent for h fixed. Actually we have
[lun[IL~(a)
1
[lu"llL,(a) < Ilu"llLOO(a) < h  ? ]lU n I]L,(a)"
(6.14)
Then, the scheme in consideration is the finite difference m e t h o d (FDM), implicit in linear parts, n
. n+l
=
+ (u;) 2
he
Tn
(6.15)
for j = 1, 2 , . . . , N  1 and n = 0, 1 , 2 , .  . with
u~=u}=0
(n>0)
(6.16)
and
uj0 =Uo(Xj)
(l_<j < Nl).
(6.17)
Scheme (6.15) with (6.16) and (6.17) approximates the solution and the blowup time appropriately. We describe the qualitative features of the approximate solution in details. For this purpose we suppose that Uo(X) is symmetric with respect to x = 1/2 and N = 2m so that u 0 ( 1  x ) = uo(x) and Xm = 1/2. We also suppose that Uo(X) is not constant and monotone increasing in [0, 1/2]. Then it follows that
u~_j = u2
(j=l,2,...,m;
n=0,1,2,...)
and n
(j=
1,2,..,mi;
n=
1,2,..).
In particular, we have and
n ?~m
l  I n a x
n uj.
j~=m
We suppose that the solution of system (6.15) with (6.16) and (6.I7) blows up so that oo
lira
n* (x)
II'u'~IIL~(~) tc~
and
~
w,, < +o<~.
Then we have the following theorem. Note that'unm_2 = maxjr . . . . +1 u~.
213
6.2. Semilinear Parabolic Equations Theorem that
6.3.
Under those circumstances, three points blowup occurs to the solution so
lim urn_ ,n
~+OO

+oc
(6.18)
and limsupu~_ 2 < +oc
(6.19)
n+(X)
hold. Proof: Let "~n ~ % h 2. Then equality (6.15) is written as 9

Uj
k'u,j_ 1
"71 ' u , j + 1
) "JF Tn
(Uj)
,
or
(1 + 2An)ujn + l
__ /~n \U,j_ {~ n + l1 Jr 'C~j+ . n+l~ n n 1 ) + ?.l,j Jr Tn (Uj) 2
(6.20)
In particular, we have un+~ = /~n (?jn+_l + u n + l ) _~ 
[1 + Tnun_i] un_I
(6.21)
1 + 2A,~
and n+l n Unm+l  2/~n'U,m_ 1 I [1 H Tn,l~n] ~trn 1 + 2)~n
(6.22)
Those equalities imply u,~+ 11 > 

n "~nun+l It Urn1
n+l
and
l+2)~n
um
> 
n Urn l+2A,~
,
respectively, and hence
~u~/I ~IIL~(~) + (1 + 2.Xn)u~_ 1 (1 + 2:~.) ~
~:+1 >_ ~ . ~ + (1 + 2 ~ . ) ~ _ ~ (1 + 2)~n)2 follows. Because
b~ ~
, n ~m u~[ LP(fl)
>_1,
we have n
~21 >
,~ + u r n _ 1
(1 + 2s
2
6. Nonlinear Problems
214 so t h a t relation (6.18) follows as II
lira inf U mn _ n~oo
1
= lim inf u~ +11 _n 1. _ > ~ lira inf A + u,~_l ~_ A + lira inf um n.o~ n~oo (1 + 2An) 2 n~oo
In use of (6.21) and (6.22) we have
=
An (U~n+'l + U~n+') + [1 + %U~n_,] U~n_, 1 +2An (1 + 4An) Anun+21 Jr An [1 + "rnu~] u,~ (1 + 2A,~)[1 +
(1 4 2An) 2
+
TnU,nn_l] tt,nn_l
(1 + 2A,~)2
This implies l t n t 11 <

A~ [1 + %%~,] u ~ + (1 + 2An)[1 + 7,~un_l] u~_ 1

(6.23)
1 + 3An
From the assumption, we have I1..I1~.(~) > 1 for n large, and then T,, = r/I1~11~(~) follows. Letting an 
, n /tm_ 1
n
UTn
'
we have by (6.23) that u~+ll an+ 1

(0.24)
7tn+ 1
(1 + 2~,,)
2A,, + [1 + T,,'u~] l t mn ,/ l t mn+l  1
A. [I + ~,,,~] u~ + (1 + 2;~,,)[I + ~',,~,m_,] '~ Urn,

Irt
i,~
n
i
n t 2A,, [1 t T ,,.~m~ln 1 ] "ltm.(1 + A,,)[1 § T,u~] u m " 1
A (1 + rb,,) + (1 4 2A,,) [1 + rb,,a,,.] a,,
(1
+
~..) (1
+
~b..)
+
2~.. [1
+
(~.25)
~..~,...] a..
In other words, we have
*"('+'~") + (1 + 2A,,)[1 + rb,~a,~] a,,
an+ 1 ~
(6.26)
a,,.  (1 + A,,.) (1 + rbn) + 2A,, [1 + rb,~a,r a,. for n large. To get a reverse inequality we reduce (6.21) and (6.22) as )',n'~'n+ 1 . . m . {
1 JrTnH,,nn_ 1 ILm_ 1
1 + 2A,,. 2 .%u.,,._, 2, n+l
, n 'u;',,. § (1 + + A,,~ [1 + r,,um]
2,~,u)
[1 §
'" 1 Tn ll'Tn. 13"II'm
(1 + 2A,,.) 2
> 
A.. [1 + ~..,,........]. u .... " +(1+
2A,.) [ l + % u ' "..... , ] ' 'bl'm 1
1 4 4A,, + 4A~,
....
215
6.2. Semilinear Parabolic Equations Similarly, we obtain an+l
n Um1 un+l
=
(1 + 2/~n) un+21
~+~1 2,~ a m_
+
[1 + r
n um
a~ [1 + ~n~ n] ~ + (1 + 2a~) ~ [1 + ~ n ~ _ , ] ~ _ ,
>

(1 + 2A~)[1 + ~ , ~ ] ~m + 2A~ [1 + ~ ; ~  1 ] ~  1 an (1 + ~b~) + (1 + 2a~)[1 + ~b~a~] a~ (1 + 2An)(1 + 7bn) + 2An [1 + Tbnan] an
for n large. This means t h a t an + l ~
an
ar~
 (1 + 2An)(1 + Tbn) + 2An [1 + Tbnan] an
(6.27)
We can deduce t h a t {an} is m o n o t o n e decreasing for n large by (6.26). Actually, t h e n 2 {)~n (1 + 7bn)+ (1 + 2/~,~)[1 + 7bnan]an}{(1 + )~n)(1 + 7bn) an + 2~n[1 t 7bnan]an}
= )~n (1  an)[1 + Tbb ~ 2an (1 + Tbnan)] ~ Tanbn (an  1) is negative. Because
0 < an < 1
and
1 1 _< b~ _< b = h 5,
(6.28)
this t e r m is d o m i n a t e d from above by
7n (1  an) [Un_l ~ 3h 2 (1 nt Tb)]. We have proven (6.18), so t h a t there exists k satisfying k > Urn1  3h2(1 + Tb) > 2h 2. Relation (6.21) implies
un41m'i>  
1+
n
TnUm1 n UrnI"
l+2)~n
Therefore, we conclude Unmt21 ~ U n _ l ~ 3 h  2 ( 1 + Tb)
for n >_ k by an induction. T h u s {an}n>k is m o n o t o n e decreasing. Let a = l i m n _ ~ an C
[0,1). By (6.28), there is a subsequence {bin} C {bn} converging to some constant c _> 1. Then, (6.25) gives t h a t
a < 1 +7ca I+TC
.a.
6. Nonlinear Problems
216 This implies a = 0 and hence lim,~__.~ b,~ = b. Furthermore, lim An = lim "rbnh 2 ~~ a~ ~~o~
( [lt~m, _ l )
1
~ 0
by (6.18). In use of (6.26) and (6.27) we obtain lim bn  lim a~+l 1 E (0, 1). n<x~ an 1 + ~b
(6.29)
n*cx~
In particular we have oo
oo
Ea,~
< +ec
E(l+Can)<+oc.
and
n:O
n=O
Equality (6.20) gives ~
A,~(u,~223 +
=
urn_l) n+
1
+
(1 +
Tu~_2)urn_ 2 n
1 + 2A,~
~n ( < % + < % ) + (1 + ~,~_~)~_~ 1 +2A,~ and hence n+l .n+l
, n
~ 2
'tt, m,_
~m2
l+ln

In use of (6.23) we get ~ n n + l 2 < Anum_ 2 + J~n, 'tim_ 

where
fin  1 + "rnu,'~_2 < 1 + Tbn unn2 < 1 + Tba,~  A,~ 1 + An
'u~

and
B.
_ <
n + A. (1 + 2An)(1 + ~  ~ nm  , ) ' / t i"n A~ (1 + ~,~<~)~'m ( 1 + A,.) (1 + 3A~) Ab(1 + Tb)[h2T,~ + ( 1 + 2A)a,~]  Bn.
1
This leads to (1.54) as n 
u,Z_2
_<
i=0
The proof is complete.
1
n 
o
Um2 E Ai + E i=0
2
n 
1
Bi Ii Aj + Bn_I <_ AI. 3=i+1
[]
6.3. Degenerate Parabolic Equations
6.3
217
D e g e n e r a t e Parabolic E q u a t i o n s
In this section, t2 denotes a flat torus for simplicity, where a, b > 0:t2 = R2/(a.Z x bZ). Let f 9 R ~ R be a nondecreasing continuous function satisfying f(0) = 0. The problem
ut Af(u) = 0
in f2 x (0, T)
~1,=0 =
with
~0(x)
(6.30)
describes several p h e n o m e n a including free boundaries inside; diffusion in porous media for
ul'~lu
f(u) =
(6.31)
with "7 > 1, very fast diffusion for the same nonlinearity with "V E (0, 1), two phase Stefan problem in the enthalpy formulation for
f(u):
oz(u+l) 0 fl(u1)
(u<l) (l 1)
1)
with constants c~,/3 > 0, and so forth. Wellposedness in X = LI(Q) of (6.30) has been established in the early 1970's via the nonlinear semigroup theory. For the differentiation L =  A acting on X with the domain D(L) = {v E Wl'l(ft) ] Lv E X } , the nonlinear operator A in X is defined by Av = L f ( v ) for v C D ( A ) with
D(A) = {v e X lf(v) ~ D(L)}. Problem (6.30) is reduced to the evolution equation
du + dt
Au = 0
with
u(0) = u0
(6.32)
in X. Here, the operator  A is hyperdispersive in X. This means that any constant )~ > 0 admits an order preserving contraction (I + AA) 1 in X. Abstract theory of M.G. Crandall and T. Liggett now guarantees the convergence
S(t) = slim m,oo
(
I +
)m
t_ A T~
(6.33)
and thus the nonlinear semigroup {S(t)}t>_o is defined on D(A) = X . Then, u(t) = S(t)u0 is regarded as a solution of (6.30). Semigroup {S(t)}t>o has the properties of order preserving and L 1 contraction, described as (6.34)
218
6. Nonlinear Problems
for Uo, fto C X and t >__ 0, where v+ = max{v,0}. (I + AA)(X) = X, and
This is a consequence of (6.33),
where v,~ 9 D ( A ) and A > 0. Time discretization of (6.30) was studied by A.E. Berger, H. Brezis, and J.C.W. Rogers. Based on the nonlinear Chernoff formula, they developed the L 1 theory. Here, we take into account of the space discretization, and consider the finite element analogue for (6.32). Decomposing ~ into simplexes with the size parameter h > 0, we write Th for their totality. Furthermore, Xa denotes the set of continuous functions on f/, linear on each T r rh, provided with the topology induced by LI(~). From the L 2 theoretical point of view, it is natural to take d d~(Uh, ~h) + (Vf(u,,), V~a) = 0
with
as a semidiscrete approximation, where r
(uh(0), f#,) = (Uo, Ch)
(6.35)
r Xh. The solution
uh 9 C I ( [ O , T ) , X , ) may be regarded as a finite element approximation of u. In terms of the operator theory developed in the previous chapters, scheme (6.35) is represented in the following way. First, the finite element approximation Lh" Xh ~ Xh of L =  A and the Ritz operator Rh" V + Xh for V = HI(U) are defined as follows, where A(u, v) = (Vu, Vv)" (i) L,,vh = fh r (ii) Rhv = v,, r
A(V,,, ~ h ) = (fh, fa,,)
for fa,, e Xh ;
A(v, fh) = A(v,,, gab)
for fa,, 9 X h .
Then, (6.35) is written as
duh
dt+ LhR,,.f('u,,)= 0
with
"u,,(0)= P,,u0,
(6.36)
where Pt, " L2(f ~) + Xh denotes the orthogonal projection. If f 9 IR + IR is locally Lipschitz continuous, scheme (6.36) is welldefined, because then vh E X,, implies f(v,,) E V. Scheme (6.35) is conforming only in this case; otherwise it is not so. We propose another scheme, provided with the properties of L 1 contraction and order preserving. Such a viewpoint is not always regarded so significantly in the linear case f ( u ) = u, but becomes important in the degenerate case as we shall see. It is actually realized by replacing the Ritz operator by that of interpolation, and adopting the technique of lumping of mass for the linear part. Let Fh be the set of vcrtices subject to rh. For a E Yh, the function wa C Xh is defined by
W a
1
ata
0
at b E F h \ {a}.
~
6.3. Degenerate Parabolic Equations
219
Then, {Wa I a ~ v . } forms a basis of Xh and the interpolation operator 7rh: W = C(f~) Xh is defined by
7ChV= E v(a)Wa. a6"Ph Remember that ~ C R 2 is a fiat torus so that is compact by itself. We adopt the m e t h o d of lumping of mass described in w Actually, each a 9 )2h takes the barycentric domain Da. From the periodic extension, it is identified with a subset of ~. Let
(x 9 Da) (x 9
1
Wa(X)
0
,
and denote by X h the vector space generated by {~a [ a C ~2h}. The lumping operator Mh " Xh ~ X h is defined through We H ~a. The adjoint operator M~ 9Xj~ ~ Xh is associated with the L 2 inner product, and we set /(h = M ~ M h " Xh ~ Xh. Let u0 E W. The scheme studied in this section is described as d dtKhUh + LhZChf(Uh) = 0
with
Uh(0) = 7rhUo
with
(Uh(0), Wa) = (TChUo,Wa)
(6.37)
in Xh. In the weak form, it is written as d dt (Uh, Wa) + (VWhf (Uh) , Vwa) = 0
for a E 12h, where Uh = MhUh. Because 7chf(Uh) E Xh, it is conforming for any continuous nondecreasing function f satisfying f(0) = 0. Furthermore, the linear operator 7rh " W = C(f~) ~ Xh is order preserving and so is the mass lumping Mh 9Xh ~ Xh. This is desirable for our purpose. We suppose that any T E Th is a right or acute triangle. This implies m a x i m u m principle and L 1 contraction for the discretized linear part. In fact, any result stated in w is concerned with the homogeneous Dirichlet boundary condition, but similar arguments assure them for the periodic b o u n d a r y condition. Similarly to the proof of Theorem 5.1, we can show the properties
0 <_ f~
6
Xh, ,X > 0
==*.
(Ih + I K / 1 L h )  l f h
_> 0
(6.38)
and
with the welldefinedness of (Ih + AIrhlLh) 1 " Xh ~ Xh. On the other hand, Lhl~h is welldefined for X:h 9 Xh with [X:h]  If~1 fa Xh = 0 by Poincar~Wirtinger's inequality, similarly to the continuous case. Letting E v = v  [v], we have
IILhlEKhPhlBLI([.~),WI,q(~~)~ Cq, lim IGLj1EKsP~v ii i~ h~0

