A MATHEMATICS

4 AND s COMPUTERS

IN STATION ELSEVIER Mathematics and Computers in Simulation 42 (1996) 595-605

A proof of convergence for the combination technique for the Laplace equation using tools of symbolic computation *

H. Bungartz *, M. Griebel, D. Roschke, C. Zenger

Institut fiir Informatik der Technischen Universitiit Miinchen, Arcisstr. 21, D-80290 Miinchen, Germany

Abstract

For a simple model problem -the Laplace equation on the unit square with a Dirichlet boundary function vanishing for x = 0, x = 1 and y = 1, and equaling some suitable g(x) for y = O-we present a proof of convergence for the so- called combination technique, a modem, efficient and easily parallelizable sparse grid solver for elliptic partial differential equations that recently gained importance in fields of applications like computational fluid dynamics. For full square grids with meshwidth h and O(h-2) grid points, the order O(h’) of the discretization error was shown in (Hofman, 1967), if g(x) E C*[O, 11. In this paper, we show that the error of the solution produced by the combination technique on a sparse grid with only O((h-’ log,(h-‘)) grid points is of the order O(h* log,(h-I)), if g E C4[0, 11, and g(0) = g(1) = g”(0) = g”( 1) = 0. The crucial task of the proof, i.e. the determination of the discretization error on rectangular grids with arbitrary meshwidths in each coordinate direction, is supported by an extensive and interactive use of Maple.

Keywords: Combination technique; Sparse grids; Elliptic partial differential equations; Order of discretization error; Symbolic computation.

1. Introduction

The combination technique [5] is an up-to-date method for the efficient solution of elliptic partial differ- ential equations on sparse grids. In comparison to the standard full grid approach with meshwidth h in each

direction, the number of grid points can be reduced significantly from O(hmd) to O(h-’ (log2(h-1))d-‘) in the d-dimensional case, whereas the pointwise accuracy of the approximation is only slightly deteriorated from O(h*) to 0(h2(log2(h-‘))d-‘).

In the last years, the combination technique has been applied to different types of elliptic partial differ- ential equations including three-dimensional problems and problems originating from computational fluid dynamics. Its excellent parallelization properties have been verified on different parallel architectures (see W,61).

??Supported by the Bayerische Forschungsstiftung via FORTWIHR - Bayerischer Forschungsverbund fur technisch- wissenschaftliches Hochleistungsrechnen, and by the Deutsche Forschungsgemeinschaft in its Sonderforschungsbereich 342.

* Corresponding author, Tel. : 49 89 289 22018; fax: 49 89 289 22022; E-mail: bungartz@informatik.tu-muenchende.

0378-4754/96/$15.00 0 I996 Elsevier Science B.V. All rights reserved PII SO378-4754(96)00036-5

596 H. Bungartz et al. /Mathematics and Computers in Simulation 42 (1996) 595-605

For a proof of the approximation properties mentioned above, in the two-dimensional case the existence of an error splitting of the type u(x, y) - ui,j(x, y) = 0(/z:) + O(h$) + O(@z~) has to be shown. Here, u(x, y) is the exact solution of the given boundary value problem, and ui,j (x, y) denotes the solution on a rectangular full grid with meshwidths h, = 22’ and h, = 2-j. Up to now, this has just been presupposed. We present a proof for the existence of such an error splitting in the case of Laplace’s equation on the unit square, if the Dirichlet boundary function satisfies certain smoothness requirements. At several steps, the proof profits from the extensive properties of modern symbolic computation tools like Maple.

2. The combination technique

Let L be an elliptic operator of second order. Consider the partial differential equation Lu = f on the unit square 52 =]0, l[* with appropriate boundary conditions. Furthermore, let Gi,j be the rectangular grid on fi with meshwidths h, := 2-’ in x- and h Y := 2-j in y-direction (i, j E N). In [5], a technique is introduced which combines the solutions ui,j of the discrete problems Li,jUi,j = fi,j associated to Lu = f on different rectangular grids Gi,j:

u; n := C C Ui.j- Ui,j - (1) i+j=n+l i+j=n

The resulting solution U& is given on the so-called sparse grid G n,n with O(2”n) grid points instead of the usual full grid G,,, with O(4”) points (see Fig. 1).

