1D-grid generation by monotone iteration discretization

Computing 34, 377-390 (1990)

~ [ ~ , i [ ~ 9 by Springer-Verlag 1990

1D-Grid Generation by Monotone Iteration Diseretization Mansour A1-Zanaidi, Safat, and Christian Grossmann, Dresden

Received April 10, 1989; revised August 10, 1989 Summary - - Zusammenfassung 1D-Grid Generation by Monotone Iteration Discretization. On the basis of the monotone discretization technique, we propose in this paper a new feedbackgrid generation principlefor weaklynonlinear 2-point boundary value problems. By means of available estimations resulting from lower and upper solutions the grid can be refined automatically. The monotonicity of the method is guaranteed by principles of monotone iterations. The convergenceproperties of the proposed algorithm are analyzed.

AMS Subject Classifications: 65L10, 65L50, 65L60 Key words: differentialequations, boundary value problems, enclosures, grid generation. EindimensionaleGittergenerierungdurchmonotoneDiskretisierungs-Iteration.In der vorliegendenArbeit

wird ein Gittersteuerungsprinzip auf der Basis yon monotonen Diskretisierungs-Iterations-Verfahren und der damit erzeugten LfsungseinschlieBungenbei schwacb n~chtlinearen 2-Punkt-Randwertaufgaben vorgeschlagen. Mittels verftigbarerSchranken wird das Gitter automatisch erzeugt. Die Monotonie des Veffahrens ist dabei durch Prinzipien der monotonen Iteration gesichert. Es werden die Konvergenzeigenschaften des vorgeschlagenenVerfahrens analysiert.

I. Introduction

In nonlinear boundary value problems the distribution of grid points plays an important role to make a discretization technique efficient. This becomes essential in the case when local singularities occur as discussed in [9], [14] e.g. There exist various principles to estimate the influence of the location of the grid points on the accuracy of the numerical solution of the BVP (see [7], [8], [9], [15], [18] e.g.). In the case when lower and upper bounds are available for the solution, one can directly use this information to control the grid generation. In [13] a feedback grid generation principle based on monotone discretization has been proposed. The two-sided bounds in [13] are generated by an iteration technique and the bounds are obtained in the sense of a limit only. In an alternative approach the monotone iteration discretization (MID) technique proposed in [5], [6] realizes the required enclosure in each finite dimensional substep. In the present paper we base a feedback grid generation on the MID-principle. In contrast to [13] this makes the grid generation completely implementable. We prove that the proposed algorithm gives convergence. Finally, some numerical examples are given to demonstrate the practical behaviour of the M I D grid generation.

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Mansour A1-Zanaidi and Christian Grossmann

2. Grid Refinement Using MID Iterations In this paper, we deal with the numerical solution of weakly nonlinear boundary value problems -u"+g(.,u)=O

in

f2:=(0,1) (2.1)

u(0) = u(1) = 0 where the function g: ff x ~ ~ ~ is continuously differentiable and satisfies the following monotonicity condition g(x, t) < g(x, s)

for any x ~ f2,

t < s.

(2.2)

Let V denote the Sobolew space V = H~(I2) and V* the related dual space. We define mappings L, G: Hl(I2) ~ V* by (Lu, v) := [ ^ u'(x)v'(x) dx for any u, v ~ V

(,

:= j~ g(x, u(x))v(x) dx. Now, problem (2.1) is equivalent to the operator equation u E V,

(L + G)u = 0.

(2.3)

The weak formulation (2.3) can be used for a wider class of functions g(', "). This is essential for covering the auxiliary problems which are also generated in MID itself itself because there g(., .) is replaced by piecewise continuous functions. We observe that the operator L is linear and coercive in V, i.e. > 7lluJl2

for any u ~ v

holds with some constant 7 > 0. The MID algorithm relies on the following three facts: i) Application of monotone iteration schemes similar to the approach used in [19] to problems (L + O)u = (O - a)u

(2.4)

with appropriate operators D: V ~ V*. ii) Modification of the right hand side of equation (2.4) such that the generated problem can be solved in some finite dimensional space possessing a base which is directly available for numerical calculations. Additionally, this modification is made in such a way that bounds for the solution of the original problem are generated. iii) Using estimators for the modification and keeping the process monotone by additional intersections. Furthermore the iterations are accelerated by updating the operator D such that a linearization occurs approximately.