L1Ev
W 1 , q (~"~)
=0
(6.40) (6.4~) (6.42)
220
6. Nonlinear Problems
for q E [1, 2) and v E Ll(f~), corresponding to the results obtained in Theorems 1.12 and 5.4. Inequality (1.48) is also replaced as [[LhTrhL1E[[L~(a),L~(a) < C, and hence
I Kff I LhzrhLlzll.(~)...(~)
_< c
(6.43)
follows from (5.7) with p = 2. Property (5.4) also keeps as Uh C X h ~
u
in
LP(f~)
MhUh ~ U in
===:=v
LV(f~).
(6.44)
Wellposedness We set
AhV = I(h 1LhTrhf (v) for v E W = C(ft), which can be regard as an operator in Xh. We have the following. Theorem
6.4. It holds that
IIMh~h[~h  ~h]+ll~'(~)  IIM ,. Iv,.
+ AAhvh aA,~'Oh]+ll~,(~),
(6.45)
where Vh, Vh E Xh and A > O. Furthermore, (Ih + AAh)Xh = Xh follows. Similarly to the continuous case, the nonlinear semigroup theory guarantees the unique solvability of (6.37) globally in time by Theorem 6.4, because IIMhTrhl'l Ll(ft) provides a norm to Xh. The solution is given as uh(t) = Sh(t)Trhuo for Sh(t) = lira m,oo
(
Ih + t A b
(6.46)
7Yt
Inequality (6.45) now gives for Vh, 77h C Xh and t _> 0 that
I M,.zrh[S,.(t)v,. Sh(t)~,,]+llL,(~) < IlMhZr,,[Vh '&]+IIL'(~)
9
Therefore, u0 >_ 'u0 implies 'uh(t) _> 'Sh(t) for u0, '~),0 E W, where Uh = uh(t) denotes the solution of (6.37) and ~h = ~h(t) that of d d~I(hfih + Lhrchf(ith) = 0
with
/th(O) = 7rhftO.
In particular, u0 > 0 implies uh(t) > O. Furthermore, L 1 stability of the approximate solution holds in the following sense:
IIMh~'h '~'h(t)lllL,(a) 4 Mh~',~luolll,,,(a)" Because {rh} is regular, we have II~,~IXhIIILI(T) ~ Inequality (5.2) with p = 1 can be replaced by
IlxhlIL,(T)
for T E Th and Xh C Xh.
c' I xh lI L, (T) _< IIMh~h xhlIIs,(T)< c X,,IIL,(T)
(s.47)
6.3. Degenerate Parabolic Equations
221
and hence
II[s,,(t)vh  sh(t)~]+ IIL,(~ ) _< c II[~  ,~]+ll~,(~)
(6.48)
follows for Vh, 9h C Xh and t _> 0. Those properties are preserved in the time discretized equation, the backward difference finite element approximation
I(h [u;(t + T)~_ uT,(t)] + LhTrhf (u~(t + T ) ) = 0
(6.49)
with u~,(0) = 7rhuo, where t = rnT with m = 0, 1, 2 , .  . . In fact, its solution is given as u[,(t) : ( ~ + ~ A , , )  m ~ u o for t = m~, and formula (6.46) reads as
Uh(t) = lira u'h(t ). ~I0
We have
(6.50)
IlM~h bT,(t)  ~T,(t)]+ IIL,(~) _< IIM~,~,~ [~o  ~,o]+1}~,(~) for u0, ~0 E W, where U~h(t) denotes the solution of (6.49) with u~(0) = 7chuo, and 5T,(t) that with ~!~(0) = 7Ch50. We proceed to the proof of Theorem 6.4. Variation and lattice are key structures. Talking about the variational structure, we note that both the solution and test functions are taken from Xh. The lattice structure is also nontrivial, as Uh E Xh does not necessarily imply [Uh]+ E Xh. Given Vh C Xh, we take F~: = {a E Fh I +Vh(a) >_ 0}. It holds that
'V=~~ 7rh [Vh]:l== nt E Vh(a)Wa, aeV~h where [  ] + = max{0, + . }. Then, 0 <_ V~h e Xh and Vh = v~  Vh follows. This implies (Ih + A K h l L h )  l v ~ >_ 0 by (6.38). Therefore,
[(Ih Jr,~KhlLh) 1 Vh]+ <_ (i~ + a K : I L ~ )  ' ~ : and hence
~h [(Ih + "~KhlLh)I Vh]+ ~ (Ih + /~KhlLh)l?2~
is obtained. Because Mh : Xh ~ X h is also order preserving we have
1
~ Mh (i h ~/~KhlLh)i Vh_/_.
Therefore,
f~ MhTrh [(]h~ /~I(hlLh)lvh]+ ~ ~A/lhVh= ~MhTrh[Vh]+ holds by (6.39). We have the following fact comparable to Kato's inequality for L =  A .
(6.51)
222
6. Nonlinear Problems
L e m m a 6.5. The inequality
(~.52)
s MhrCh [(KhXLhrrhV) 9sgn+v] > 0 holds for v C W = C(f~), where
sgn+ v = ~1 to
(v>O) (v < o).
Proof." Given u, v E W, we have
7rh (u. sgn+v)
~ u(a)wa aeyhn{v>O}
= =
~
u(a)wa = rrh (u. sgn+Trhv).
ae Vhn{ rrhV>_O}
Therefore, f
f
Ja Mhrrh [(K171Lhrrhv) 9sgn+v] : Jn Mh~rh [(Ir
9sgn+vh] ,
where v~ = rrhv E Xh. Writing uh = (Ih + AKt71Lh) 1 vh for A > 0, we have Yh
Uh

=
)~ ( I h 2t / ~ / ( / 7 1 L h ) 1 IV h1
Llz~tJh"
This implies A j~ Mhrrh [(Ih+ kKt[1Lh) 1K[[1Lhvh " sgn+vh]
= s Mh~,,[(~,, ~,,,). ~n+~,,] = s ~,,,~,, [.,,,,]+ s A,,,~,,[.,,,,~n+~,~] >_ s ~',,~,, [v,,]+ s Mh~,, [~,,,]+ >_0 by (6.51). Letting A I 0, we obtain
.s M,,~,, [(<7'L,,~,,,) . ~g~+,,,,] > o, or equivalently, (6.52). We now give the following.
6.3. Degenerate Parabolic Equations
223
Proof of Theorem 6.~: First, we show t h a t
I I M ~ [v  O]+IIL~r < IlMh~h [v  ~ +
~A~v

AA~O]+ IL~(~)
for v, ~ r W and ,k > O. We may suppose t h a t f : R ~ IR is strictly increasing by taking f~(u) = f ( u ) + su and later making s I 0. Letting u = v + and ~ = ~ + )~I([~lLhTChf(v), we have
AK~iLh~hf(v)
[IMh~h [V  ~]+ll.(~)
= s Mh~h [(~  ~)sgn+(~  ~)]
= /o M ~
[(~ ~). sgn+(~  ~)]
 ~ f a Mh:rh [KhlLhTrh [f(v)  f( 9 Because f is strictly increasing, for w = f ( v ) 
sgn+(v  ~5)].
f(~) we have
sgn+w = sgn+(v  ~?). Therefore, L e m m a 6.5 assures
J~ MhTrh [t(hlnhTrh I f ( f )  f(?))]. SgIl+(V  ?))] = ]~ Mh7rh [I(hlLh7rh w.
sgn+w]
>0
so t h a t we obtain
I Mh:rh[ v  ~?]+lln,(fl)
<
.~o Mh:rh [(u  ~ ) . sgn+(v  ~)]
_< /~ M.Tr. [u

/~]+
as is desired. Now we proceed to the proof of (Ih + AAh)Xh = Xh. Given Uh E Xh, we show the existence of Vh C Xh satisfying Vh + AAhVh = Uh. In fact, Ta = Ih + AAh is a continuous m a p p i n g on Xh, a finite dimensional vector space. Note t h a t IIMhTrh I 9Illnl(~) provides a norm to Xh. In use of (6.45), we can take an open ball (9 C Xh sufficiently large, satisfying uh q~ Ta(OO)for any A > 0. We may suppose t h a t uh C (9. T h e n the topological degree d(T~, Uh, (9) is welldefined. Its h o m o t o p y invariance implies
d(Ta, Uh, (9) = d(Id, Uh, (9) = 1. Therefore, Uh C Th ((9) follows. The proof is complete.
L~176 Scheme (6.37) has the property of L ~  s t a b i l i t y furthermore, stated as follows.
[]
6. Nonlinear Problems
224 Theorem
6.6. The semigroup {Sh(t) } in Xh defined by (6.~6) satisfies
IIS,,(t)~h~011~o~(n) <_ II~h~011~oo(n)
(6.53)
for t >_ O and uo E W. Proof: Thanks to (6.33), inequality (6.53) is reduced to
II',~,~llLoo(~)_ II(Zh + AAh),~hll~oo(~) for uh E Xh and A > 0. Let vh = (I + AAh)uh and b =
(6.54)
I1~,~11~(~).The relation
(~h, Xh) + A (VTrhf (Uh), VXh) = (Vh, Xh)
(Xh 6 Xh)
(6.55)
holds for Uh = Vh = +b E Xh. This means + + ~ h  (Z,~ + ~m,~)' (+b) = +b and inequality (6.54) is reduced to the following lemma of comparison. consequence of Theorem 6.4 and the proof is complete.
It is a direct []
L e m m a 6.7. Given vh, 9h 9 Xh, let Uh, ith 9 Xh be the solutions of (6.55) and
(~h, Xh) 4 A (VTrhf(~h), VXh)  (~h,Xh)
(Xh 9 Xh),
respectively. Then, Vh <<"vh implies Uh <_ iZh in f~.
Time Discretization We have proven convergence (6.50) for the backward difference approximate solution u~ of scheme (6.49). Here we study another scheme based on the nonlinear Chernoff formula of Brezis and Pazy. We suppose that the nonlinearity f is locally LiI)schitz continuous. Let # > 0 be the Lipschitz constant of f on
[II~h~,0ll~oo(~),I ~h~011LOO(~)], and take the regularizing parameter c = c , , 0 as 7 $ 0 satisfying #~/a~ _< 1. Under such a preparation, the schcme is given as
~(t
+ ~)  ,,,,;(t)
+
1  e~K,Y1LhI
7Chf (",,,,,(t)) T
=
0
(6.56)
with u,~(O) = 7rhuO
(6.57)
6.3. Degenerate Parabolic Equations
225
for t = m r (m = 0, 1, 2 ,  .  ) . Here, {e~ denotes the linear semigroup in Xh generated by K~ 1Lh. We can show tile convergence lira sup IlMhTrhlu;(t)  uh(t)lllLi(a) 0 Ts
(6.58)
0
for T > 0, where u~(t) and uh(t) denote the solutions of (6.56) with (6.57) and (6.37), respectively. In fact, it holds that
ttt:(t )
=
[fh
(tml)] m (TrhttO)
for t = m r with the nonlinear operator Fh(r) " Xh ~ Xh defined by
[F~(~)] ( ~ ) = x~ + L [L~  ~ ' / q ~ , h f ( x ~ )  ~ f ( ~ ) ] J O"r
9
(6.59)
Applying Theorem 6.6 for f(u) = u, we have
eaKhlLh7rhY+ IILOC(f.t) _< I1~~+ I~(~)
(6.60)
for a > 0 and v C W. A quick overview now guarantees the estimate
II~(t)ll~(~)
_< II~h~oll~(~)
inductively by #r/o., _< 1 and (6.60). We may assume t h a t f = f(u) is Lipschitz continuous with the Lipschitz constant # in R by replacing f(u) for lul >_ 117rhuoIL~(n)if necessary and then
r If(r)
GT
f(s)l + ( r  s)  L ( f ( r ) GT
f(s)) = I r  sl
(6.61)
follows for r, s E R. Theorem 5.1 assures L1 (f~)
for v C W. Equality (6.61) gives for Xh, Oh C Xh t h a t
IIMhTrh Ifh(T)Xh  fh(T)~JhlllLl(t2) T < =lMh~hIf (Xh)  f (~,,)lllL~(~) (7 T
O"m

IIMh~,~
Ix~  r
L 1 (Q)
(6.62)
Convergence (6.58) is a consequence of the nonlinear Chernoff formula. Namely, it suffices to show the convergence of the resolvent in the sense that lim TiO
(
Ih +  ( I h  Fh(r)) 7
)1
Xh = (Ih +/~Ah) 1Xh,
(6.63)
226
6. Nonlinear Problems
where A > 0 and Xh C X h. Letting Ch = (Ih + )~Ah) 1 Xh, we set ~
r
=
)1
~,~ + 
(~
r,~(~))
Xh
T
and
X~=r T
In use of xh = r
+  (h,  Fh(~))Cj:, T
we have
(
IIMh~h ICh
1+

Chlllc,(.) < IIMhTrh IXh  x;~lllL,(~) + IIMhTrh I ~  ~hlllLl(a) T
by (6.62). This means
IIMh~hICZ  ~hlllL,(~) ~ IIMh~hI ~ h  ~,SIIIL,<~> 9 Here we have x~r
=
C h _ A[
eaT
K,: I L h T r h f (r
 zrhf (~Ph)]
Crr
~ r
+ AK[[1LhTrhf (r
= r
+ AAh~h = Xh
by (6.59). Hence limTl0 ~Pt: = Ch and (6.63) follows. The proof is complete.
C o n v e r g e n c e of R e s o l v e n t TrotterKato's theorem assures that convergence of semigroups is a consequence of that of resolvents, and under that spirit, we study the convergence of resolvents now. Here, we take the simplest case that {rh} is uniform" each vertex is shared with 6 triangles and the following property holds in rh, where oz(z) = x + h~ei for a basis {el,e2} of R 2 and uh < h i , h 2 <_ h with a constant u > 0: T E rh
==*
ai(T) C rh
(i = 1, 2)
Under such a situation, we can show tile following. T h e o r e m 6.8. Under the assumption stated above, it holds t/tat lim II(Ih + AAh) 17rh'v (1 + AA) l VllL~(m = 0 hlO
f o r A > O and v E W .
(6.64)
227
6.3. Degenerate Parabolic E q u a t i o n s Proof." Given I > 0 and v E W, we set vh = 7rhv and Uh : (Ih + / ~ A h ) 1 7rhV.
We have I(hUh + ALhTChf (Uh) = I(hVh
(6.6s)
II~ll~oo(~) ~ IIv~ll~oo(~)~ II~llLoo(~)
(6.66)
and (6.54):
Therefore,
III~t~*hllL~(~)< C II~IIL~(~)fonows from (5.6). Similarly, IIKhvhllLoo(a) < C Ilvll~(~)
holds, and we get IILh~hf(uh)llL,(~) <_ Ca ~
117rhf(u,~)llw~,~(~) 
II~IIL~(~) by O(1)
(6.65). The relation
(q ~ [1, 2))
now follows from (6.40), because 117rhf(u,~)llL~i~) < c is obvious. Later, we shall show the following. L e m m a 6.9. The f a r o @ {uh} c L~(f~) is relatively compact as h I O. This lemma assures that any hk ~ 0 takes a subsequence { h; } satisfying 7rh;f(Uh;) ~ W weakly in w ~ ' q ( a ) and Uh,k ~ u strongly in Ll(f~) with some w E w l ' q ( a ) and u E L l ( a ) . Those convergences can be also almost everywhere in f~. We have w = f(u)
(a.e. in f~).
(6.67)
In fact, given s > 0, we have a measurable set f~ C f~ satisfying If~ \ f~l < s and uh; ~ u uniformly on f~ by Egorov's theorem, where I" I denotes the Lebesgue measure. This implies that 7rhf(Uh,k) ~ Z(U) uniformly on a~ and w = f ( u ) holds in f~. Therefore, (6.67) follows by making s I 0. In this way, 7rh,kf(Uh,k) ~ I(U) weakly in w l ' q ( a ) and Uh,k ~ u strongly in Ll(f~) with some u C L l ( a ) . It holds that
IlL; 1EZ~hUh  L 1 Eullwl,q(f~) ]IL;1EKhPh (U h <_ C l l u h  u
by (6.41). Similarly,
u)llwl,q(f~ ) tIlL;1EI~hrhu
L~(n) + [ l L h l E I ( h P h
u  LIEullw~,q(n)
L1Eullwl,q(f~) "
Letting h = h~ ~ 0, we see that the righthand side converges 0 by (6.42).
6. Nonlinear Problems
228
follows from Ilvh vllL~(~> ~ O. We have [Lh:rhf(Uh)] = 0. This implies L h l K h E u h + A:rhf(Uh) = L h l K h E v h and hence
L1Eu + Af(u) = L1Ev follows. On the other hand, [Kh,kUh,k] : (Mh,kUh,k, Mh,kl ) = (Mh,kUh,k, 1) * [U] by (6.44) and similarly, [KhVh] * IV]. This implies [u] = [v] by [KhUh] = [I(hVh] and hence u + A L I ( u ) = v is obtained. We have u = (I + A A )  l v and convergence (6.64) now follows. Theorem 6.8 is reduced to Lemma 6.9. []
Proof of L e m m a 6.9: We show that any c > 0 admits ~ > 0 such that
o < h < e~, lyl d~ < ~
===:~
~ luh(x + y )  uh(x)l < e.
(6.68)
Then the assertion is proven. In fact, we have the following properties by (6.66):
1) sup II~hllg,<~) < + ~ h
2) Any e > 0 admits # > 0 such that A C f~ with [A[ < ~ implies suPh
~ luhl< c
Let hk I 0 a n d r > 0. We have k. such that k_> k. implieshk < 5. On the other hand uh,, u h 2 , ' " , Uhk._, are uniformly continuous on f~. Making ~ > 0 smaller if necessary, lyl < ~ implies
f l ~ ( x + y)  ~(~)1
dx <
(k= 1,2,..)
by (6.68). FrechdtKolmogorov's criterion now holds and {uhk} C Ll(f~)is proven to be compact. We recall the translation operator ai " x H x + hei. Let 2 = (or1, a2} be the transformation group generated by cri (i = 1, 2), and
II~ll = sup ~(~)  xl xEIR 2
f o r c r E 2 . We s e t a z ( x ) = x + z f o r z E R 2. G i v e n y E R 2 , w e h a v e a E E a n d z E R I~11 < lyl and Izl _< h such that T + {y} = az (a(T)) for any T E rh. We shall write cr*ut~ for uh o or. Tile first term of the righthand side of
2in
229
6.3. Degenerate Parabolic Equations f
is represented as I,, [Uh(X + z)  ~(~)1 d~ with I~l _< h by a transformation of variables. We shall show that this term is estimated from above by a constant times
sup~ep~~ I~*~(~)  ~(~)1 &. IIoII_
This implies
s
~,(~ § y)  ~h(~)l d~ ~ C sup s
I~*~,(~)  ~,~(~)1 d~
aEE
IIoII_
sup~ s
(6.69)
I~*uh(x)  Uh(x)l & .
IIoI1_<1~1
To see this, we take T E % and let its vertices be {al, a2, a3}. Suppose a3 = o 1 ( a 2 ) and al = a2(a3), without loss of generality. The inverse assumption (actually uniformity of %) guarantees uh(a3)  ~h(a~.)l _< I ~ ,
Similarly, w~ h~ve I~h(al)  ~ ( a 3 ) l
IlWhll~(~) ~
C h ~
 ~ 1 1 ~ ( ~ ) ~ c h ~ I1~1~  ~,11,(~) ~
Ch ~
9
II~r~h  ~hll~x(~) ~nd hence
sup IIo*~ UhIILI(T) ~ Ch 3 ~ aEE
IIo*~ ~11.(~)
(6.70)
aEE
Iloll_
IIoll___h
follows. Let {T/}~I C rh be the set of triangles which share one or two vertices of T E rh. Letting T = Ui1_21Ti, we have
I~(~ + ~)  ~.(~)1 _< h IIW.ll~(% for I~1 __ h and x E T. Applying (6.70), we have IIW~ll~oo(%
=
sup IIW,~ll>o(~,)
1
<_ Ch 3 sup 1
E
II~*u~ ~llL'(a)
aEE
Ilal[_
< ch ~ ~
I1~*~~11~).
aCE
II~ll__h
This implies
aEE
Iloll___h
6. Nonlinear Problems
230 and hence
~ luh(x + z) Uh(X)ldx < 12C' ~
a*Uh(X) Uh(X)IIL,(n)
o'C Y?,
II~ll
a*uh