For a detailed introduction to sparse grids, see [ 1,9]. Note that we have to solve IZ different problems Li,iui,j = fi,j, i + j = n + 1, all with 0(2n+‘) grid

points, and 12 - 1 different problems Li,jui,j = fi,j, i + j = n, all with O(2”) grid points, and to combine their bilinearly interpolated solutions. Furthermore, note that these 2n - 1 problems are independent from one another and can be solved in parallel.

Assuming that the error of the solution ui,j obtained on the rectangular grid Gi, j is of the form u (x, y) - ui,j(x, y) = O(hi) + O(hf,) + O(hzht) with h, = 2-’ and h, = 2-j for each inner point (x, y) E J2, it has been shown in [5] that

u-u;. = 0(4-S), (2)

which has to be compared to the 0(4Vn)-error of the full grid solution u~,~. In this paper, we deal with the question, which smoothness requirements have to be fulfilled by the Dirichlet boundary function in order to receive such an error on Gi,j. For the remainder, we concentrate on the Laplace equation

. ?? ? ? ? ?

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? ? ? ?

? ?

? ? ?? ? ? ??

? ?

?? ? ?

? ?

? ? ? ? ? ? ? ?

.

.

.

.

.

.

.

.

. on

. . . . *.....*..a....*..

. : . : . :

. . : . . ?? : ?? . . :

. : . : . :

. . . . :***.***:.......: .

I : ?? : . . . : . . ??

. . . . . ?? ? ? ? ?? ? ? ?. . . : . : . . . . :..*.....*..*....

d

Fig. 1. Sparse grids G3.3 and e.4,~

H. Bungartz et al. /Mathematics and Computers in Simulation 42 (I 996) 595-605

huh, y) = 0,

with the Dirichlet condition

u(x, Y) = 1 g(x), (x, y> E SC y = 0, 0, (x, y) E ss?, y > 0.

Throughout the paper, we assume that

g(x) E C4]0, 11, g(0) = g( 1) = g”(0) = g”( 1) = 0.

597

(3)

(4)

(5)

3. Explicit solutions

In this section, we are looking for explicit representations of the solutions u,ui,j, ui,m and Urn,,, of the Laplace equation (3) and the following three associated discretized problems with meshwidths h, = 22’ and h, = 2-j:

ui,j(x - h,, Y) - 2ui,j(xv Y) + ui,j(x + hxt Y)

G + ui,j(x, Y - hy) - 2ui,j(x, Y) + ui,j(x, Y + hy) = o

% (6)

Ui,co(X - hx, Y) - 2Ui,co(X,.Y) + Ui,ce(X + h.x, Y) + a2Ui,co(X, Y) = o

hZ ay2 ’ (7)

um,j(x, Y - hy) - 2uco,j(x, Y) + um,j(x, Y + hy) + a2um,j(x, Y) = o

h: ax2 (8)

Thus, ui,j solves the discrete problem resulting from the use of central finite differences in x and y direction. Accordingly, ui,a and uo0.j denote the solutions of the problems discretized only in x- or in y- direction, respectively.

We start with two definitions:

T(t, y) := sinh(tm(1 - y))

sinh(tn) ’ (9)

1

b, := 2 s

g(x) sin(nrx) dx.

0

From [8], we know that

(10)

u(x, y) := E b, sin(nnx)T(n, y) n=l

solves (3) and fulfills (4). The three discrete problems (6)-(8) are solved by

(11)

ui,j(x, Y> := 2 bn sin(nm)T(pn, y), fl=l

(12)

598 H. Bungartz et al./Mathematics and Computers in Simulation 42 (1996) 595-605

(13)

u,,j(x, y) := E b, sin(nm)T(L, y>, II=1

(14)

where

2j+l cLn(i, j) := -arsinh(2’-j sin(nn22-I)),

u,(i) := jlLmW”,,(i, j),

(15)

(16)

h,(j) := lim pn(i, j). i+w

(17)

In each sum (12), (13) and (14), sin(nnx) T (., y) solves the corresponding problem (6), (7) and (8), re- spectively. The series converge, because T (t, y) 6 1 and because our smoothness assumptions (5) concerning g allow four times partial integration of g(x) sin(nrx), leading to the relation

lb, I < cgln4, (18)

with some constant cg depending on g. Furthermore, ui,j, ~i,~ and um,j fulfill all the same boundary condition (4) as u does. This results from the fact that we always use the continuous Fourier coefficients b Il.