1D-Grid Generation by Monotone Iteration Diseretization

379

A special realization of problem (2.4) is u s e d i n [19] for proving existence and for generating bounds for the solution of BVPs. Analogously to the principle used in [19] we have the following Lemma 2.1: Let G: C ( ~ )

--*

L2(g2) be a continuous

mapping. We assume functions u,

~ V to exist such that u <_fi and (L + G)u_ <_ 0 <_ (L + G)~. Further, let some constant r > 0 exist with u <_ u <<_v <_ ~

=~

(rl - G)u <_ (rI - G)v.

Then some u ~ V, u < u <_ ~ exists solving problem (2.3).

Here and in the sequel " < " means the natural ordering in V and in V*, respectively. For further estimations we need some inverse monotonicity and a weak maximum principle. Similar to [11] the following lemma holds. Lemma 2.2: (i) I f u, v ~ Hi(t2) are functions satisfying the conditions

u(O) < v(o),

u(1) <_ v(1)

and (L + G)u <_ (L + G)v then we have u(x) < v(x)

for all x ~ I2.

(ii) I f u, v ~ Hi(g2) satisfy the operator equations (L+G)u=O

and

(L+G)v=O,

respectively, then we have the estimation

[[u - viler0,11 < max{lu(0) - v(0)[, [ u ( 1 ) - v(1)[}. The interested reader is referred to [5] for further details of the theory and computational realization of MID. In the present paper we combine the MID principle with grid refinement on the basis of available bounds and we concentrate our attention on the differences in the analysis of M I D grid generation in contrast to MID on fixed grids.

3. Mid Grid Generation Algorithm To obtain iterations in finite dimensional subspaces of V, V* we generated recursively grids X k on s These grids are characterized by their grid points, i.e. X k = x g~Nk }i=0 on g2 satisfying

380

Mansour AI-Zanaidi and Christian Grossmann a = xok < x ~ ' " < x~k_ 1 < x~k = b.

We denote h i := x~ - Xi_l,k f2 ik := (X~-I, X~), i = I(1)Nk. To avoid double indices let X denote some grid of this family. Let s, t 9 X denote any grid points with s < t. Then we define a partial grid X [s, t] by X [ s , t ] := { x i 9 1 4 9

[ s , t ] , i 9 {0. . . . . N}}.

The partial grid has the step size h ( X [ s , t ] ) := max (xi - xi-1) Qi c [s,t]

The mesh size of the complete grid is given by h(X) := h(X[0, 1]). As finite dimensional subspaces of V* we select spaces Pro(X) of piecewise polynomial functions with possible discontinuities at the grid points of the actual grid X only, i.e. Pro(X) := {V: Via, is a polynomial of maximal degree m}

These spaces are embedded in Loo(Q) and we equip them with the usual norm in Lo~(s We use notations similar to those in [5] and we repeat some essential facts only to make the paper self contained. Intervals S in Lo~(~2)are characterized by its lower and upper bound s and ~, respectively in the form s = [_s,~-] := {s 9 Lo~(~): s _< s __ ~}.

Related to subspaces W c L~(f2) we define sets of intervals I ( W ) := { [_s,g]: _s,g 9 W}.

We refer to I-4] e.g. for further information on intervals. First we give some motivation of the proposed estimators to make the definition given later more transparent. The fundamental idea of the MID-technique consists in combining monotone iterations similar to [19] with appropriate simplifications of the iteratives and of the right hand side of the differential equations by piecewise constants and by piecewise linear functions, respectively. Due to [19] an analytical iteration process can be derived from (2.4) by (L + DU)u_k+' = (D k - O)u k, (L + DR)~ k+~ = (D k -- G)~ k

to generate sequences {u_k}, {fig} of lower and upper, respectively, bounds. We enclose _uk, gk in each step of the iteration by a piecewise constant bound tk ~ uk ~ ~k ~ ~-k

to make the estimations more simple. Moreover the operators D k - G are supposed to be monotone on the actual interval [t, ~-k]. This property is given in a).