( Ih + )~Ah ) I 0" , Y h
with Vh = 7rhv and (6.45) gives
<
I~1. I ~ * ~  ~1 ~ ( , ) .
Inequality (6.69) is reduced to
s I~,,(~ + y)  ~,,(~)1 d~ _ C' sup la*v II~ll

~11~o~(~) ace IIGII
and p r o p e r t y (6.68) is a consequence of tile uniform continuity of v C W. T h e proof is complete. []
Convergence of Semigroup Convergence of the semigroup is a consequence of tile resolvents. Uniformity of essential for the following t h e o r e m to hold. Theorem
Th
is not
6.10. If f is strictly increasing and rh is uniform, we have lim sup IISh(t)~rhuo  S(t)uo ILI(~) = 0
(6.71)
hi0 0 < t < T
for T > O and uo E W . Similarly to the s t a n d a r d theory, we make use of the Yosida approximation to derive the convergence of semigroups from that of resolvents. Here, we recall the following facts. Given A > 0, let J~ = (I + AA) 1 and Aa = ( I  ,J~)/A. We take d d~UA + Aaua = 0
with
'ua(0) = u0
(6.72)
6.3. Degenerate Parabolic Equations
231
in X = Ll(f~). We can verify t h a t Aa is an maccretive operator and hence  A a generates a contraction semigroup. Any T > 0 admits a contraction (I + TA,x) 1 on X:
I1(1 +
_< IIv 
7A,x)  i v  (I F TA,~) 1~1
~?llLl(a)
(v, ~? E X)
(6.73)
Obviously, Aa is (2/A)Lipschitz continuous in X because J~ is a contraction. Therefore, (6.72) is uniquely solvable; we get the solution UA e c l ( [ o , oo), X). It is known t h a t u~(t) is regarded as an approximation of u(t) = S(t)uo. In fact, the inequalities 
<
2("~2+ / ~ t )
1/2
IIA'~ollc~(~)
(6.74)
and ~ ( x / ~ + A)IIAuollLl(~)
(6.75)
IlS(t)uo sa(t)uo Ll(f]) __~ 3(v/,~t + )~) Auollc,(~)
(6.7s)

are known. In particular,
holds for u0 E D ( A ) . It is natural to introduce the corresponding operators on Xh as Jh,~ = (Ih + AAh) 1 and Ah,~ = (Ih  Jh,~) /)~. The problem d ~uh,~ + A~,~uh,a = 0
with
uh,a(0) = 7rhuo.
(6.77)
has a unique solution Uh,~ E C1([0, oo), Xh). Our first task is to derive lim sup Iluh,a(t)  ua(t)llLl(a ) = 0 h~0 0
(6.78)
for ,~ > 0. However, in the framework of nonlinear semigroup theory, ua(t) E W does not follow from u0 E W in spite that ua(t) E L~176 actually follows from u0 E L~176 Because the interpolation operator 7rh works only to continuous functions, this causes a technical difficulty. To avoid it, we take time discretizations and deduce an analogous result first. We prove the following. L e m m a 6.11. of
Given )~ > 0 and ~ > O, let u~(t) E X and U~h,~(t) E Xh be the solutions
~X(t + ~)  ~ (t)
+ A:~u~ (t + T) = 0
with
+ Ah,:~u~h,~(t + T) = 0
with
u~(O) = "Uo
and
~L~( t + ~)  ~L~(t)
U~h,:~(0) = 7ChUo,
232
6. Nonlinear Problems
respectively, where t = m~ with m = O, 1, 2 , . . . . the continuous time t >_ O. Then, it holds that
Combining linearly, we extend them to
lim sup l u ~ , ~ ( t )  u~(t)ilL,(n) = 0
(6.79)
hi0 0
forT>O. Proof: Because f is monotone increasing, we can show that v E W implies u = (I + w A ~ )  l v E W . In fact, this relation is equivalent to + 1 u =  v + J~u

T
(6.80)
T
and hence 1 +

IlullL~(a)
T
IIv Ig~(a) 
u + (u
W
v)
T
= I&ullg~(a) <_ IlullL~(a) L~(f2)
follows from (6.54). This implies II,ulino~(fl) _< IIv ILo~(fl) < +oe. Now (6.80) means A u = u + (u
T
A v)+ ALf(u + (u
T
v)),
~ d c f (~ + ~(~, v)) c c ~ ( ~ ) fonows from ~, ~ z c ~ ( ~ ) . This implies f
u+(uv) "r
E W=C(f2)
from the elliptic regularity. Because f C C (R,R) u+A(uv)/wE W and h e n c e u C W b y v C W . From the property stated above, we get u[(t) C W (I + "rA,x)~Uo for t = mr. As a consequence of Theorem 6.4, the operator on Xh equipped with the norm ]MhTCh 1" ]]]L'(n); it
is strictly increasing, this implies for t > 0 by u0 C W and u[(t) = Ah,~ has the following properties is (2/)~)Lipschitz continuous and
(I + rAh,~) 1" Xh * Xh is a contraction with respect to ]]MhTrh 1" ]]]gl(n)" If t = m r , we have ,u rh,~ (t) u~(t)
=
(Ih [ TAh~) , m 7rh'Uo  ( I + ~A ~ ) m Uo
=
(I + TAX)mTrhUO (I + "rA,x)muo
m + E[(I
+ rA~) (me) (Ih + rAh,~)'
~=1
  ( I + TA,x) (me+l) (Ih + TAh,~)(e1)]TrhU0 by the associative law of operators. In use of (6.73), Llnorm of the second term of the righthand side is estimated from above by m
Z II[(1 + T A ~ ) ( I h e=l
+ TAh,,x)  e 
(Ih + TAb,A) (fl)] 7[hUOIILI(f2)
6.3. Degenerate Parabolic Equations
233
This is equal to
m g=l m g=l
We obtain from (6.73) that
][u~,~(t) 
u~(t)llLl(a) ~
[[(Trh /h) U0[ILI(~)
m g=l m
(6.81) g=l
In use of the Lipschitz continuity of Aa, the second term of the righthand side of (6.81) is estimated from above by
2~ m
)~
~]
g=l
< A Ee:l (Ih + TAh,~)eTrhUo (I + TA~)euo +7 ~
(I + ~A~
~o (I +
Ll(a)
~A~) ~
g1
m <
2"]5Z
g1
2t
T (~f) Iluh,~  u~,T ( ~ T ) l l L 1 (~) "JIi I1(I
~h) UollL,
(~)
6. Nonlinear Problems
234
To handle with the third term of the righthand side of (6.81), we note that
[
3
[Aa (I + TAa)  e  Ah,a (Ih + tAb,a) e] 7rhUo = A~ (I + rA,x)errhuo A,x (I + rA,x)euo +Ax (I + TA~)euo Ah,~ (Ih + rAh,~)eTrhuo = [Aa (I + T A A ) g rch A,x (I + rA~)ej] UO 1
+ ~ [(1 + rAa
)e
e  ( I h + TAb,a) 7r,,]no
1
[Ja (I + rAa)  e  Jha (Ih + rA,.a)eTrh] Uo A = [Aa (I + rAa)err. Aa (I + rAa) el UO + ~1 [(I + rAa )e  ( I h + tAb,a) err,,.] Uo 1 )~
[& _
Jh'~1
(I + ~A~) ~
1[ A Jh,aTrh(I + TAa)  e 
UO
]
Jh,,~(Ih + rAh,a)eTrh Uo.
Therefore, in use of the Lit)schitz continuity of A~ and (6.73), tile third term of the righthand side of (6.81) is estimated from above by
2 7"?Tt
,~
T
m
II(~h I) UollLl(~) + ; ~
T
r
I[""'~'(e") ~"~(e")llL'(~)
g=l m
+ A
e:l
CT
gl(~)
m. g=l
2t _<
A
m I(r:h
1) uoll~,
(~) +
(C + 1) ;T
Z I1'" (e~)  ,4(e~)II~,(~) g=l
m
+ ~
Uo
Ll(f~)
g=l
CT
m g=l
6.3. Degenerate Parabolic Equations
235
We can summarize them as
[l(Trh I)uo[lLl(~)
CT,~ E [[Tj,~7. h,,, (g,.]_) __ U~(gT_)[ILI(~) _31_ 1 + ~ g=l CT
m
7.
T
g=l
m
7.
g=l
for t = m r . Because of the discrete version of Gronwall's lemma, this implies
~uv II~L(t)  ~i(t) I .(~) < ~ / ~ ( 1 + _c~_)II(~.  I)~0 I~(~)
0
m _FeCr/,xC7 ~ Zm [ (~h  I ) u 7.(eT) x ][Ll(a ) _31_e c r / x ~a ~~.I[[Jxg=l g=l
Y.x~h]uX(g7)llCl(n) ' '
where T = m r . As is noted, u0 E W implies u~(lr) E W. Therefore, the righthand side of the above inequality converges to zero as h I 0 by (6.64). The proof is complete. [] Now, we show the following. L e m m a 6.12. Given A > 0 and uo E W , let u~ = u~(t) C Cl([0, o c ) , X ) and Uh,~ = Uh,~(t) e Cl([O, o c ) , X h ) be the solutions to (6.72) and (6.77), respectively. Then the convergence (6. 78) follows. Proof: The semigroup {T~(t)} generated by A~ is a contraction in X = Ll(gt). This implies [[u~(t)I]Ll(a ) < [lU0[ILl(a), and hence
IlAAux(t)  Axux(s)lln,(~)
2
<_ A ] l ~ ( t )  ~(S)IILI(~>
< V4 It ~l II~ol LI(~) (6.82)
,~ follows. Because of
u~(t + T)  u~(t) + TA~u~(t + T) = 0 and uA(t + T)  u~(t) +
l
t+T
AAu~(s)ds  O,
,It
the error function eT.(t)  u~(t)  u~(t) defined for t = m T (m = O, 1,. ) satisfies (I + ~A~)u~(t + "r)  (I + TA~) 'u:~(t + 7) = e~(t) +
f
,It
t+T
[n~u~(s)  A~u~(t + "r)]ds.
236
6. Nonlinear P r o b l e m s
This implies
_< I e'(t)llc~(a) +
f
t+T
Aau~(t + ~)  A~ua(s)llLi(a) ds
dt
m[.
by (6.73), and hence
l e'(t)llLl(n) < ~
1)7
f=l
follows for t = mr. In use of (6.82) we obtain m 4
II~(t)llLi(~) ~
T2
2tT
V ~ 2 II~0
i
ii
/=1 or
2TT
sup
0
IluX(t) u~(t)l L,(n) < VIlu011L,(~)
for T > 0. Similarly, we have 2T7
sup
I~;,x(t)  ~h,x(t)[[]~(~) ___~
0
IIMh~h I~h~olll~(~)
and hence CTT
sup 0
I uh,~(t)  u,~(t)ILI(f~ ) ~ ~
~ I'U,0 L~(Ft) Jr s u p I l t h , A ( t ) 0

ttA(t) llLl(a),,
follows. Now, we send h I 0 and then r I 0. We get (6.78) by (6.79). The proof is complete. [] Now we complete the following. Proof of T h e o r e m 6. i0:
We shall show that (6.83)
lim sup [ uh(t)  u(t)llLl(gt)  O. hJ.O
0
Consider, first, the case that Uo E W satisfies f (u0) E C2(f~). Inequality (6.76) is a consequence of the general theory. We also have sup o
IIMh~h lu~,~(t)  uk(t)lllL,<~) < 3 (\ ~
+ A)]lMhTCh AkTrh'uO /
by Theorem 6.4. Here, in use of LhZrh [f(u0)] = 0 we have
IIMh~h IAh~h~ofll~l(~)=
IIM,~,~ IK;' fh~hE f
(~o)lll~,(~)
< c I I K ; ' L ~ L  ~ E . Lf (~o)11~(~) < C IILf (~o)11~(~)
Commentaw
to Chapter 6
237
by [L I, E] = 0 and (6.43). We have
<_ <
sup 0
Ilu~(t)
sup O
I ~th(t)  Uh&(t)l[Ll(f~) + sup []Uh,l(t)  ~tA(t)l Ll(f~) I sup
sup
0
u(t)llL,(~)
O
I u~,a(t)
 ua(t)llLl(a) + C ( x / ~
O
+
ua(t)  u(t)llLl(ft)
A) IILI (Uo)IL=(~) 9
This implies
uk(t)
limsup sup hi0
u(t)
Af_ ,~)
LI(~~)_~ C ( ~  T
ILf(u0)ILk(n)
0
by (6.78), and conclusion (6.83) follows by making A t 0. It remains to extend it for the general case of u0 E W. We again make use of the strictly monotonicity of f for this purpose. P u t t i n g v0 = f(uo), we take a sequence {vk} C C2(f~) N W satisfying lim Ilvk  v0llc~(a) = 0 and set uk = f  l ( v k ) . Then it holds that f (uk) < W N Cg(f~) and lira ][uk  u0 c~(a) = 0. We have proven lira sup
h$0 0
II&(t)Trhuk 
S(t)uklIL~(~)
: 0
for k = 1, 2 , .   . In use of (6.34) and (6.48), on the other hand, we obtain
IIS(t)uoIIS(t)uo
Sh(t)TChUOIILI(~) s(t)u~llc,(~) + I I s ( t ) u ~  Sh(t)TrhUkllLl(~) + II&(t)Trhuk Sh(t)TrhUOIIc,(~)
_< Iluo  uklIL,(~) + IIS(t)uk  &(t)~h~kllL,(a) § C II~hUk  ~OIIL~(~) < C I1~  ~OIILoo(~) + IIS(t)Uk  &(t)TrhUkllL~(a) 9 This implies sup 0
II&(t)~huo
 S(t)uol
LI(~) !
If we make h ~ 0 and then k ~ complete.
Commentary
to Chapter
C 1]'~/r  Uo] Lee([]) 1L s u p
0
II&(t)Trhuk