Now, let N < 2i be arbitrary, but fixed. Then

N-l

ii(x, y) := c b, sin(nnx)T(n, y) n=l

(19)

solves (3), too and fulfills t(0, y) = li(l, y) = C(x, 1) = 0 for all x, y E [0, 11. But instead of g(x), we obtain a different boundary function &j(x):

N-l

ii@, 0) = c b, sin(nnx) =: j(x). n=l

(20)

The same holds, if we neglect all terms with n 2 N in (12), (13) and (14), getting fii,j, Lii,oo and G,,j:

N-l

ii,j(x, y) := C b, sin(nTx)T(p.,, y), (21) n=l

N-l

ii,~(x, Y) := >: bn sin(nTx)T(un,y), n=l N-l

(22)

G,,j(x, y) := C b, sin(nnx)T(h,, y). n=l

(23)

We can now write

U - Ui, j = (U - ii) + (ii - iii, j) + (ii, j - t4.i. j). (24)

H. Bungartz et al. /Mathematics and Computers in Simulation 42 (I 996) 595405 599

Since we want to determine the order of u - U~,J in 52, note that 0 < x, y -=z 1 for the rest of the paper. First, we deal with u - li. u - Li is of arbitrary order NPk (k E N, see [7]):

]U - G] = 2 b, sin(nvx)T(n, y) < 2 IbnjeFnny, n=N n=N

since T(n, y) 6 e -‘*y. Therefore, we obtain

because e- ’ < k!/xk for each integer k > 0 and each x > 0. Next, we look at Lii,j - ui,j:

1ii.j - ui.jl = 2 bn sin(nnx)T(pLL,, Y> < 2 IhI. n=N n=N

Here, 1 b, I < cg /n4 leads to

(25)

(26)

In the following section, we will develop a technique to treat the remaining difference ci - iii,j.

4. Error splitting

In this section, we reduce our problem of determining the order of Li - Gi,j to the problem of finding bounds for certain differential quotients. The bounds and the necessary smoothness requirements for g will be determined with an extensive use of Maple in Section 5.

Now, let us write

l2i.j - 22 = fi,j + 4") + q"',

where

fi,j I= l2i.j - iii,~ - Li~,j + ii!, r,(l) ._ ,. _ * I

.- Lli.00 - U, r!2’ ._ -

J .- u c0.j - 24.

We want to learn under which circumstances

c,j =

hold, where 2-j.

We begin

D(x, y, hx, h,)h:h;. I+) = Cl(x y h,)h2 (2) 2 I , , X’ rj = C2(xt Y, hy)hy

ID 1, I Cl I and ]C2 I are bounded from above by some B(x, y). Remember that h, = 2

with fi,j. Let US define A .

yi,j I= Ui,j - ui,j+l - ii+l,j + Gi+l,j+l.

Then

2 2 Yi+sl.j+sz = fi,j. s2=0 s, =o

-i

(27)

(28)

(29)

hy =

(30)

(31)

600 H. Bungartz et al. /Mathematics and Computers in Simulation 42 (1996) 595-605

Thus, if we assume now that

Yi,j = h:h$d(x, Y, hx, hy)

with Id@, y, hX, hy)l < b(x, y), we obtain

c,j = hzh; 2 2 4-(s’+s’) d(x, Y, h,2-S’ , h,2P2) =: h;h$D(x, y, hX, hy) s2=o.q=0

with 1 D 1 bounded from above:

ID(x, y, hx, hy)l < b(x, y) 2 f+“+~‘2) = ;b(x, y) =: B(x, y). s2=oq =o

Consequently, it is sufficient to show

Yi,j = h:h;d(x, y, hx, hy) with Id(x, y, hx, hy)l < b(x, y).