1D-Grid Generation by Monotone Iteration Discretization

381

By means of mappings B, Z given in c), d) we construct piecewise linear bounds _zk, ~k of (Dk -- G)u k, (Dk - G)~ k, i.e. z_k < (O k _ G)u_k,

(D k _ G)~k < -~k.

Instead of solving boundary value problems of the type (L + Dk)u k+l = z_k,

(L -t" D k ) u k+l = ~k,

finally, we only estimate the related solutions by some piecewise constant functions _sk, 3-k. This property is summarized in the estimator S(:, .;X). We refer to [1] for various realizations of such estimators satisfying property (3.3) described later. Now, we introduce mathematically some mappings motivated before to estimate derivatives as well as the solution of a simplified linear problem with a piecewise linear right hand side. Unlike the appr/oach in [5], the dependence on the actual grid X is to be taken into account. We assume g(.,.) to be continuously differentiable and suppo/se the following mappings to be available: a) d(.;X): I(Po(X)) ~ Po(X) such that u, v E T, u < v

implies

(D(T;X) - G)u <_ (D(T;X) - G)v

(3.1)

for any T ~ I(Po(X)). Here, D(-;X) is related to d(.;X) by [D(T; X)] (x) = diu(x ),

x ~ g2i,

i = a(1)N

(3.2)

and dl denotes the value of d(T; X) in the subinterval s of grid X. b) S ( . , ' ; X ) : P o ( X ) x I(PI(X)) ~ I(Po(X)) such that (L + D ) - I W

(3.3)

= S(D, W ; X )

for any D e Po(X), W e I(P,(X)). c) B(.; X) " I(Po(X)) --, I(Po(X)) satisfying --g~(., T) = B(T; X )

for any

T ~ I(Po(X)).

(3.4)

d) Z ( ' , ' , ' ; X ) : P o ( X ) • I(Po(X)) • I(Po(X))--*I(PI(X))with Z(O, T,B; X) = [_z,5], z_(x) = d~t_~ - g(x~_~, t_3 + b_~(x - x~_~),

x e ~

(3.5)

~(x) = d,r, - o ( x , _ , , T , ) + b,(x - x~_~)

and D ( T ; X ) e P o ( X ) , respectively.

B(T;X)=[b,b]eI(Po(X))

defined in a) and

To simplify the right hand side of equations U ~ I(V),

(L + D)U = (D -- G)U

we replace (D -- G)U by an interval Z E I(P 1(X)) as sketched above. By definition d) the inclusion

c),

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Mansour Al-Zanaidi and Christian Grossmann

Z ~ (D - G)U

(3.6)

can be guaranteed. In combination with lemmas 2.1, 2.2 this forms the fundamentals of the MID principle. M i d Grid Generation Algorithm Step 1: Select some initial grid X ~ and select intervals U ~ e I(V), T O e I(Po(X~ that U ~ c T O and with D O = D ( T ~ 1 7 6 B ~ = B ( T ~ 1 7 6 Z ~ = Z ( D ~ T ~ 1 7 6 that (L + D ~

~

(3.7)

~~ Z~

holds. Select some 0 e (0, 1), and set k := 0. Step 2: Determine U k+l e I(V) by (L + D k ) u k+l = Z k.

(3.8)

T k+I := T k c~ S(D k, Z k, X k ) .

(3.9)

Step 3: Define

The bounds of the interval are denoted by T k+l = [t k+l, ~-k+l] Step 4: Define ~k+l := lit *§

-

(3.10)

t *§ II

and determine Ik := {ie {1. . . . . Nk}: I~#+1 -- _tk+l[--> O(~k+l}.

(3.11)

Set X k+l := X k U {xk-1/2 }i~ rk with xk_ll2 := (xL1 + xk)/2.

(3.12)

Set Nk+ t := Nk + card I k and denote the points forming the new grid X k+l by x k+l, i = 0 , . . . , Nk+ 1. Step 5: Define D k+l e P0(Xk+l), B k+l e I(Po(Xk+I)) by D k+l := min {D k, D(Tk+I; xk+t)}, B k+t : = B k (3 B ( T k + I ; x k + t ) , Z k+l = Z ( D k+l, T k+l, B k+l ; X

TM

).