S(t)ukllLl(n).
oo, convergence (6.71) is obtained and the proof is []
6
6.1. The monograph Suzuki [369] is devoted to the theory of semilinear elliptic boundary value problem. There, the existence (unique, multiple, and non) of solutions and their qualitative features are described. To see that any nontrivial solution u of (6.1) with (6.2) is unstable, note that the linearized operator is given as  A  p lul p1. Then, its first eigenvalue is negative by
llVul[~(n)
P
U IP+I
'L~+ (n)
(p
I)
2
<
238
6. Nonlinear Problems
and Rayleigh's principle. It indicates the linearized instability. See Henry [173] for details of this notion. In many cases, the simple iterative sequence defined by (6.3) becomes monotone, which makes its dynamics simpler. For this topic, see Amann [5], P.L. Lions [248], and the references therein. However, the following fact essentially due to Fujita [136] is worth noting: Let u. is a nontrivial solution, positive in f2. Then, I'uk L~(a) + +oc and IlukllL~(a) + 0 according to u0 > u. and 0 _< u0 < u., respectively. Therefore, scheme (6.3) is not suitable to get unstable solutions. See also Suzuki [369] for the proof of this fact. Ambrosetti and Rabinowitz [9] obtained infinitely many solutions for (6.1) with (6.2) by tile variational method of minimax type. See Berger [32] and Rabinowitz [320] for various methods of variation and applications to nonlinear partial differential equations. Concerning the Morse indices of tile minimax solutions, see Bahri and Lions [22] and the references therein. Uniqueness of tile least energy solution for (6.1) with (6.2) is proven by C.S. Lin [240] when [2 C R 2 is a convex smooth domain. Nehari's variational principle was introduced by [289], [290], [291] for the twopoint boundary value t)roblem. Applications to partial differential equations were done by Coffman [90], [91], Struwe [361], and others. Its equivalence to the mountain pass lemma was noted by Ni [294]. However, to our knowledge, Nehari's iterative sequence was introduced by Coffman and Ziemer [92] in the study of quasilinear elliptic boundary problems, although a related idea can bee seen in Ding and Ni [113]. Mizutani and Suzuki [268], [269] gave a systematic study on that method of iteration, generalizing nonlinearities, unifying the method of inverse powers, and so forth. There, tile numerical computation was done by the use of finite element method. See also Suzuki [369] for some theoretical aspects of this sequence. An example of numerical experiments is presented in Figure 6.1. The Nehari manifold is closely related to the stable and unstable sets in the theory of double wells. For this topic, see Ikehata and Suzuki [189], [190] and the references therein. The standard bootstrap argument valid for the subcritical nonlinearity in elliptic boundary value problems is presented in Suzuki [369]. Actually the elliptic estimate utilized in this section is indicated as (A)IL2(f2) C H2(f2). It holds because f2 is a convex polygon. Recently, several trials have been made to provide a proof of the existence of the solution with the aid of numerical computations. Among others, Nakao's method works to the fixed point equation u = r ( u ) on a Banach space X. For instance, if T is a (nonlinear) compact operator, then Schauder's fixed point theorem is applicable. If one can construct a bounded convex closed set U satisfying T(U) C U, then there is a fixed point of T in U. For this purpose, we construct a finite dimensional set Uh and its complement U ~. If T is split as T = Nh + M~, satisfying Nh(g) c Uh and Mrs(U) C U 1, then T(U) C U follows for U = Uh  U 1. Here, numerical verifications of Nh(U) CUh and Mh(U) c U 1 are done differently, Newton's method and norm estimate, respectively, for instance. For more details, see Nakao [276], [277], [278], [279], Watanabe and Nakao [404], Yamamoto and Nakao [408], Nakao and Yamamoto [280], Yamamoto [407], Nagatou, Yamamoto, and Nakao [272], Nakao, Yamamoto, and Nagatou [281], Nagatou [271], and the references therein. 6.2.
Nonexistence of the classical solution globally in tittle for some kind of parabolic
C o m m e n t a r y to C h a p t e r 6
239
Figure 6.1: Shapes of approximate functions Uk,h of minimizing sequences to Au = u 2 with ulon = O. The detail of the scheme is described in [269]. The domain f~ is assumed to be Lshaped with vertices (0,0), (1,0), (1,3/16), (3/16,3/16), (3/16,1), (0,1)and is divided into 2784 elements (equal right triangles).
equations was noticed by Kaplan [196], Fujita [135], [136], Tsutsumi [392], [393], and others. A lot of remarkable works have been done up to now. See Levin [238], Suzuki [369], Samarski, Galaktionov, Kurdyumov and Mikhailov [334], Deng and Levine [109] and the references therein. Blowup phenomenon for (6.12) was first observed by Fujita [137]. See Kohda and Suzuki [225] and the references therein for later developments. One point blowup is first noted by Weissler [405]. Later, Friedman and McLeod [131] and X.Y. Chen and Matano [74] refined the study. Theorem 6.3 is due to Y.G. Chen [75]. It also studied more general nonlinearities and showed that the exponent p = 2 in the nonlinear term u p , is critical for the three points blowup described here. Related studies are done by Nakagawa [273], Nakagawa and T. Ushijima [274] and Chen [76]. See also Berger and Kohn [33]. The validity of scheme (6.15), especially approximating the blowup time, was recently studied by T.K. Ushijima [399]. Another important application of FDM to the theoretical study on the semilinear parabolic equations was done by Tabata [376]. It studied the Euler approximation to
240
6. N o n l i n e a r P r o b l e m s
Figure 6.2: Behavior of blowup solution to ut = uzx + u 2 with u[x=o,1 = 0. The time discretization is subject to rn = r . min (1, IlunllL ~ ) with r > 0. We take as the initial value uo(x) = #x(1  x)(x 2  x + 3/10) with a constant p. We note that uo(x) has two peaks. These figures show the behaviors of blowup solutions when (a) # = 1000 and (b) # = 5000. In the farmer, the blowup takes place at only one point, but in the latter, it occurs at two points.
one space dimensional problem under tile Neumann boundary condition for general nonlinearity, and showed that if r / h 2 < 1/4 then the property of tile monotone decreasing of the number of peaks of the solution arises, where h and r denotes tile space and the time mesh size, respectively. Its continuous version was provided by Matano [260], and Angenent [11] showed sharp profiles of the movement of peaks. Stability of the FDM to the reactiondiffusion system was studied by Mimura, Kametaka, and Yamaguti [265]. Then, many works have been devoted to the discretized semilinear parabolic equations. See Thomde and Wahlbin [385], Thomde [383], and the references therein. 6.3. Concerning physical backgrounds of (6.30), see Friedman [130]. The hyperdispersivity of operator  A associated with the degenerate parabolic equation was proven by Br~sis and Strauss [65]. It was Crandall and Liggett [96] that proved the fact that a hyperdispersive operator in a Banach space generates a nonlinear semigroup by formula (6.33). For this abstract theory and also the regularities of the solution, see Barbu [26] and Miyadera [266]. The work by A.E. Berger, H. Brdzis, and J.C.W. Rogers was published in [31]. Figure 6.3 shows an example of numerical computation by their scheme. Nonliner Chernoff formula was proven by Brezis and Pazy [64]. Recently, Evans [120] and Barles and Georgelin [27] observed that a scheme by Bence, Merriman, and Osher [28] for constructing mean curvature flows is justified by that formula in the framework of the levelset approach due to Osher and Sethian [310], Evans and Spruck [121], and Chen, Gig& and Goto [77]. See Crandall, Ishii, and Lions [95] for basic notions of the levelset approach. Scheme (6.35) was studied by Rose [328], Nochetto and Verdi [304] and Kacur, Handlovicovg, and Kacurovg [195] for the porous media and tile Stefan nonlinearities, ~ d by L ao x [2351 and Lesaint and Pousin [236] for (6.31) with ~, E (0, 1). It is known that nonlinear semigroup associated with the degenerate parabolic equation (6.30) is generated also in H~(t2). See Nochetto, Savard, and Verdi [305] and the references therein
C o m l n e n t a l y to C h a p t e r 6
241
for numerical analysis based on that point of view.
Figure 6.3: Shapes of approximate solutions to ut = A f ( u ) with u[oa = 0, where f ( u ) = u + 1 (u < 1), = 0 (1 < u < 1), and = 0 . 8 ( u  1) (u > 1). The time discretization is done by the scheme (6.56).
Provided with the properties of order preserving and L 1 contraction, scheme (6.37) gives reliable numerical solutions very rapidly. It is also simple enough to perform actual computations, while theoretical studies are performed in the framework of the theory of nonlinear semigroups. See [63] and so forth for Poincar~Wirtinger's inequality. Kato's inequality for  A was given by [204]. Concerning the topological degree and its properties see the standard monographs on nonlinear functional analysis, Nirenberg [295], Berger [32], Zeidler [416], Deimling [108], and so forth. The nonlinear Chernoff formula was presented by Brezis and Pazy [64]. Scheme is a finite element analogue of the one by [31]. Several versions of nonlinear TrotterKato's theorem are known since the original paper Trotter [387]. A typical form, making use of the Yosida approximation, is described in Brezis [62]. For the linear theory, see Kato [205] and the references therein. Ushijima [394] applied it to the finite element approximations. We adopt the method of B~nilan, Crandall, and Sacks [29] for the proof of Theorem
242
Z Domain Decomposition Method
6.8. See Yosida [410] or Brezis [63] for FrechdtKolmogorov's theorem. If t2 is a polygon, Theorem 6.10 follows if f is strictly increasing, convergence (6.64) holds for ~ > 0 and v C W, and t2 has no corner with angle 2re. For the fundamental properties of the Yosida approximation for maccretive operators such as (6.76), see Miyadera [266]. There, inequalities (6.74) and (6.75) are indicated in the formulas below (4.5) and above (3.49), respectively. If t2 is a polygon with any vertices not to have angle 2re, the elliptic regularity necessary to the proof of Lemma 6.11 is assured.
Chapter 7 Domain Decomposition
Method
Basic idea of the domain decomposition method (DDM) is found by the work of H.A. Schwarz in the 1860's. The domain in consideration is divided into several parts, and the approximate solutions are computed separately and succecively on them, provided with suitable interaction conditions on artificial boundaries. This method fits parallel computations, domains with complex shapes, and coefficients with strong inhomogeneties. Target problems in this chapter are the Poisson and the stationary Stokes system. Tools necessary for our program are; square root of selfadjoint operator, Dirichlet to Neumann map, variational principles, reflection principles, infsup constant, and so on. Several monographs have been devoted to the mathematical theory of DDM such as Chan and Mathew [73], Smith, BjOrstadt and Gropp [352], Quarteroni and Valli [319] and Gu [165]. Especially, the problems which we are going to consider are discussed in [319] in more generalities. Howerver, operator theoretical analysis enables us to gain sensitive relationships between the rate of convergence of iterations and the shape of decomposed domains, as we are now describing. We mainly describe the continuous case, because the essence of the problem does not depend on the way of discretization.
7.1
Dirichlet to Neumann (DN) Map
This section is devoted to the Dirichlet to Neumann (DN) map, which plays a fundamental role in the analysis of DDM. We take the case that f~ C R 2 is a bounded domain, that OFt is composed of two smooth curves 7 and F, and that ~ and F are intersect transversally at two points. Those intersections make corners of f~. The Dirichlet to Neumann map S is defined formally as follows. First, given a function defined on 7, let h solve Ah=0
inf~
and
h=
Then, S is defined by
s ~ = ~~ 243
I,
{~ [0
on4/ on F.
(7.1)
7. Domain Decomposition Method
244
where O/On stands for the differentiation along the outer unit normal vector n, and "iv the trace to 7. This definition, however, has a usual sense only when h is appropriately regular, say, h E H2(f~). One obstruction is the corners of ft. The condition h E H2(f~) does not follow even if ~ is sufficiently smooth. Another is the requirement of later sections to regard S as an operator in L2(7). Those difficulties are overcome by tile representation theorem concerning bilinear forms on Hilbert spaces. Examining the wellposedness of (7.1), we take more generally that
A,u=f
inf~
with
u=
[~ 0
l
on7 on F,
(7.2)
where f E L2(f~). We introduce a closed subspace I l l ( a ) of HI(f~) by
 {v e H ' ( ~ ) I
Ir
v r  o}.
Poincar~'s inequality is still valid for functions in KI(Ft), and the Dirichlet norm is equivalent to the H l ( f i )  n o r m in I(l(ft) 9
IIV.llg=(~)
~
I1 I.,(~)
The underlying Hilbert space in our consideration is X = L2(7). The standard L2(7) inner product and norm are denoted by (., ")x and I I Ix, respectively. We set
Holo/2 ( ~ ) = { ~ e H1/2 (~) ~ p 1~2 ds < + ~ } with
II~llglg:<+ =
I1~11~,,~(+
+
p'~
ds
,
(7.3)
where p = dist,(., 07) denotes the distance along 7 from tile end points of 7, and ds the r4~/2 (7) becomes a Hilbert space equipped with the norm line element of 7. The space :oo 1/9
(7.3). P u t t i n g V = H00 (7) and[ lie  II .11.~o~<~), we thus obtain the triple V C X C V' of Hilbert spaces. In the present chapter, symbols C and C' are used to denote positive constants depending only on the domain in consideration. The values of these constants may differ from each other in a single context. When we want to specify the dependence of such constants on a parameter p, which may9 not be a number, we write Cp or C'p" In spite of the corners of [~, trace theorem keeps to hold in the following way. Proposition
7.1.
We have the following:
(i) Every ~ E Kl(f~) admits ~ = "~h~ E V satisfying I~llv < C I ] ~ ) l l [ [ l ( ~ ) . (ii) Every ~ E V admits ~# E t(~(f~) satisf~ling ~ ~ = ~ and I1~1 +,,(~) _< c ' I~llv.
245
7.1. Dirichlet to Neumann Map
Consequently, problem (7.2) takes a unique solution for given ~ E V and f E L2(f~); we have a unique u E Kl(f~) satisfying u], = { and
(Vu, VV)L~(a) = ( f , V)L~(a)
(7.4)
for any v 9 H~(a). In fact, if u is a solution for ~ = 0 and f = o, then u E H~(f~) follows. The relation IIVullc2(n) = 0 holds by (7.4) with v = u. This implies u = 0 from Poinca%'s inequality, and thus the uniqueness follows. On the other hand, given ~ E V, we take its extension ~( E Kl(f~) satisfying [~)~[[H'(f~) ~ C [[~[[v" Because the linear functional
v E H~(f~)
~
(f, V)L2(a)  (V{;~, VV)c2(a)
is bounded, we have w E Hl(f~) satisfying (Vw, Vv)c:(a) = (f, V)c2(a)  (V~e, VV)L:(~)
(7.5)
for v E H~(a). T h e n u = w + r solves (7.4). If f = 0, u is nothing but the harmonic extension of ~ into f~. This case is referred to as u = ~ . Then we have ]lVw La(a) < IlVCelln~(a) by (7.5). The following inequality holds by Proposition 7.1 and Poinca%'s inequality, where ~ E V:
c' IIg Iv < IV~gll~(~)_< c I1~11~
(7.6)
In fact, we have
c ' I~llv <
I ~ll,,~(r~)~ IlWll~(~)= I I V ~ l L~(~) < IIW, ~(~) + IIV~,~ll~(~) < 2 IIV~l ~(~) < 2 I1r HI(~) ~C II~llv.
We define a bilinear form ,.7 9V x V ~ R through the following relation, and call it the J  f o r m pertaining to (f~, 7)"
j ( ~ , .) = ( v ~ ,
v~,)~.(~)
Because of (7.6), it is bounded and coercive; we have
J(~, ~)>__ c '2 II~ll~,
[fl(~, ~)l _< c 2 I~ Iv II~llv
and
(7.7)
for ~, r/E V. Therefore, a positive selfadjoint operator S 9D ( S ) C X . X is introduced, associated with J :
J ( ( , r~)= (S[, r])x
(~ E D ( S ) C 17, r]E V)
General theory now guarantees the following. P r o p o s i t i o n 7.2. We have D ( S 1/2) = V and J ( ~ , ~) = ( $ 1 / ~ , S1/~r])x for ~, r]E V. The mapping S 1/2" V >X i8 an isomorphism. In particular, we have
c' I ~llv < Ilsl/~Jl~ _< c II,~llv for~e
V.
(7.8)
7. Domain Decomposition Method
246 If h = 7{~ is suitably regular, we get
f
ob
for r/ E V. This means S'~ = (Oh~On)Iv in the sense of distributions, and in this way the Dirichlet to N e u m a n n map is realized as a selfadjoint operator in L2(',/). We write S = S(t2, 7) and call it the D N map pertaining to (t2, 7), to express this correspondence. The o p e r a t o r S 1/9 can be characterized by the variational principle. In fact, the variational principle for the harmonic equation claims t h a t
IIV~ L=(~)~ IlVv L~(~) for ~ E V, where v E Kl(f~) is so taken as v[~ = ~. This, together with Proposition 7.2, yields the following.
Proposition 7.3. Given~ E V, we have
Is,/=
IIV,, L=(~), o~ equivalently J ( ~ , ~ ) <_
[IVv[l~2(~), for v E I('(t2) satisfying vl~ = ~.
7.2
Dirichlet
to Neumann
(DN)
Iteration
Studying the typical domain decomt)osition algorithm, we assume t h a t fi C R ~ is a b o u n d e d d o m a i n with the Lipschitz b o u n d a r y F = 0g~ and take the problem Au=f where f E L2(f~).
~=~
in t2
with
u=0
onF,
(7.9)
The exact solution of this taryct problem is henceforth denoted by
H~(a).
We divide the target domain t2 into two disjoint subdomains t21 and t22 by an artificial boundary, a smooth simI)le curve denoted 1)y 7: n
f~ = f~l U [22,
fil N ~22 = O.
We assume t h a t 7 connects transversally two points oil F. The outer unit normal vector is generally denoted by n, but we specify the one to 7 outgoing from t21 by u if necessary. We put F1 = 0t21 \ 7 and F~ = 0t22\"/. Dirichlet to N e u m a n n (DN) iteration is as follows. Take a function it (~ E V as the initial guess. Then, for k = 0, 1, 2 , .  . , we sllccessivcly generate:
,tt(1k) :
the kth a I)proxiInation to 'u,[~2~;
u~k ) : p(k+l) .
the kth api)roximation to 'u,[~2" the (k + 1)th aI)I)roxima,tion to "~.]~
247
7.2. Dirichlet to N e u m a n n Iteration
by solving
~''~1
/Xu~ k) f
in
Au~k) = f
in f12
with
u~k) __ l 0
with
{
on 1'1
#(k)
on 7
on F2
u~k/= 0
Ou~k) On 
Ou~k) cgv
on 3'
The value of p(k) is adapted by #(k+l)
_._
(1  0)# (k) +
o~)1~,
(7.10)
where 0 is the relaxation parameter subject to 0 < 0 _< I. This procedure is rigorously understood as follows: I) Take p(0) E V and put k = O.
2) Find u~k) e /(1(~~1) satisfying
~")1~=
,(~) and
(~7"U'k)' ~ VVl)L2(f~I)


"
( fial , Vl)L2(al)
for vl E Hl(f~l). 3) Find u~k) C Kl(f~2)satisfying
(vu~ k), VV2)L~.(~) 
V2)L~(n:) 
(fln~,
(Vu~ k), V~)I)L2(f~I) ]
for v2 e K'l(a2), where 7./)1 E Kl(f~l) stands for an extension of
(flf~l' ~)1)L2(al)
~l~.
4) Renew p(k) as (7.10). Set k = k + 1. Go back to 2) until convergent.
Amplification Operator Taking attention to the convergence, we introduce error functions ~k) = u~k)_ Ulal, ~k) = u2(k)  u]a2, and ~(k) = #(k)  '51~. We obtain A~p~k)  0
in ~1
with
Ar k) = 0
in f~2
with
@[k) = { 0 ~(k)
on F1 on 7, on F2
0
On and =
(1 
&,
on 7,
7. Domain Decomposition Method
248
Letting ,5'1 = E(~I, "y) and S2 = S(f~2, 7), we have Sly(k) _ 0~)~k)_. J _ 0%, "r
01/)~k)
__ $2 (~z~:k)[7)
On
and hence ~b~k)l~ =  S [ ' & ~
(k) follows. The (k + 1)th error ~(k+l) can be expressed, at least in the formal manner, as {(k+i) = (1  0){ (k)  0S;~&~ (~/ This formula is justified by the following. L e m m a 7.4. The operator Ho = 321S1 with its domain D(Ho) = D(S1) admits of a bounded extension H into V, explicitly written as
H
S21/2(S~/2S21/2) *.q'l/2 J,a 1 ,
(7 l l )
where * denotes the adjoint in X . Proof: It is easy to verify H0( = H ( for ~ E D(Ho). T h e boundness of H follows from ql/2 ql/2 S~ 1/2 E s V), (S~/2S~1/9) * E s and ~'1 E s In fact, since ~1 S21/2 C L;(X), the adjoint operator (S~/2S21/2) * is also b o u n d e d in X. [] Replacing $21El by H, we get the recursive expression
~(o) e V,
~(k+l) _ Ao~(k)
(k  O, 1 , 2 , . . . )
for Ao = (1  0)I  OH. We call it the amplification operator for the error of the DN iteration. A l t h o u g h H is not sclfadjoint with the usual X  i n n e r product, we can treat it as a selfadjoint o p e r a t o r in V. For this purpose, we introduce a special inner product in V in terms of the DN m a p as follows:
((~ , 77))= (S~/2e , s~,/'e ~)~
(~, ~ c v )
The space V again forms a Hilbert st)ace with this new inner product. corresponding n o r m II1" Ill = ((', .))1/2 is equivalent to I1" IIv in V by (7.6).
(712)
In fact, the
L e m m a 7.5. Under the new inner product (7.12), the operator H defined by (7.11) is selfadjoint in V.
Proof." Since H is b o u n d e d in V, it suffices to verify t h a t H is symmetric. This is seen from
((H~, 7]))  ((X~/2E21/2)* S~/2~, ~b2r~l/27])XX= (X~/2~, Ell/2T]) x, ((~, HI]))  (S~/2~, (S~/2S21/2)*S~/2I])x  (Sl/'2~, S~/2I])x. The proof is complete.
[]
7.2. Dirichlet to Neumann Iteration
249
Since H is bounded and selfadjoint in V with the inner product (7.12), Ao has the same properties. The recursive formula for ~(k) implies (7.13) for k = 0, 1, 2 , .   , where ro(Ao) denotes the spectral radius of Ao. The spectral mapping theorem now guarantees the following. L e m m a 7.6. We have
ro(Ao)=
sup [ 1  0  0 s l , sEa(H)
where or(H) denotes the spectrum of H. Proof." Because Ao is expressed as a linear function of H, we have
~(Ao) :
sup ~6a(Ao)
Ir
=
sup I1  0  Osl, sEa(H)
where a(Ao) stands for the spectrum of Ao. The proof is complete.
C o n v e r g e n c e w h e n 7 is a L i n e S e g m e n t The spectral radius of Ao is evaluated in accordance with the geometry of f~i (i = 1, 2). First, we shall deal with the case that 7 is a line segment on the z2axis. Furthermore, we say that the condition (I) is satisfied if 7~Lf~2 C_ f~l, where 7~c denotes reflection with respect to the z2axis: 7~r: ( x l , x 2 ) ~ (  x l , xg). L e m m a 7.7. If condition (I) is satisfied, then we have 0 <_ H <_ 1, so that 0 <_ s < 1 holds for any s E or(H).
Proof: We take Jforms ,.71, J2, and ,72* pertaining to (f~l, ~'), (f~2, 3') , and ( ~ * 2, ~), _I1/2 respectively, where f~29 = 7~rf~2. Let ~ E V = /00 (~/). We have ,72(~ , ~) = ~ , *2 (~ , ~) from the symmetry. We show J l ( ~ , ~ ) _< J2*(~,~) In fact, let h~ be the harmonic extension of ~r into f~, and v its zero extension into f~l" 1,
Ah; = 0 = ~h~ V
[0
inf,;
with
h;=
~~ 0
[
in f~ in f~l\f~
Then the variational principle Proposition 7.3 implies
on7 on0f~;\~,
250
z Domain Decomposition Method
Therefore, we get
o _< ((S~, ~)) <_ II1~111=
by Oq'l(~, ~)  ((H~, ~)) and f12({, {) = II1~111~. The proof is com)lete.
[]
Lemmas 7.6 and 7.7 imply r,(Ao)<_ sup I i  O  O s l = ? ( O ) .
0<s
This, together with (7.13), gives
II ~(~)111___~(0)lll~(~')lll _< ~(0)~111~(~ Because we have positive constants c~) and c~ determined by t22 satisfying
Iv < (O) co II (~
c0
by (7.8), we get the following theorem with Co = cg/c'o. T h e o r e m 7.8. If condition (I) is satisfied, then there exists a constant Co > 0 determined only by f~2 such that
I1 ( )11 _< co#(0) I1(~
~
(7.14)
for k = 1 , 2 , 3 , . . . .
Here, we have
f(0)=
1 0 201
(0 < 0 _< 2/3) ( 2 / 3 _ < 0 < 1)
and hence 0 < f(0) < 1 follows if 0 C (0, 1]. Furthermore, (7.14) is equivalent to
I1~~ 
~1~1I~ _< ~o~(0)~1I# ~~ ~1~I1~.
This implies the convergence of the approximate solutions {u}k)}. Theorem
7.9. Under the same assumption of Theorem 7.8, we have
[lUlk )  ?~ll2,IIHl
C1r
k [U~0 )  ?~[f~lI[Hl(f~l),
11'4~> ~[~211n1(~2) _< c~(0)~11~5 ~
~ ~2llHltm),
(7.15)
(7.16)
where Cl and c2 are domain constants. Proof:
Inequality (7.15) is a consequence of Proposition 7.1 and (7.14). Let '/3 be the I
harmonic extension of fi = 'v~k)[ into f~l" 1
7
L 0 on (~1 \'~.
251
7.2. Dirichlet to Neumann Iteration In use of
we obtain
IlVv~k)~
vv~ ~)
Wll~
This implies
In a similar manner, we get
I Vv[ ~) IL~(~)<_ CliVv~)llg~(~). Inequality (7.16) follows from (7.15).
[]
We proceed to a natural modification. Given m > 0, let
We say that condition (I,~) is satisfied if 7~LTmf~2 C_ ~1 holds, where Tmf~2 denotes the image of f~2 by Tm. L e m m a 7.10. If condition (I,~) is satisfied, then we have 0 <_ H < m. Proof: The Jform pertaining to (fY, 30 is denoted by J ' , where fY = TmQ2. Similarly to the proof of Lemma 7.7, we have Jl(~,~) < J ' ( ~ , ~ ) for ~ E V. Taking the harmonic ~xt~n~ion h~ of ~ into a', w~ p~t ~ ( ~ , ~ ) = h ~ ( ~ l , ~ ) for ( ~ , ~ ) e a'. We h~v~ v E KI(Q ') and
_< m & ( ~ , ~ ) . Y'(~ , ~) _< IIVvll~(a,) ~ This implies
Oq"l(~, ~) ~ mJ'2(~, ~), and the proof is complete.
[]
Similarly to Theorems 7.8 and 7.9, we have the following. T h e o r e m 7.11. If condition (Ira) is satisfied, then we have (7.14) with ~(0) defined by 10
( 0 < 0 < 2)  m + 2
(7.~7) (m+1)01 and a constant co > 0 determined by Q2.
m+2
<0< 1 
7. Domain Decomposition Method
252 Theorem
7.12. Under the same assumption, we have (7.15) and (7.16) with f(O) defined
in (Z1"7). This time, ~:(0) C (0, 1) follows from 0 < e _< 1 and 0 < e < 2/(m + 1). Exchanging the roles of ~ and ~2, we define the condition (Ie) in accordance with (Ira), where g > 0: 7ELTe~I C_ ~2. Then, conditions (Ira) and (I e) imply J l ( { , { ) < rnJ2({,{) and J2({,{) _< gJl({,{), respectively, for { C V. We get the following, where the case g = oo is permitted under the agreement of g1 = 0. L e m m a 7.13. /f both conditions (fl) and (In) are satisfied, then we have g1 ~ H < m. 7.14. Under the same assumptions, inequalities (7.14), (7.15), and (7.16) hold with f (0) defined by
Theorem
1_(1_+_e_1)0(0<0 (m+l)O
Optimal
~1
2
 m + ~ 1 + 2
1
)
<0<1 77l[~
If 0 < 0 < 1 and 0 < 0 < 2 / ( m +
<
1
{2


"
+ 1), this ~(0) is in (0, 1).
Rate of Convergence
Theorems 7.8, 7.11, and 7.14 imply tile following. Theorem
7.15. As an optimal value ropt of f(O), we have the following.
1~ /f condition (I) is satisfied, then ropt

1/3 is assumed by 0 = 2/3.
2 ~ If condition (Ira) is satisfied, then fopt = m / ( m + 2) is assumed by 0 = 2/(m + 2). 3 ~ If both conditions (fl) and (In) are satisfied, then ~opt = (m + g~)(m + g1 + 2) is assumed by 0 = 2 / ( r n + g1 + 2). One can show that the choices of 0 in the above theorem are really optimal in the following sense. Namely, let ~ be a rectangle and 7 a line segment parallel to the lateral sides of the rectangle with 0 < a1 _< a2 and b > 0:
~1
=
{(x,,x2) [  a l < :I:l < O, 0 < .T2 "~ b } ,
a~
=
{(*l,~)10
w h e r e T = { ( 0 , x2)[ 0 < : r 2 < b } . A~
\
b ]
< * ~ < a~, 0 < ~ < b},
We note that and
qZ,,(x2)=
sin x/~,,x2
253
7.2. Dirichlet to Neumann Iteration are the eigenvalues and the eigenfunctions, respectively, of
 d 2 ~ / d x ~ = A~
with
~ ( 0 ) = ~ ( b ) = 0,
where n = 1, 2 ,   . . Given { E V, we have O(3
' E Cn~n n=l
for cn = Cn(~) = (~, ~Pn)x. The harmonic extensions hi of { into f~l is written explicitly. It holds that oo
Ohi
= ~
~ X 1 Xl0
x / ~ c , coth \
b /~,(x2)
n=l
where coth s = (e s + eS)(e ~  eS) 1. Therefore,
( na l Tr"~
holds by cj = (qPn,~j)x = an,j, and A(1) = x/~coth(na17r/b) (n = 1 , 2 ,  .  ) are the eigenvalues of SI. Similarly, A(2) = ~ c o t h ( n a 2 7 r / b ) (n = 1, 2 , .  . ) are the eigenvalues of $2, and 9n is an eigenfunction of $1 and $2 corresponding to A(1) and A(n2), respectively. It can be shown that H is a compact operator in X. The spectrum of H coincides with the set of its eigenvalues, {/~(nl)//~(n2)}n~176 1. B e c a u s e /~(ni)//~(n2) is nonincreasing in n, we have ((H~, ~))
sup ~ II1~111  ~ ~
)~I)
A~2)
tanh(Tra2/b) _ tanh(Trai/b) = r(al, a2, b)
and inf ((H~' ~)) _ inf
II1111
= 1
This implies 1 < H < 7(al, a2, b). Both conditions (I e) and (Ira) are satisfied with f = 1 and m = a2/al. assures
1 < H < a__~2.
L e m m a 7.13
(7.18)
al
Because T(al, a2, b) , a 2 / a 1 as b ~ +oc, estimate (7.18) is really optimal. This (f~, 7) satisfies conditions (I z) and (Ira), and the really fastest speed of convergence is arbitrarily close to ro~t. Similar tendencies are observed for other (f~, 7)'s numerically; the universal optimality is realized by the choices of 0 in Theorems 7.157.15 if (f~, 7) satisfies (I ~) and (Ira) . We call them the general optimal choices of 0 in this meaning.
254
7. Domain Decomposition Method
Convergence
w h e n 7 is a C i r c u l a r A r c
We proceed to the case that 3' is a part of a circle with radius R and with its center at the origin of the coordinate plane. We take the circular reflection with respect to the circle, where (r, ~) denotes the polar coordinates"
We say that the condition (R) is satisfied if 77.cf~2 c f~l. Let f~ = 7~cf~2 and take Jforms ,.72 and J2* pertaining to (f~2, "7) and (f~, "7), respectively. Then f12(~, [) = J2*([, [) holds for [ E V from the reflection principle, and the following facts are proven similarly. L e m m a 7.16. If condition (R) is satisfied, then we have 0 < H < 1. Theorem
7.17. Theorems 7.8 and 7.9 remain valid with the condition (I) replaced by
(R). The condition comparable to (Ira) is given in terms of
T~,R" (r,
~)
~
/~ + ~ ,
~
m
We say that the conditions (Rm) and (R e) are satisfied if 7r f~2, respectively.
C_ f~l and TCcTe,Rf~I c
L e m m a 7.18. If condition (Rm) is satisfied with m > 1, then we have 0 < H < m. If
(R e) is satisfied with g >_ 1 furthermore, then f1 < H < m follows. Proof: First, we take the case that f~2 is adjacent to 7 from outside, and hence r > R holds on f~2. Let J ' be the Jform pertaining to (~', 7), where s = T,,,,Rfl2. Then we have o"7"1(~,~) J ' ( { , { ) for ~ C V. We take the harmonic extension h.2 C I('(f~2) of ( into s and put v(r', qp) = h2(r, qp) with r' = r + (r  R ) / m . We have v E I ( ' ( ~ ' ) and ,


L'2(fl,) = .
~
,
k, Or'J
+ 777 ~
(Oh.2 2r' )  9rdrdqp + L
1
(
r'dr'd~
c3h2
)
2
r 9 l 9rdrdqp
_< mS~(<,,~), because 1/(m 2) _< r'/r _< 1 on ~22 from the assumption. This implies J~((, <) _< m,.7:,2((, () and the first assertion follows. The second assertion is a consequence of the reflection principle. Let, J~* and J2* be the Jform pertaining to (s V) and (f~,~,~/), respectively, where s = Tr163 and ~ = TCcFZ2.
7.3. Dirid]let 2 to Neumann 2 Iteration
255
We have J l ( ~ , ~ ) = J~*(~,() and J 2 ( ( , ~ ) : J2*((,() for ~ e V and J2*(()  eJl*({) similarly. This implies the conclusion in this case. The other case that r < R on 91 is treated similarly by the reflection transformation described above. [] We can argue similarly to the case that 7 is a line segment by L e m m a 7.18. Theorem
We have the following.
7.19.
1~ Theorems 7.11 and 7.12 remain valid with condition (Ira) replaced by (Rm). 2 ~ Theorem 7.14 remains valid with conditions (~) and (Ira) replaced by (R e) and (Rm),
respectively. 3 ~ Theorem 7.15 remains valid with conditions (I), (1t) and (Ira) replaced by (R), (Rt),
and (Rm), respectively.
7.3
Dirichlet 2 to Neumann 2 (DDNN)
Iteration
For the same target problem and decomposition of the domain as in the previous section, we take a simultaneous algorithm, Dirichlet 2 to N e u m a n n 2 (DDNN) iteration. Namely, letting #(0) E V be an initial guess, we solve succecssively the following, where p and q are the relaxation parameters subject to p >_ O, q > O, pq # O, and 0 < p + q <_ 1: Au~k) = f
in ~"~1
with
u{k)= { 0
it(k)
k u~k) = 0
Au~k) = f
in f~2
in ~'~1
on F2
(Ou~k) On =
Ou~k) Ov
v?)={ ~
onr
with
with Av{ k) f
{
on FI on ~,
with
k #(k) on 7, v{k) = 0 on F1 OV~ k) OV~k)
{
.(k+l) __ (1  p  q)p(k) + pu~k)l~ +
on 7,
Ov =
qv{k)l~"
On
on 7,
Introducing the error function ((k) = #(k)_ ~[~ defined on O/, we can derive formally that ~(k+l) __ (1  p  q  F S 2 1 S l
 q S 1 1 ~ 2 ) ~(k).
Rigorously we have ~(o) E IS,
~(k+l) = Bp,q~(k) (k = 0, 1 , 2 ,  . . ),
256
7. Domain Decomposition M e t h o d
where Bp,q = (1  p  q)I  p H  qH 1. The operator Bp,q may be called the amplification operator for the error of the DDNN iteration. Expressed as a rational function of the operator H, it is bounded and selfadjoint in V with respect to the inner product (7.12). We have the follwing. P r o p o s i t i o n 7.20. If both conditions (I e) and (Ira) are satisfied, then we have sup
ra(Bp,q) <
11pqpsqsl[,
~l<s<m
where ra(]Bp,q) denotes the spectral radius of t~p,q. Proof:
guarantees
We have ~1 _< H _< m by Lemma 7.13. The spectral mapping theorem now
ra(Bp,q)
sup
sup I i  p  q  p s  q s  l l <
]l  p 
q  ps  q s  l [ .
e 1<s<m
sea(H)
The proof is complete. For the moment, we pick up the case that p = q and 0 = 2p = 2q. The following theorems are proven similarly to the DN iteration. T h e o r e m 7.21. Suppose both conditions (I e) and (In), and let g <_ m without loss of generality. Suppose, furthermore, that there is a 0 satisfying 0 = 2p = 2q and 0 < 0 < 4 m / ( m 2 + 2m + 1). Then, we have
IIv  coe(~
v
for k = 1, 2, 3 , . . . , where Co > 0 is a domain cortstant and 1
~(0)=