Goto step 2 with k + 1 instead of k.

-

-

Remarks: Taking into account D k e Po(Xk), Z k ~ I(Po(Xk)) problems (3.8) can be solved exactly by means of exponential splines (compare [5], [12]). The coefficients of these splines can be efficiently determined by a tridiagonal linear system.

Because grid points are added only by the method given above we have (Po(X ~) c Po(Xk+'). Thus we can identify D k as an element of Po(X k+l ) also.

1D-Grid Generation by Monotone Iteration Discretization

383

Finding intervals such that inclusion (3.7) holds forms an important question for application of the MID-algorithms. The selection of initial intervals has been discussed in [5]. -

Theorem 3 . 1 : F o r intervals U k, T k, k = O, 1. . . . generated in the M I D - a l g o r i t h m given above we have the inclusions

"'" U k+l c

U k ~ ... c

U 1 ~ U O,

9"" T k+l ~ T k ~ ... ~ T 1 c T ~

U k ~ T k,

k = O, 1. . . .

and each o f the intervals U k contains at least one solution o f problem (2.3). P r o o f : Using mathematical induction similarly to that used in [5], we prove

(3.13)

( L + D k ) u k ~ Z k ~ (D k - G ) U k

to hold for all steps of the algorithm. Here the right inclusion is a direct consequence of definition (3.5) of the mappings Z ( - , . , . ; X). The induction for the left part of(3.13) is different from the one associated with the fixed grid case. Let us consider some subinterval s k+l of the grid X k+l. Then a unique index j e {1 . . . . . Nk} exists such that ~e-~k+ 1 k i C ~r'~j 9

F o r x ~ s k+l we obtain from (3.5), (3.8) --U

~- d ;=g k+l .

zk_ = d ; t ; - ~4,'~(xkj-1,_j t k ', nt- b_fi(X - X ; _ I ) .

Now, two different situations can occur: i) x~_+( = X]-l. This leads to - u k+'' + #u_ ~+1 = d ; t ; - o(x,-1,k+' t;~) + b_](x - x~_+;).

ii) xi-1k+a = X]_m" In this case we obtain -u_k+~" + d)u ~+~ = d ; t ] - g ( ~ y ? , t ~ )

+ b~(~ - ~_+~) + g(~_+?,t;)

-- g(xjk_l, t;) "-[- b;(xk_+(

-- 41

).

With (3.4) in both cases the following inequality holds. - - Uk + l t ' _

"4-

d;u_ k+l _< dfi]_ - g(x~+?,t_y) + b](x ,k+l,:~,_,~

O.k+l . By the constructions made in the algorithm we have for any x e .., t~ _< t~'+~ _ u k+l

in

s '+~

and b_; < b~ +1

d k > d k+l ,

Using (3.14) this results in _ u k+l" + d~+Xu k + l_.

< d~+lt~ + x ._.

--

g(xff+( _tk+l) -]- _bk+l(x

+ (d) -- d~+X)(t~ + 1 .9

- u TM)

--

X~_+1)

(3.14)

384

Mansour Al-Zanaidi and Christian Grossmann k + t - g-e. ,k+lxI + b~+t( -< d k9+ l t_t x ik+t - 1 , ~-i x _

in

= z k+t

O k+~ --t

- - A,i_ 1 .k+l',l

"

In an analogous way we obtain __~-k+l,, ..[_ d k + l g k + l >__~-k+l

in

~?~+t.

This proves that the left part of (3.13) holds. The remaining part of the proof, in particular, the complete induction can be realized in the same way as in [5]. Lemma 2.2 guarantees the intervals U k to be decreasing. The existence of the solution can be derived from lemma 2.1. and (3.13). 9 We remark that the theorem is quite standard if the operators D and the grids would not be changed. This is reflected in the proof given above where we concentrate our attention to the non trivial part only.