[
O
1 ( 1 )
1 is in
(
20
+~ ~ +
0I
(
4m
1
)
<0<1)
mP + 6 m + 1 
(o, I).
T h e o r e m 7.22. Under the assumption of Theorem 7.21, the general optimal choice of O is attained by 0 = 4 m / ( m 2 + 6m + 1) with the optimal value m 2  2m + 1 7"opt ~ m2 I6m + 2"
T h e o r e m 7.23. Theorems 7.21 and 7.22 remain valid with conditions (I e) and (Ira) replaced by (R e) and (Rm), respectively, for g, m >_ 1. We only state the optimal choice of p and q for tile general case, because the convergence result for the DDNN iteration is complicated then.
257
7.4. Robin to Robin Iteration
T h e o r e m 7.24. Suppose that both conditions (I e) and (Ira) are satisfied with g < m.
Then the general optimal choice of p and q for the DDNN iteration are attained by p = 2g/am,l and q = 2m/c~m,l with the optimal value
me 2x/~+ 1 ropt =O~md
where a,,~,e = me + 2(m + g + ~ )
+ 1.
T h e o r e m 7.25. Theorem 7.24 remains valid with conditions (Ira) and ([) replaced by
(Rm) and (Rl), respectively, for g, m >_ 1.
7.4
R o b i n to R o b i n ( R R ) Iteration
Again we are concerned with problem (7.9), and keep the assumption and the notation of w on the domain f~ and its decomposition. The iterative scheme in this section, which is called the Robin to Robin (RR) iteration, uses the Robin boundary conditions. Given u ~ C H2(f~), we take u[ ~ = u~ and u~~ = u~ as initial guesses. We successively solve {u~ k) = 0
Au~ k)  f
cgu~k) +
in f~l,
o n 1' 1
ou~k )
gV.
0,tt~kx) =
t OU~k  l )
on 7
On
and
{
u~k) = 0
Au~k) = f
Ou~k)
in f~2,
on ['2
Ou~k) _
(:gu~kl) ~ OU~k  l )
+

on 7
O~
with the relaxation parameter 0 > 0. Convergence of scheme is reduced to the case of f = 0. The following argument is due to Q. Deng. First observation is the identities
vui
I dx
~r]i tyui
.(q
i
for i = 1, 2, where
~?~k)_ cgu~k) 
07
Ou~k) +
and
~ k ) _ cgu~k) 
Ou~k)
o7 +
_(k+l) = 20u~k) _ @k) for i r j. This implies bk+ 1 = b k  4Oak for Here, we have qi
1
2
7. Domain Decomposition Method
258 and hence
1 0 _< ak = ~~(bk  bk+l) follows. The nonnegative sequence {bk}k is nonincreasing sequence in k and the limit lim bk exists. Consequently, ak , 0 as k , oo. We obtain the following. T h e o r e m 7.26. Sequences {u~ k)} and {'u~k)} generated by the R R iteration converge to the exact solution ~t of (7. 9) in the sense that
I1~~) ~la, II~l(a,)
+
Ilu~~) ~ln~ll~l(a~) ~ 0
as k, c<).
7.5
Exterior
Problem
DDM is effective to the equation on u n b o u n d e d domains. We take the exterior domain, so t h a t f~ = R 2 \ O with O C R 2 being bounded. Consider Au=f
inf~
with
u{=O ~
on O 0
o
as Ixl~ +oo,
where f e L2(f]). We employ the polar coordinates (r, ~), and assume that O contains the origin. Taking R > 0 sufficiently large, we put ER = { (r, g)) I r > R} and divide f~ into f~l = En and f~2 = f~\En. Thus, 7 = OER is regarded as an artificial boundary. Letting p(0) be an initial guess, we take Au]k)= f
in f~l
with
t" UI /
(k) ]~ 0
{
= p
~k)
~L~~) = 0
in f~2
 A u ~ k)  f
with
#(~+1) = (1  o)# (~) + o~)1~,
10'u~k)
R Or
as r ~ + o c on 7,
10u~k) =R
Or
o n OO
ouT,
where 0 E (0, 1] is the relaxation parameter. T h e Green's function for the first problem is explicitly given by 1 R 4 + r2r ' 2  2rr'R 2 c o s ( g )  ~') G(x, .T.') = ~ log R 2 (r 2 + r' 2 _ 2rr' cos(~  g)')) ' where x = (r, ~) and x ' =
~ / ( ~ , ~) =
(r', g/). We have
r 2  /~2 f 2 ~ 27r
Jo
#(k)(~9, ) R 2 + r 2  2Rr c o s ( ~  ~')
dqp' + 9f~ G ( x , x ' ) f ( x ' ) dx' 1
7.5. Exterior Problem
259
and hence
RlOu{k)Or(qo) = ~ 1 ~2~ It(k)(qJ)sin 2 ~~' &;' +~ ,~lj~ [~r G(z, z, )] f(z') doe' 2
(7.19)
1
follows in the second problem. We can avoid the unboundedness of the domain, under the cost of the singularity of the integral in this way. Convergence analysis is made similarly. Let V = H1/2(7 ) and ts
= {v E H1(~2) ] rio 0 = 0}.
The DN maps S1 = S(Ftl, 7) and $2 = S(~9, 7) are defined as before. We have J 1 (~, ~)
IlVhl 2
=
2
and =
S1/2~
2
with the harmonic extensions hi c HI(~I) and h2 E _~1(~2) of ~ E g into ~1 and ~2, respectively. The error function ((k) = it(k) _ ul~ satisfies ~(k+l) = (1  0)~ (k)  0 S 2 1 S 1 ~ (k),
and hence the amplification operator is given as
Ao = ( 1  0 ) I  0 H with the bounded extension H of S~1S1 into V. Provided with the inner product (7.12), we have
r,(Ao)=
sup [1  o  o s [ ,
se~(H)
where ro(Ao) and a ( H ) denote the spectral radius of Ao and the spectrum of H, respectively. For the moment we study the case (.9 is a disc and hence ~2 is a circular ring. L e m m a 7.27. We have
R ~ + R~ 1 <_ H _< R2  R~)
(7.20)
i f O = {(r, qo) I r < R0} withO< Ro < R. Proof: We have FZ2 = {(r,~)[ R0 < r < R}. Given { c V, we take the harmonic extensions hi and h2 of { into ~'~1 and t~2, respectively. In terms of the Fourier series, = {(q;) is written as oo
~(~) =
~g~(~) n1
Z Domain Decomposition Method
260 for gn(~) = an cos n7) + bn sin n~. This implies oo
(r)n
n=l
and
h~(~, ~) = ~ n ~ n  ~  no~R ~g~(~) n=l Because of
7r
n=l
n=l
we have oo
,.71(~, ~) = E
oo
n(a~ + b~)
and
n1
n=l
where R~ R n
~ R O n R n
<
R 2 [ R 02
O~n  D
by R o / R < 1. Hence
R 2 + R~ &(~'~) < n ~  ng J`(~'~) follows. On the other hand, h~(r, ~) = hl(R2/r, ~) is nothing but the harmonic extension of into (9 from the reflection principle. The variational principle now guarantees al(~,~)
= IlVhlll ~L2(O) < I I v ~ l l ~ , (M) =
&(~,~),
where It2 denotes the zero extension of h2 into (9. Those inequalities imply (7.20) similarly to the proof of Lemma 7.7. Now we proceed to the general case. L e m m a 7.28. Suppose that (9 C R 2 is a bounded domain containing the origin, and let
R o  i n f { g > 0 I (_gCBe} with Be = {(r, ~ ) [ r < g}. Then, inequality (7.20) follows.
261
7.6. The Stokes System
Proof." Let hi and h2 be the harmonic extensions of ~ c V into ~'~1and ~2, respectively. The inequality Jx(~, ~) _< &(~, ~) follows similarly to the proof of Lemma 7.27. On the other hand, Lemma 7.27 implies R2
+/702
IlVk~ll ~L2(f~2) < R2__ R02 Jx({, {) where h2 denotes the harmonic extension of ~ into
(~ = {(<, ~ ) l R o
< < <
~}.
Combining this with the variational principle
we have R 2 + R~ S2(~, ~) ~ /~2 _ ~_~'Sl ~ (~, ~)
The proof is complete. Lemma 7.28 implies the following. T h e o r e m 7.29. The error ~(k) decays exponentially if 0 < 0 < 1. More, precisely, we have
I1(~ IIv
(k) IIv for k = 1, 2, 3,., where
,~(0) =
1R2
+ R~
0<0<

2R2 + R~
( R~ + R~ ) 2R 2 + R 2~ <_0< 1 .
201
T h e o r e m 7.30. The general optimal choice of 0 for the exterior DN iteration is attained by 0 = (R ~ + R~) / (2R ~ + R~) with the optimal value ro~ = R ~ / ( 2 R ~ + R~).
7.6
The Stokes System
In this section we study DDM applied to the Stokes system Au+Vp=f
and
V.u=0
in~.
{7.21)
7. Domain Decomposition Method
262
It describes the motion of a viscous incompressible fluid with slow velocity. Here, t2 C R 2 denotes a bounded domain, and u = (u 1,u2), p, and f = (fl, f2) stand for the flow velocity, the pressure, and the external force, respectively. Density and viscosity of the fluid are assumed to be the unity. Homogeneous Dirichlet boundary condition is treated in w u = 0
on 0t2
(7.22)
This assumption indicates that the fluid is nonslipping on the boundary. There exists a unique ( u , p ) E Hi(t2) 2 x {L2(i2)/IR} satisfying
A(u, v)  (p, V . V)L2(~) = ( f , V)L2.n)2 and (V.
u, q ) L ~ ( ~ ) = 0
for any ( v , q ) E H I ( ~ ) 2 x L2(~), where A ( u , v) =
Vu  Vv dx =
dx.
i,j=l Oxi ~ The second requirement is equivalent to V . u = 0 a.e. In fact, we get IIv 9UllL~(a) = 0 by taking q = V . u . The problem is reformulated as to find (u, p) C H(~,o(a) 2 x {L2(a)/R} satisfying .A(u, v)  (p, V . V)L2(~) = (f, V)g2(n)2 for v E H(~(f~)2, where H~,o(f~)2 = H~(f~)2 N HJ(f~) ~ with H~(f~) 2 = { v E H l ( f ~ ) 2 I V . v = O } . Its unique solvability is a consequence of the Poincar(~ inequality on H~(Q) and the Helmholtz decomposition Hg(f~) 2 ~ H~,o(l)) 2 x {L2(I))/R}.
Inhomogeneous Boundary Value Problems As in w we take tile case that cOt2 is composed of two smooth curves ~/and F, with and P intersecting transversally at two points. We study the inhomegeneous boundary condition = ~
t
0
As before, n stands for the unit outer normal
on ~/ on F. to F. Let
I(l(t2)= {vEHl(t2)2[ K~(f~)={vcKl(f~)l
for V
=
(7.23)
Vir=0}, Vv=O},
/41/2 "~00 (7) 2. The solenoidal version of Proposition 7.1 is described as follows.
263
7.6. The Stokes S y s t e m
P r o p o s i t i o n 7.31. We have the following. (i) Every v e K I ( ~ ) admits ~ = vl. Y e Vo satisfying II~llv < C (ii) Every ~ C Vo admits v E KJ(•)
IIvlIH~(~)~.
satisfyin v[~ = ~ and IlvllH,(~)= < C' IIr
Similarly to the Poisson equation, lifting argument now applies to inhomogeneous (7.23) for (7.21). Given (~,f) E Vo x n2(~) 2, we have a unique (u,p) E K J ( ~ ) x L~(~)satisfying ul~ = ~ and (7.24)
A ( u , v)  (p, V . V)L2(a) = ( f , V)L~(~)2
for v C H~ (~t) 2. Here and henceforth, p C L02(~) indicates that p C L2(~) and f ~ p dx = O. We make use of the following lemma due to I. Babugka and A.K. Aziz to prove Proposition 7.31. L e m m a 7.32. Given F C L~o(~t), we have ~ C H~(~) 2 satisfying V. 9 = F
in ~
A(~, ~) <  ~ IIFll ~L 2 ( g t )
with
,
(7.25)
where 7 is a domain constant. Proof of Proposition 7.31: The first case is an immediate consequence of Proposition 7.1 and the divergence formula of Gauss: O = J~ V . v dx = L ~ v . n ds
The second case is shown as follow. Given ~ C Vo, we take its extension ~p C Kl(~t) according to Proposition 7.1. Here we have F  ~ . ~ E L~(~) by LF
dx= LV'r
dx= L~'nds=O.
Lemma 7.32 gives ~ e H~(~) 2, provided with (7.25). Then v = ~  r satisfies the desired properties and the proof is complete. [] We proceed to the mixed boundary condition ~n  pn = g u=0
on 7 on F,
(7.26)
where ~~  pn = ( V u ) n  pn =
nj Ozj
 phi
. i
264
7. D o m a i n D e c o m p o s i t i o n M e t h o d
Smooth u and p solving (7.21) satisfy, by Green's formula, that Vu
p ( V . v) dx 
~
 pn
. v ds =
f . v dx
for v 9 K~(fl). Problem (7.21) with (7.26) is formulated as to find ( u , p ) 9 K J ( f l ) x L2(fl) satisfying A ( u , v)  (p, V . v) = L g . v ds + ( f , v)
(7.27)
for v C K l ( f l ) . To study the unique solvability of this problem, we introduce the operator G : L2(fl)+ KI(Q) by (7.28)
(Gq, V)H,(a)2 = (q, V . v)
for v 9 Kl(ft). This is assured by Riesz' representation theorem. L e m m a 7.33. The range of G denoted by n a n (G) f o r m s a closed subspace of K l ( a ) and the orthogonal decomposition K l ( f l ) = R a n (G) @ Kl~(fl) holds. Furthermore, the operator G is an i s o m o r p h i s m between n2(fl) and R a n (G) = Kla(fl) L. Proof: Taking v  Gq in (7.28) implies liGqllg,(~)2 <_ C liqilL~(n) and hence the boundedness of G : L2(a) , H ' ( a ) 2 follows. Given q 9 L2(f~), we have ~? 9 K l ( f i ) satisfying
V. ~ = q
in f~
II~llm(fl)= ___ c IlqllL=(fl) 9
and
(7.29)
In fact, we have w 9 H2(ft) and r / 9 H1/2(Of~) 2 satisfying Aw = q
in
f~,
Ilwllg~(n) < c Ilqllc=(n),
and
~1~ :
f
0,
/
Jo f~
~ . ~ d~ : ~o # 0,
I1~11,,,~(o~)~ ~ C Ilqll L2(f~),
respectively. The value of r / o n 7 is not specified, and such a choice is possible. Then, rl,r = V w i o a + ecrl E H1/2(Oft) 2 satisfies
/o
7]~ . n d s = 0
for ~ =
n
1L
77o
V w. nds.
n
Therefore, we have v C HI(f~) 2 satisfying Vv  0
in ~,
vlo~ 
w~,
IlvllHl(~)~_< c IIw~ll.,~=(o~)=
by Proposition 7.31. Relation (7.29) now follows for ~ = Vw  v. Taking v = ~ in (7.28), we have Ilqllt=(~) < C IIGqlIH,(~)=. This imI)lies the closedness of R ( G ) .
7".6. The Stokes System
265
The relation Ran (G) • = Kl(ft) is easy to prove. It is left to the reader and the proof is complete. [] Problem (7.27) admits a unique solution if (g, f) C L2(F) 2 x L2(f~) 2. In fact, then the linear functional
E KI(~)
~~
fr g" Fds + (f, ~)
is bounded, and we get a unique u E K I ( ~ ) satisfying
.A(u, ~p) = fr g " ~ ds + (f , ~) for ~ C KJ(f~). For this u, the linear functional V
E
is also bounded, and we have ~ e Ran (G) satisfying
r(r
= (~, r
for r C Ran (G). Theorefore, Lemma 7.33 assures the existence of p E L2(f~) satisfying F ( r = (p, Vr However, F vanishes on K~(f~), and we have F(v) = (p, V  v ) for v 9 K I ( ~ ) , or equivalently (7.27). So far, we have observed that the Dirichlet form A(u, v) on Kl(ft) x Kl(f~) induces the first equation of (7.26) as a natural boundary condition. If we adopt g(u, v) = 2 E
[
1_
ei,y(u)ei,j(v) dx,
(7.30)
d~t.
for A(u, v), then the corresponding boundary condition becomes a(u, p)n = g on 7, where
1 (Ou ~ cOu3~ eid(u) = 2 \Oxj + cOxi,]
and
a(u,p) = (PSid + 2eid(u)) 9
We note that e(u) = (eid(u)) and a(u,p) indicate the deformation rate and the stress tensors, respectively. In particular, a(u, p)n forms the stress vector.
SDN
map
If (w,p) e K~(fi) • L~(fi) with wl.~ = ~ solves (7.24) for v E H~(f~) 2 with f = 0, then w and p are called the Stokes extension of ~ c Vo into ~ and the accompanying pressure of w, respectively. It is easy to verify that
c' I1~1 ~ _< .A(w, ~) < C I1~11~ 9
(7.31)
7. Domain Decomposition Method
266 A solution h E Kl(ft) of
(
Ah=0
inl2
with
h= ~
[0
on'7 on F
is called the (vectorvalued) harmonic extension of ~ C V into f~. Babu~kaBrezziKikuchi's infsup condition is expressed as
(q,V'v)i2(a) fl < inf sup gg(a)vcH~(a)2 IqjJg2(a)IW, II
(7.32)
where fl > 0 is a constant and IlVvll = v/A(v, v). Here, the optimal value of fl is assumed, called the infsup constant, and thus it is a constant determined by f~. L e m m a 7.34. Given ~ E Vo, we have
.A(h,h) <_ A ( w , w ) <_ (1 +fl~)2.A(h,h),
(7.33)
where w and h are the Stokes and the harmonic extensions of ~ into f~, respectively. Proof." We shall write II" II for II 9Iig2(n) or II" Iig2(a)2, in short. The first inequality is a consequence of the variational principle for harmonic functions. If p C L02(f~) denotes the accompanying pressure of w, then we have
Ilpll ~
sup
(p, v . v) =
sup
(Vw, v~) ~ IVwll.
In use of A(w, w) = A(w, ~  h) + A(w, h) = (p, v
( w  h)) + A(w, h)
< ]IPII" V h l l + l l V w l l 9 IVhll< ( 1 + 1 ' ] IIVwll 9IIVhll, /9/ k the inequality I Vwll_<
(1) 1+~
]lVhil
follows and the proof is complete. We put Xr = {~ C X I (~ , n ) x = 0}, where X = L2(7) 2 and (.,.)x denote the usual L2(7) 2 inner product. The bilinear form J : V~ x V~ , R is defined by J ( ~ , 7]) = A(w, v), where w and v denote the Stokes extensions of ~ and r/, respectively. We call it the (Stokes ) Jform pertaining to (f~, 7). Similarly to the case of the Poisson equation, we get a selfadjoint operator S : D(S) c Xo ~ Xo, satisfying J ( ~ , r/) = (S~, r/) x for (~, 77) r D(S) x Vo. It holds that J ( ~ , r/) =
267
7.6. The Stokes System
(S1/2{,Sll2rl)x for {, 77 E D ( S 112) = Vo and S 1/2 : Va + Xa is an isomorphism. Finally, variational principle holds and the operator S 1/2 is characterized by
for v C K~(f~) satisfying vl~ = ~. The operator S is called the Dirichlet to Neumanu map for the Stokes system, or simply the SDN map pertaining to (f~, 7). We shall often write S = S(f~, ~/) to express this correspondence. Given ~ E Vo, let w be the Stokes extension of ~ into f~. We take P0 C L~(gt) as the accompanying pressure of w. Assume their suitable regularities, say, (w, po) C H2(f~) x H I ( ~ ) . Even in this case, an additive constant c of p = P0 + c is specified for the function Ow/On  pn defined modulo R to belong to Xo. In fact, we have
where I~[ denotes the measure of O'. Therefore,taking 1 L
Ow
.
we ensure that O w / O n  pn E X~. This procedure is automatically done in the abstract setting mentioned above.
Dirichlet to N e u m a n n ( D N ) Iteration The target problem in consideration is (7.21) with (7.22). Keeping the notation introduced in w we study the following scheme, which may be called the DN iteration. We take a vectorvalued function #(0) on ~/first, satisfying
L
(v.34)
/~(o). u ds = 0
and then generate successively {u[k), p~k)},
{u~k), p~k)},
{v(k+')},
and
through V  u ~ k) = 0 in ~1, {  A u ~ k ) + V p ~ k ) : f, u~k) = 0 on F1, u~k) = p(k) on 7,
Aura)+ Vp~k) : f, u~k ) = O
onF2,
Vu~ k)=O
On
inf'2,
Ou
+p~k)u
ony,
7. Domain Decomposition Method
268 and
#(k+l) = (1  0)# (k) + Ou2 ~, where 0 E (0, 1] denotes the relaxation parameter. Relation (7.34) and the equation of continuity in t22 imply
f p(k). u ds = 0
(k = 1 , 2 , . .  ) .
Therefore, compatibility condition for the first problem holds at each step of the iteration. The accompanying pressure p~k) of the velocity u~k) is taken uniquely by fal P~k)dx = O. One may take p~k) = 0 at some point in a l . Then, the accompanying pressure p~k) of u~k) is uniquely determined subject to the choice of p]k). Let $1 = S(t2~,3') and $2 = S(t22,3') be the SDN maps pertaining to (t2~,3') and (t22, 3'), respectively. We get the following facts similarly to the Poisson equation. 1~ The operator Ho = S~1S1 admits a bounded extension in Vo, provided with the expression H = S~/2(S~/2S;1/2)*S~/2, where* denotes the adjoint in Xo. 2 ~ The space Vo forms a Hilbert space under the inner product
((%~,'1']))= (S1/2~ ~2 c,1/2 r/jx x
(~, ~ ~ vo)
(7.35)
with the corresponding norm II1" III = ((, .))~/2 equivalent to I1 liE. 3 ~ Under (7.35), the operator H is selfadjoint in Vo. 4 ~ The error on 3' is successively generated in accordance with ~(o) C Vo
and
~(k+l) = Ao~(k)
for k = 0, 1, 2 , . . . , where Ao = (1  O ) I  O H denotes the amplification operator for the error, selfadjoint and bounded in Vo under (7.35). 5 ~ We have
ro(Ao) =
sup
Aea(H)
11  0  0~1,
where a(H) and ro(Ao) denote the spectrum of H and the spectral radius of Ao, respectively. For the Stokes system, the spectral radius of the amplification operator is estimated in terms of tile infsup constants /~1 and ~2 corresponding to t21 and t22, respectively. L e m m a 7.35. Under the condition (I), we have 0 < H < (1 +/31) 2 for/3 = max(/31,/32).
269
7.6. The Stokes System
Proof: Let f~; = ~Lf~2 The Jforms pertaining to (f~i, "/), (f~2, ~'), and (f~, ")'), are denoted by if1, if2, and if2*, respectively. Given ~ E Vo, we take its harmonic extension hi into f~l and put Ql(~,~) = A n l ( h l , hl),
where Aal ( , ) denotes the Dirichlet form defined on f~l. Lemma 7.34 implies
Qi([,~) ~ ~]'1(~,~) ~ (1 + z~l) 2 Q l ( [ , [ ) . The bilinear forms Q2 and Q~ are defined similarly, and admit the corresponding inequalities. In particular, we have
j~(~, ~)< (1 + 9;~) ~ QI(~, ~) < (1 + 9 ~ ) ~ &(~, ~)
(7.36)
by Q,(~, ~) < Q;(~, ~) _< &(~, ~). Let w~ be the Stokes extension of [ into f~ and take its zero extension v to f~l \ g/~. Then, the variational principle implies
y~(~, ~) _< A~(~, ~)= Y;(~, ~). Similarly, we have Q1 (~, ~) _< Q~([, ~) by Proposition 7.3. In use of
Q;(~, ~) = Q~(~, ~) < &(~, ~), we have
~]'1(~, ~) ~ J2*(~, ~) ~ (1 + 321) 2 Q;(~,~) ~ (1 Jr~2i)2~2(~, ~). Combining this with (7.36), we thus obtain
Jl(~, ~) _< (1 + 9~) ~ &(~, ~). This inequality implies ((H~,~)) _< (1 + ~1)211~111~ because of ((HI, I)) = IIs~/~11~ = ffl([,[) and Ill,Ill= = IIS~/~ll} = &(~,~). The proof is complete. [] Letting c~ = 1 + (1 + fl1)~, we get r~(Ao) <_
sup
O
[1OAO[
=max{[1O[,[1o~O[}=?(O).
This implies the following. Theorem
7.36. Assume condition (I) and take 0 E (0, 1). Then, we have
for k = 1, 2, 3 ,   . , where
~(0)=
I1~
(~)
~10
(7.38)
[ c~01
(0<0_<2/(oz+1)) (2/(c~+1)_<0<2/c~)
and co > 0 is a constant determined by f~2.
270
7. Domain Decomposition Method
We see that 0 < ?(0) < 1 follows from 0 E (0, 2 / a ) . The following facts are obtained simlarly to the Poisson equation. Theorem
7.37.
Under the assumption of the previous theorem, we have
(k) u~  ~ ~ l l H ' ( ~ . )
< c2~(O)kll~ ~
UI~IIH~(~)
for k = 1 , 2 , 3 , .   . T h e o r e m 7.38. Under the saute assumption, the general optimal choice of the relaxation parameters is attained by 0 = 2 / ( a + 1) with the optimal value ropt = (a  1 ) / ( a + 1).
Commentary
to Chapter
7
7.1. The space H~/2(3') is familiar among domain decomposition researchers. In other contexts, it appears as a concrete characterization of the real interpolation space [H~(3'), L2(y)]l/2 (Lions and Magenes [247]), and the domain of L 1/4, where L denotes a minus Laplacian on 3' with the zero Dirichlet boundary condition (Fujiwara [150]). If 3' is a line, rrl/2 00 (3') has another expression. Let 3' be a line segment on the xaxis. Then oo rrl/2 (3') is coincident with the set of ~ = }~n=lan7),, C L2(3') satisfying E
a n2 /~ n1/2 <
ntO0 ,
n=l
where )~n and ~an denote the eigenvalue and eigenfunction of de~ dx 2 = ) ~
in 3'
with
~=0
on 03'.
We also have the equivalence
I1,~11ttoo,/~ ('~)
~
an~ ntn= 1
a 2n)~n~/2
,
(7.39)
n 1
See Saito and Fujita [332] for the proof. By this form one can prove t h a t H is a compact operator on X. Proposition 7.1 is a particular case of Theorem 1.5.2.3 of Grisvard [164]. If f~ is a polygon, an L p version of Grisvard's result was presented in Arnold, Scott and Vogelius [12]. An elementary proof of Proposition 7.1 is also given by [332] in use of (7.39). The Dirichlet to N e u m a n n map S is sometimes called the SteklovPoinca% operator. It is regarded as S : H1/2(Of~) ~ H1/2(Of~) if fi is a Lipschitz domain. An extensive analysis was made by Quarteroni and Valli [319] based on the study of harmonic extensions. There, S is regard as an operator from V into V'. Characterization of D ( S 1/2) by the quadratic
Commentary to Chapter 7
271
form was done by Fujita [141]. The key fact for Proposition 7.2 to hold is the closedness of ,.7({) = J ( ~ , ~ ) ; the conditions ~,~ E V, {,~ + {0 in X, and ,.7({n ~m) ~ 0 imply {0 E V and J ( { n  {0) + 0. We refer to Kato [205] for the abstract theory, particularly Chapter VIw If f~ is a rectangle and "y is one of sides of the rectangle, formula (7.39) implies D(S) = H~(7). Concrete characterization of D(S) for the other case is not known. Algebraic properties of the discretized Dirichlet to Neumann map are discussed in Agoshkov [4]. 7.2. The DN iteration was proposed by Furano, Quarteroni, and Zanolli [152] and MariniQuateroni [258], [259]. The latter papers adopt FEM, and the DN iteration is described as a Richardson iteration relative to the Schur complement matrix. Actually, the Schur complement matrix is a sort of finite dimensional analogue of the DN map. See [319] for further discussions. We just note that Bj0rstadt and Widlund [37] proposed a conjugate gradient method to solve a system on the Schur complement matrix. A similar idea was given by Bramble, Pasciak and Schatz [50]. Descriptions of this section are based on Fujita [141], Fujita, Fukuhara nd Saito [142], and Fujita and Saito [146]. See Yosida [410] for the spectral mapping theorem. 7.3. The idea of DDNN iteration was given by Bourgat, Glowinski, LeTallec, and Vidrascu [43]. Mathematical analysis on both discrete and continuous problems was done by Chu and Hu [80]. We followed the arguments of Fujita, Fukuhara, and Saito [142] and Fujita and Saito [146] based on the operator theoretical analysis. 7.4. The RR iteration was proposed by P.L. Lions [251]. The convergence is also discussed there. Theorem 7.26 was due to Deng [110], but the convergence rate is not known. Douglas Jr. and Huang [115] illustrated a simple example illustrating this issue. Also, Douglas, Jr., Paes Leme, Roberts, and Wang [116] derived a rate for a modification of the RR iteration combined with the mixed finite element method. The RR iteration is successfully applied to the Helmholtz equation defined in an exterior domain. See Desp%s [112] and Liu and Kako [253]. Some remarkable works on the Schwarz alternating method were done by Lions [249], [250] and Chan, Hou and Lions [72]. 7.5. The scheme describe in this section was proposed by D.H. Yu [413], [414 I. Theorems 7.29 and 7.30 were also obtained there. Related works were done by Yu [412], [415]. Numerical computation of the singular integral (7.19) was discussed by Yu [411]. 7.6. Standard monographs for mathematical theory of the Stokes and NavierStokes equations are Constantin and Foias [93], Galdi [153], [154], Ladyzhenskaya [229], Lions [252], and Temam [380]. Numerical approximations are discussed in Girault and Raviart [159], [160], Glowinski [161], and Quarteroni and Valli [318]. The Helmholtz decomposition is described in those literatures. Lemma 7.32 is shown in Babugkaaziz [19] for the smooth domain and in [159], [160] for the Lipschitz domain.
272
7. Domain Decomposition Method
Our proof of the unique solvability of the mixed boundary value problem (7.21) with (7.26) follows the argument of SolonnikovSeadilov [355]. Given 9 C H~/~(~) ~ 
(0
1/2 2 o (~)
)'
,
we have a similar result; the unique solvability of the problem to find (u, p) E I(~(f~) x L2(f~) satisfying .A(u, v)  (p, 27. v) = (9, v) + (f, v) for any v E KX(Ft) holds, where (., .) denotes the duality between r41/2 ~00 (.y)2 and H1/2(7)2. Another proof via the mixed weak formulation is described in [319]. For weak formulations using (7.30), see [229], [319]. Marini and Quarteroni [259] studied the finite dimensional version of the DN iteration for the Stokes system. It proved an anologous result of Lemma 7.34. An operator theoretical analysis in the Hilbert space Xo was done by Saito [331]. Lemma 7.35, Theorem 7.36 and Corollary 7.40 were presented there. Condition (7.32) was introduced independently by Babu~ka [18], Brezzi [66], and Kikuchi [218]. The finite dimensional counterpart of this condition guarantees the wellposedness of its mixed finite element approximation as is described in w We have seen that the convergence factor of the error depends explicitly on the infsup constant under condition (I). To evaluate the value of the infsup constant is of importance. In this connection, the following wellknown lemma is useful, where D C R 2 denotes a simplyconnected bounded domain with Lipschitz boundary. See Girault and Raviart [159] or Velte [400] for the proof. L e m m a 7.39. The relation 1//3 = v/~/2 holds for the infsup and Korn's constants/3 and ~, respectively. Here, Korn's constant arises in Korn's (second) inequality
A(~, ~) _< ~E(~, ~) for v E H i ( D ) 2 satisfying
/D( OX 0~22 0721) dx = 1 ~X;
0.
(7.40)
Here, g denotes the bilinear form defined by (7.30). The optimal value of ~, which is again denoted by the same symbol, is called Korn's constant. The following facts are known concerning Korn's constant (Horgan [180], Horgan and Payne [182])" 1~ For any bounded domain, ~c > 4 and the equality is attained if D is a disk. 2~ Let D be starshaped with the boundary represented in plane polar coordinates by r = 9(0). Then we have h: _< 2(1 + maxoC(O)), where
I~'(0)1 G(O) =
1+ \ 9 ( 0 ) j
+ 9(0)
In particular, when D is a regular nside polygon, we have ~ _< 2/(1 sin(Tr/n)).
Commentary to Claapter 7
273
3 ~ If fit is a square, we have ec _< 8 + 4x/~. As a corollary of Theorem 7.36, we have C o r o l l a r y 7.40. If condition (I) is satisfied and moreover that both f21 and f22 are squares, then, as long as 0 < 0 < 0.1423, the exponential decay of the error as stated in (7.37) is guaranteed. Furthermore, in this case, we get ropt < 0.8671. The numerical value of the infsup constant plays important role also in other areas such as the theoretical analysis for the NavierStokes equations (Horgan and Wheeler [183], Ames and Payne [10]) and numerical verifications of finite element solutions for the Stokes equations (Nakao, Yamamoto and W a t a n a b e [282], [283]). Horgan and Payne [182] conjectured that ~ = 7 if f~ is a square. Concerning further discussions about Korn's constant, the survey paper Horgan [181] is useful.
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Index   regularity, 21, 45 stability, 21, 45
(I)o, 155, 163, 164, 170 (i)o, 146, 147, 148, 151, 153, 170 (II)o, 147, 155, 156, 159, 170 (ii)o, 146, 148, 150, 153, 155, 170 (III)o, 154, 155, 159, 162, 170 (ill)o, 146, 147, 148, 150, 153, 155, 170
adjoint, 248, 268 form, 6, 20, 61, 120, 124, 127, 151 operator(s), 117, 219, 248 atiine space, 183 amplification operator, 247, 259, 268 for the error of the DDNN iteration, 256 for the error of tile DN iteration, 248 artificial boundary, 246, 258 AscoliArz~la's theorem, 191 asymptotic   behavior, 35 expansion, 46
(P, S), 154, 155, 156, 159, 160, 162, 163, 164 wlimit set, 209 A acceptability, 146, 169, 170 acceptable, 145, 146, 170 Aoacceptable, 146, 148 Hol/2 o (7), 244, 249, 262, 270 Jform(s), 245, 249, 251,254, 269 Stokes, 266 L~ stability, 44, 77, 204 structures, 172