4. C o n v e r g e n c e A n a l y s i s o f the M I D G r i d G e n e r a t i o n A l g o r i t h m

In theorem 3.1. we showed that the MID grid generation algorithm produces a decreasing sequence of intervals U k E I(V), k = 0, 1, ... as the original MID algorithm did. In this section we investigate the convergence of these intervals to the unique solution u of problem (2.3). Lemma 4.1: There exists some constant c > 0 such that

II_ukllcl~) <_ c,

II~kticl~ <_ c

f o r k = 1, 2 . . . . holds. P r o o f : Because of d k > d k+l > 0 and B k+l c B k some c 1 > 0 exists such that Ilz_kllL2(a) < Cl ,

II~kllL2(a)< Cl,

k = O, 1. . . .

Problems (3.8) and the coercivity of the operator L lead to

~lluk§

2 _< ( L u k + l , u k+l) < ( ( L + D)uk+l,u k+l) = ( z k + l , u k + l ) .

This results in Iluk§ have the estimation

_< cl/~. Because of the continuous embedding V ~ c ( ~ ) , we Iluk+tHc(~) < c2,

k = 1, 2 . . . .

with some constant c2 > 0. Thus dku k+~ ~ L2(f2) holds and we obtain IIdku~+l [[L2(O)--< tldkllL=(a)IIuk+l

IIC(~) 9

Next, we use (3.8) in the form L u k+l

=

Z k

_

dku k+l "

This results in

Ilu k+l IIH=(a) < c3(llPIIL2r

+ IId~llL=~a)I1u~+x IIcr

ID-Grid Generation by Monotone Iteration Discretization

385

Using the continuous embedding H2([2)(Cl(~) and the previous estimations this proves the statement. 9 Before proving the convergence of the M I D grid generation technique, we introduce some realization of a mapping S(', "; X) satisfying (3.3). As assumed the mapping S(., .; X) should realize a piecewise constant underestimation and overestimation, respectively, of the solution of (3.8) on the actual grid X. This means for given z ~ PI(X) some _s, ~-~ Po(X) have to be determined such that the solution u of the following problem

(L + O)u = z is bounded by _s< u and u < ~, respectively. Let e~, ~ be defined by e i := min{z(xi_ 1 + 0) -- diui_l,z(xi -- O) - diui} and ei := max{~(xi_l + 0) - dfii-l,2(xi - O) - di~i}.

(4.1)

Based on linear interpolations of the solutions of (3.8) and bounds for the remainder (compare [5]) we define the estimator S(-,';X) = [_s,~-] ~ I(Po(X)) by _si := min{ui_l,ul} + e ih2/8 and := max{~_l,~i} + e~h2/8

(4.2)

Further realizations of mappings S(', "; X) satisfying (3.3) are proposed and analyzed in [1]. We remark that the estimator given by (4.2) locally approximates the solutions u, u of problems (L + D)_u = z,

(L + D)~ =

by some _s,~ ~ Po in the following sense Ir_u-- _SllL~(a,)--< chl,

Ilu - s-llL=(a,) -< ehi

with some c > 0. This guarantees the estimators to be close to the lower and upper solution in areas with fine grids. Greater differences can occur over larger subintervalls/2 i, especially if the solutions changes heavily. In [13] we directly used the maximal difference between the upper and lower solution to control the grid. To avoid this non strictly implementable estimations here we apply piecewise constant bounds. The idea can be improved by piecewise linear enclosures e.g. Theorem 4.1: The sequences {Tk}, {U k} generated by the M I D grid generation algorithm with a mapping S(.,.; X) accordng to (4.1), (4.2) converge in the following sense

lim [[~-k_ k--~oo

tk[lL~(.Q) = 0

(4.3)

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Mansour AI-Zanaidi and Christian Grossmann

and l i m I1~ ~ - _u~llc(~) = 0 . k~oO

Proof: We apply a similar type of proof as used in I-13], however, unlike the detailed analysis is different from [ 13] because in each of the grids only one iteration step is performed instead of a complete iteration on each level of grids.

First, we select intervals O k := (ak, bk) ~ (0, 1) satisfying Oi,

ak, bk e (3

iel k

~k i

k

and [I~-k -- -tkllL~(a~) : ~k with 6k as defined in step 4 of the M I D grid generation algorithm. By the same arguments as used in [7], [13] we have lira h(Xk[ak, bk] ) = 0.

(4.4)

k--~oo

By selecting appropriate subsequences {ak} b {bk} ~ some a, b exist with a = lira a k ,

b = limb k .

ke]~

k~f

Because the sequences {_tk}, {~-k}, {d k} ~ L2(O ) are m o n o t o n e and bounded these sequences converge in L2(O). We denote the related limits by _t, t- and d, respectively, i.e. l i m II_t~ - _tilLers) = 0 ,

l i m IIt k - tilLs(a) = 0

k---~oo

k~oo

and lim lid k - dilLs(a) = 0.

(4.5)

k~Go

By construction and by the supposed monotonicity of g(', ") we have d > 0. F r o m (4.1), (4.2), (4.4) and from lemma 4.1 we obtain l i m lit k -- _Uk IIL=(a~,b~) = 0 ,

l i m I1~-k -- u- k 11'-2(~,bk) -- 0

k~f

kel

With Theorem 3.1 the results in lim II_tk -- Uk IIL2(a.b)= 0, k~oo

l i m l i p - - U- k IIL2(~,b) = O .

k--+oo

Next, we investigate the convergence of {_zk}, {gk}. Due to (3.5) there holds zk = dktk _ gk(.

tk ) _.1_ q k

with gk(., .) and qk(.) defined by qk(x) := bik(X -- xk_l)

for any

and gk(x, S) := g(xk_l, S),

S e R.

x e O~

(4.6)

1D-Grid Generationby Monotone Iteration Discretization

387

Thus, we obtain (4.7)

z k = dt k _ g(., t k) + (d k _ d)t k + g(., t k) _ 9k(., t k) + qk

NOW, (3.4), (3.5) result in 19(x, t~) - g(x~_ 1, t~) + b{(x - x~-l)[ < 2max{l_b~], [B~l}hp

in

0~'.

Using (4.3) to (4.7) we have l i m I1~~

-

d~

+ g(',U)HL2(a,b ) = O.

ke[

Using (3.8), (4.5) this leads to for any

(~',v')+(dg, v)=(d~-o(',g),v) g(a) = ~,

veHl(a,b)

~(b) = fl

with ~ = limk_~o~~k(a), fl = limk-~o~~k(b). Here, (.,.) denotes the usual scalar product in LE(a, b). From lemma 4.1 we conclude = lira ~k(ak),

fi = lim ~k(bk).

k--*oo

k-~oo

The remaining part of the proof is similar to the one given in [13]. Using the definition of ak, bk we obtain uk(ak) -- u_k(ak) < ~06k,

uk(bk) - u_k(bk) < e6k.

(4.8)

By lemma 2.1 (ii) the following estimation holds II~ - u-llcta,bl < max{~ -- ~,fi - fl}.

(4.9)

On the other hand we have l i m I1~k - ukllcto~.b~l = l i m •k" k"*oo

(4.10)

k~oo

Because the sequence {6k} is monotone and bounded it tends to some limit 6 > 0. From lemma 4.l and (4.8)-(4.10) we obtain 6 = l i m I1~ - _ullct,.bl < e 6 . k"-~oo

with e e (0, 1), now, 6 = 0 holds. This proves the convergence (4.3). The remaining part of the statement we conclude from (4.3) by using theorem 3.1 and the resulting estimation I1~k - _u~llc(~) _< I1~-k _ gkllL~(a ).

9

5. Numerical Results

The proposed grid generation principle is implementable on computers in the strict sense because the generated subproblems (3.8) as well as the estimating mappings can be calculated by finite number of operation and by using available information only. With PASCAL-SC [17] an appropriate tool is available for realizing interval

388

Mansour Al-Zanaidi and Christian Grossmann

estimations and for performing interval operations (compare [6]). We apply the proposed grid generation algorithm to the test problem considered in [5], [13], i.e. we consider the problem

- u " + 2~ sinh u - 2a sgn[(x - 0.55)(x - 0.75)] = 0

in

(0, 1)

u(0) = u 0 ) = 1.