Babu~ka and Aziz lemma due t o   , 263 backward difference, 62, 126, 140, 145, 221,224 barycenters, 172 barycentric   coordinate, 203   d o m a i n , 172, 175, 203, 219 bilinear form(s), 4, 5, 7, 10, 50, 51, 55, 58, 61, 83, 84, 93, 104, 108, 131, 135, 144, 183, 185, 188, 192, 244, 245, 266, 269, 272 blowingup of the solution, 211 blowup, 211,240 phenomenon, 239  solutions, 240 time, 212, 239 one p o i n t   , 239 three p o i n t s   , 213, 239 bootstrap argument, 238
L1

contraction, 171, 177, 217, 218, 219, 241 estimate, 178, 204 structure(s), 172, 178, 207 theory, 218

L2 error analysis, 88 90, 91, 92 inner product, 2, 12, 142, 172, 219 projection, 16, 174, 178 realization, 50 stability, 19 topology, 118








e
s
t
i
m
a
t
e

,
L; error estimate, 20, 21 stability, 173
WI,p 303




304
Index
boundary element(s), 171   method, 190, 192, 205 Brezis and Strauss L1 estimate of , 26 charge points, 195, 205 simulation(s), 171 Chernoff formula nonlinear, 218, 224, 225, 240, 241 closed range theorem, 6, 42, 185 coercive, 42, 192, 245 strongly, 7, 9, 10, 15, 58 coerciveness, 93, 144 s t r o n g   , 8, 22, 51 collocation method, 192 points, 195, 205 method o f   , 205 compact, 179 operator, 253, 270 compatibility condition, 268 condition (Ie), 252, 253, 255, 256, 257 (Ira), 251, 252, 253, 254, 255, 256, 257 (I), 249, 250, 252, 255, 268, 269 (Re), 254, 255, 256, 257 (Rm), 254, 255, 256, 257 (R), 254, 255 conformal mappings, 206 transformation, 206 conjugate gradient method, 271 constraint(s), 182, 183, 184 continued fraction expansion, 147, 170 contraction, 48, 62, 69, 217, 231,232, 235 property, 51 convectiondiffusion, 181 convex, 2, 13, 16, 20, 21, 28, 29, 129 closed set, 238   closed subset of a Hilbert space, 42 function, 209 hull, 172 

polygon, 1, 10, 16, 21, 27, 29, 31, 35, 42, 47, 77, 135, 207, 238 smooth domain, 238 CrankNicolson approximation, 75 scheme(s), 75, 145, 153 critical point, 207 CSM, 205, 206 curved element(s), 13, 43 cyclic matrix, 196 DDNN iteration, 255, 256, 257, 271 DDM, 243, 258, 261 deformation rate tensor, 265 degenerate, 208 case, 218 parabolic equation(s), 207, 240 Descloux('s) lemma, 19, 44 lemma of , 17 Dirichlet boundary condition, 40, 53, 219 condition, 1, 42 form, 2 problem, 182 Dirichlet to Neumann iteration, 246, 267 map, 243, 246, 270, 271 for the Stokes system, 267 Dirichlet 2 to Neumann 2 iteration, 255 dispersive system(s), 2O3 dissipative structure, 171 distributions, 205, 246 DN iteration, 246, 256, 261,267, 271 for the Stokes system, 272 map(s), 243, 246, 248, 259, 271 domain decomposition   algorithm, 246 method(s), 243 researchers, 270 dual exi)onent , 29 form, 45
305
h~dex operator, 172, 183, 184, 189 space, 2, 4 duality, 272 argument, 14, 26, 28, 34, 43, 58, 93, 94, 121, 136, 151 map, 204 method, 139 Dunford integral(s), 51, 59, 63, 67, 75, 93, 109 dynamical systems, 93 Egorov's theorem, 227 electromagnetic theory, 183 elliptic   boundary problem(s), 238 boundary value problem(s), 1, 8, 43, 45, 207, 237, 238 nonlinear, 207  differential operator, 135   e s t i m a t e , 14, 42, 59, 60, 71, 74, 98, 119, 121, 123, 124, 136, 137, 144, 178, 238 operator(s), 43, 93, 95, 132, 204   r e g u l a r i t y , 5, 42, 46, 232, 242   Sobolev inequality, 60, 93, 94 uniformly, 71 energy method, 90, 94, 131, 135, 140 enthalpy formulation, 217 equation of continuity, 268 Euler approximation, 239  equation, 183 evolution equation(s), 83, 84, 93, 96, 118, 135, 154, 217 operator(s), 94, 96, 98, 99, 101, 107, 108, 118, 121, 122 external force, 262 FDM, 44, 212 FEM, 1, 171 Feyman path integral, 170 finite difference analogues, 211   method, 44, 212 operator, 46, 154
solutions, 207 finite element(s), 205 h p   , 43 p   , 43 analogue, 218, 241 approximation(s), 35, 43, 44, 45, 53, 62, 70, 83, 95, 97, 118, 135, 145, 153, 171, 172, 218, 221 discretization, 21,205 method(s), 1, 10, 19, 43, 44, 53, 93, 140, 238 scheme, 207 solution(s), 44 space(s), 70 flow, 181 velocity, 262 fluid, 262 dynamics, 181 mechanics, 205 viscous incompressible, 262 forward difference, 64 Fourier coefficients, 198, 201 series, 259 transformation, 193 inverse , 193 fractional powers, 43, 51, 58, 93, 101, 144 Frech~tKolmogorov's criterion, 228   theorem, 242 Fredholm  alternative, 182   theory, 191 free boundaries, 217 free boundary problem(s), 42, 206 fulldiscrete, 93, 140 fundamental solution(s), 171, 190, 191, 195 method, 205 Gauss
divergence formula o f   , 263 Gel'fand triple, 42, 204 general optimal choice(s), 253, 256, 257, 261, 270
306
hMex
generation theory of FujieTanabe, 99, 144 of KatoSobolevskii, 99, 144 of KatoTanabe, 104 of TanabeSobolevskii, 98, 144 KatoTanabe's, 123 Green's formula, 2, 192, 264 Green's function, 31, 40, 258 discrete, 20, 27, 37, 44, 81 regularized, 39
BabugkaBrezziKikuchi's   condition, 205, 266 inhomogeneous, 263 boundary condition, 45, 262 boundary value problem, 21, 42, 43   Dirichlet boundary value problem, 44 52, 68, 93  generators, 119 parabolic equation(s), 95, 144   problem, 70 interaction conditions, 243 interpolant, 11 interpolation, 218 inequality, 42 operator(s), 15, 43, 172, 181,219, 231 space, 270 theorem, 19, 44 theory, 60, 93 inverse assumption, 14, 15, 16, 17, 18, 19, 20, 21, 22, 27, 28, 31, 44, 53, 60, 65, 75, 77, 78, 79, 80, 88, 91, 94, 137, 153, 174, 178, 229 isomorphism, 28, 31, 172, 245, 264, 267 canonical , 184  
  e q u a t i o n ,
 
HSlder('s) continuous, 53, 68 inequality, 24, 25, 30 harmonic, 199 equation, 246 extension(s), 201, 245, 249, 250, 251, 253, 254, 259, 260, 261, 266, 269, 270 vectorvalued, 266 function(s), 266 heat equation, 47, 77 Heinz(') inequality, 9, 99, 167 theorem, 61 inequality o f   , 43 HeinzKato's theorem, 59 Helfrich's method, 85, 94 Helmholtz decomposition, 262, 271 higher accuracy, 70, 75, 93, 144, 153 HilleYosida('s) theorem, 84 theory o f   , 48 homotopy invariance, 223 hybrid finite elements, 183 hydrodynamics, 183 hyperdispersive operator, 240 hyperdispersivity, 240 hyperbolic equation(s), 47, 53, 83 systems, 203  
infsup constant(s), 243, 266, 268, 272, 273
Jordan curve, 206 Kato('s) inequality, 221,241 square root problem, 43 theorem b y   , 65 Korn's constant(s), 272, 273 inequality, 205 second inequality, 272 Lagrangian multiplier, 183, 184 law of conservation, 203 LaxMilgram's theorem, 4, 7, 42, 105 layer potentials, 205 double, 191 single , 191 least energy solution, 207, 238
307
Index
levelset approach, 240 LeviTanabe's method, 97 Lie groups, 165 linearized instability, 238 Lipschitz boundary, 190, 246, 272 constant, 224, 225 continuous, 181,188, 218, 224, 225, 231,232 domain, 45, 187, 270, 271 lumping, 180, 181,204 of mass, 171,218, 219 operator, 172, 173, 181,219 m a s s   , 219 scheme of , 171 maccretive operator(s), 231,242 msectorial, 5, 8, 51 operator(s), 6, 42, 43, 53, 61, 85, 96, 99, 108, 118, 131, 135, 144 maximum principle, 21, 45, 147, 171,174, 219 discrete, 20, 21, 22, 44, 46, 181, 182 general, 44, 45 storong, 45 mean curvature flows, 240 meromorphic flmction, 151 minimal surface, 44 mixed   b o u n d a r y condition, 263   boundary value problem, 272 elements, 171   finite element method, 271 finite elements, 183, 205, 272 weak formulation, 272 Morse index, 208, 238 mountain pass critical value, 208 lemma, 238 multistep method, 154, 155, 170 Nakao's method, 238 natural boundary condition, 265 NavierStokes
equations, 271,273 system, 205 Nehari('s) iterative sequence, 238 manifold, 208, 238 variational formulation, 208 variational principle, 238 Neumann boundary condition, 12, 44, 240 boundary value problem, 45, 205 condition, 50 problem, 3 Newmark's/3 scheme called, 94 Newton's method, 238 Nitsche's trick, 43 nonnegative type, 20, 45 numerical range, 5, 42, 54, 55, 65, 93 numerical verifications, 238, 273 







one paraeter families of subgroups, 165 orthogonal complements, 189 decomposition, 185, 186, 264   projection, 7, 8, 16, 18, 54, 59, 77, 85, 86, 118, 137, 189, 218 system of eigenfunctions, 48 Pad~ approximation, 146, 147, 170 parabolic equation(s), 47, 51, 54, 62, 93, 95, 118, 145, 153 initial boundary value problem, 101 problem(s), 46, 93 region, 8, 153 parallel computations, 243 peak(s), 211,240 movement o f   , 240 number o f   , 240 PhragmdnLindelSf theorem, 147 Poincard's inequality, 2, 3, 24, 42, 186, 188, 244, 245, 262 Poinca, r~Wirtinger's inequality, 12, 43,219, 241 Poisson equation, 1, 182, 186 porous media, 217, 240 

308 pressure, 262 accompanying   , 265, 266, 267, 268 principal value, 191 rapidly decreasing function, 193 Rayleigh's principle, 238 reactiondiffusion system, 240 real method, 93, 94 reflection, 249, 254 principle(s), 243, 254, 260 transformation, 255 regular, 11, 14, 19, 43, 53, 178, 220, 244, 246, 272 triangulations, 13, 21, 171, 181 relaxation parameter(s), 247, 255, 257, 258, 268, 270 resolvent(s), 47, 55, 93, 107, 225, 226, 230 set, 5 Richardson('s) extrapolation, 35 iteration, 271 Riesz(') representation theorem, 2, 4, 7, 31, 42, 264 theorem, 184 representation theorem of, 183 RieszThorin's interpolation theorem, 173, 204 Ritz operator, 7, 16, 19, 20, 21, 45, 119, 120, 135, 179, 182, 218 projection, 85 RitzGalerkin approximation, 10 method, 7, 189, 192 Robin to Robin iteration, 257 RR iteration, 257, 258, 271 RungeKutta method, 170 saddle point, 183, 184, 185, 186, 187, 188, 189, 204 Schauder's fixed point theorem, 238 Schur complement matrix, 271 Schwarz althernating method, 271 SDN map(s), 265, 267, 268
Index
selfadjoint, 43, 83, 84, 93, 165, 248, 249, 256, 266, 268 operator(s), 9, 43, 47, 77, 99, 100, 115, 165, 166, 243, 245, 246, 248 semidiscrete, 53, 54, 68, 70, 77, 83, 85, 93, 94, 118, 126, 131, 135, 153, 218 semigroup, 47, 51, 53, 77, 93, 145, 146, 174, 177, 217, 224, 235 (C0) , 49, 93, 145, 165, 170, 204 of operators, 47, 48 property, 49, 74, 93, 97, 121 theory, 53, 93, 145, 205 analytic , 165 approximation of, 53 contraction, 49, 169, 231 convergence of, 226, 230 holomorphic, 5, 42, 51, 93, 97, 98, 148, 154, 170 linear, 225 nonlinear, 170, 217, 220, 231,240, 241 rational approximation of, 170 strongly continuous, 49 semilinear elliptic equations, 207 parabolic equations, 207, 211,239 discretized, 240 singlestep method, 153 singular integral, 271 skewadjoint, 84 smoothing property, 47, 50, 51, 54, 62, 74, 94 Sobolev('s) imbedding theorem, 21, 45, 208 inequality, 24, 25, 29, 31, 45, 60, 79, 80, 93, 94 space(s), 1, 42 solenoidal, 262 solid mechanics, 183 spectral decomposition, 6, 43, 100, 115, 166 mapping theorem, 249, 256, 271   radius, 249, 256, 259, 268 spectrum, 5, 8, 76, 153,249, 253, 259, 268 stable sets, 238
309
Index
Stampacchia estimate b y   , 45 stationary point, 184 Stefan nonlinearities, 240 two p h a s e  problem, 217 SteklovPoinca% operator, 270 Stokes equations, 271, 273 extension(s), 265, 266, 267, 269 system, 187, 204, 205, 261, 268, 272 continuous, 205 discretized nonstationary, 205 stationary , 187, 243 stress tensor, 265   vector, 265 strong continuity, 49 convergence, 165 inhomogeneties, 243 solution, 2, 5, 53 topological degree, 223, 241 torus, 217 trace, 244 computations, 127 operator, 2, 4, 42 theorem, 40, 244 triple of Hilbert spaces, 4, 7, 42, 83, 108, 131,244 Trotter's product formula, 165 TrotterKato's theorem, 226 nonlinear, 241 type (0, M), 6, 8, 42, 51, 54, 56, 59, 97, 154 type (00, M0), 148, 153 unstable, 237 linearized, 207 sets, 238 solutions, 207, 238 upwind, 181   difference, 171
finite element method, 181 Ushijima's works, 204 variational inequality, 42 method of minimax type, 238 principle(s), 243, 246, 249, 260, 261,266, 267, 269 problem, 3, 4, 10, 42, 182, 183, 184, 204 structure, 1, 171, 188, 208, 221 weak form(s), 1, 90, 132, 172, 187, 219   formulation(s), 184, 272 solution, 2 Yosida approximation, 230, 241,242
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