H e r e , , > 0 denotes a parameter of the problem. For large values of ~ the solution of the given problem has boundary layers as well as interior layers at 0.55 and 0.75. We obtain the following numerical results for different sets of data. As initial grids we used equi-distributed ones with N o = 20 and the functions t o = - 2, ~-o = 2 have been applied as starting bounds. = 10 0 N,

deIge, deleq,,

0.9 128 7.76E-6

0.9 102 4.05E-6 2.40E-5

~ = 1000 0.5 0.9 132 125 2.74E-4 2.98E-4 6.65E-3

Here, N , denotes the final number of grid points we used. We denote by deloe~ and by dele,u the maximum error del= max {fi-i-u_i } O
for the generated grid and for an equi-distributed grid of approximately the same number, respectively. The results obtained show a good improvement in accuracy achieved with the grid generator in comparison with equi-distributed grids. Especially the generated grid is condensed near the layers.

0

0.55

0.75

x

1D-Grid Generation by Monotone Iteration Discretization

389

The figures above report the initial grid as well as the first 18 grids and the density of grid points in the final grid, respectively, obtained by the proposed method for data a = 1000, Q = 0.9. The peaks in the density of the gridpoints correspond directly to the boundary as well as to the interior layers of the solution of the example.

References

[1] Abhari, H., A1-Zanaidi, M., Grossmann, C.: Computational aspects of montone iteration discretization algorithm. Preprint 07-01-88, TU Dresden. [2] Adams, E.: Invers-Monotonie, direkte und indirekte Intervallmethoden. Bericht Nr. 185, Forschungszentrum Graz, 1982. [3] Adams, E., Ansorge, R.; Grossmann, C.; Ross, H.-G. (eds.): Discretization in differential equations and enclosures Akademie Verlag, Berlin, 1987. [4] Alefeld, G., Herzberger, J.: Einfuehrung in die Intervallrechnung. Teubner Verlag, Stuttgart, 1974. [5] A1-Zanaidi, M., Grossmann, C.: Monotone iteration discretization algorithm for BVP's. Computing 41 (1989), 59-74. [6] A1-Zanaidi, M., Grossmann, C., Monotone discretization in boundary value problems using PASCAL-SC. (Proceedings of ISNA-87, Prague). [7] Baku~ka, I., Vogelius, M.: Feedback and adaptive finite element solution of one-dimensional boundary value problems. Numer. Math. 44 (1984), 75-102. [8] Bietermann, M., Babu~ka, I.: An adaptive method oflines with error control for parabolic equations of reaction diffusion type. J. Comp. Phys. 63 (1986), 33-66. [9] Doolan, E. P., Miller, J. J. H., Schilders, W. M. A.: Uniform numerical methods for problems with initial and boundary layers. Boole Press, Dublin, 1980. [10] Esser, H., Niederdrenk.: Nichtaequidistante Diskretisierung von RWA. Numer. Math. 35 (1980), 465-478. [11] Gilbarg, D., Trudinger, N. S.: Elliptic partial differential equations of second order. Heidelberg: Springer, 1977. [12] Grossmann, C.: Monotone discretization of two-point boundary value problems and related numerical methods. In: [3], 99-122. [13] Grossmann, C., Ross, H.-G.: Feedback grid generation via monotone discretization for two-point boundary value problems. IMA J. Numer. Anal. 6 (1986), 421-432. [14] Goering, H., Felgenhauer, A., Lube, G., Roos, H.-G., Tobiska, L.: Singularly perturbed differential equations. Berlin: Akademie Verlag, 1983. [15] Lippold, G.: Adaptive approximation. ZAMM 67 (1987), 453-465. [16] Nickel, K.: The construction of a priori bounds for the solution of a two-point boundary value problem with finite elements I. Computing 23 (1979), 247-265.

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Mansour A1-Zanaidi and Christian Grossmann: 1D-Grid Generation

1-17] PASCAL-SC. Information manual, version IBM-PC/DOS. Universitaet Karlsruhe, 1986. 1-18] Rheinboldt, W. C.: On a theory of mesh-refinement processes. SIAM J. Numer. Anal. 17 (1980), 766-778. 1-19] Sattinger, D. H.: Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J. 31 (1972), 979-1000. Mansour A1-Zanaidi Kuwait University, Dept. Mathematics P.O. Box 5969, Safat, Kuwait

Christian Grossmann Dresden University of Technology, Dept. Mathematics Mommsenstr. 13, DDR-8027, Dresden