Writing now i&1,02 (x, y) instead of fii,j(X, y) with arbitrary parameters ~1, ~72 3 1 and resulting formal meshwidths 2-ar, 2-a2 ~10, i], the definition

N-l

5(x, Y, m,a2> := &,,o,b, Y) = C h Wnax)T(b(fll, 02)~ Y> (32) n=l

with fixed N < 2ar leads to a function fi differentiable in 01 and ~2, and allows us to represent yi,j as integrals over the differential quotient a2fi/LIal 8~2:

i+l j+l

ss

a2qx, y, m,a2> Yi,j = aDlao da1 dc2.

i j

Finally, if we suppose that

a2ik Y, cl, a2) aa2aa2 = 4-(ar+0~)&x, y, CJl) CQ)

with 12(x, y, or, a2)] < b(x, y), we receive

i+l j+l

yi,j = h2h2 x Y JS 4i-u1+j-U2&x, y, al, a2) da1 da;! =: h:h;d(x, y, h,, h,),

i j

(33)

(34)

where

i+l j+l

Id@, Y, hxt hy)l < Nx, Y) JJ

4i-a14j-02 da1 da2 6 b(x, y).

i j

H. Bungartz et al. /Mathematics and Computers in Simulation 42 (1996) 59.5-605 601

Thus, we have to show relation (34). Since the explicit representation of 6 (32) is a finite sum, it is sufficient to show that a*T(p.,, y)/aalau2 is of the according order:

with a suitable chosen function 4 (n, y) that fulfills

N-l

c IhlII~(~, Y)I < Q(Y) v1 < N <l-F’. n=l

The analogous steps work for $” and r(*) and lead to according aT(h,, y)/&q that have to be shown:

aT(‘4l(w), Y>

aal < 4-%#P)(n y) , ,

aT@rl(~*)~ Y)

aa2 < 4-%#P(n, y),

with functions c$(‘), c#J(~) also fulfilling (36):

5. Bounds for the differential quotients

In Section 4, we have reduced our problem to finding bounding ~$(*)(n, y) so that (35)- (39) hold. We begin with i32T(pn,,)/i3al aa2:

(35)

(36)

properties of aT(u”, y)/aa’ and

(37)

(38)

(39)

a*wn~ Y> a*w, Y> ah ah aTo, Y) a*h afllau2 = at*

~- t+ 801 aa2

+ at t=pn aal aa

ah ah f’2bn3 Y$-~ + PI&.,, Y$& 7

functions $(n, y), $(‘)(n, y) and

(40)

with

PI (t, y) := n ((1 - y) coth(tn(1 - y)) - coth(tn)) , P2(t, Y) := P:(t, Y> + am, YI

. at

Now, we have a closer look on the partial derivatives of Pi. This is where Maple comes in, as these derivatives turn out to be quite complicated. With the abbreviations (substitutions)

nTr sin 2 nn w .= - z:=2al+l, . z 7 u := 2a2+‘W’

602 H. Bungartz et al. /Mathematics and Computers in Simulation 42 (1996) 595405

we obtain

ah In 2 ~ = -&34-~1 sin 2 --- a0, 4 z3 ah In 2 _ = -,2,3,34-a2 arsinhv

aa 4 ,3-

a2b In2 2 2 un z cos z 1 p= -IT ( > w2n54-(~l +a21 -__ aDlag 4 C z3 z2 > (1 + v2)3/2.

Thus, the right-hand side of relation (40) equals

a2uh Y) aal au2

= T(pn, y)n54-(al+flz’) (yT2)2w2 (!g _ 79

1

I?~~ )

1 312 + Pl (CL,, Y) (1 + up> .

Since our upper summation bound N is less or equal 2”’ (36) we have 0 < n -C 2Ol. Therefore, determine the following intervals:

z El& 7F/2c, sin 2 ~ - 7 ??]8/,~, 1/3[,

z3 w E 12/n, l[.

Thus, with the definitions

1 QI (n, Y, 01, ~2) := TUn, y)f’l bn, y) (1 + u213,2,

Q2(n+ Y, ~713 02) := T&x, Y)~‘~(PU,, ~1~s (

arsinhu 1’ ~ -

u V3 > V2&-T-7 ’

we get

a2wn3 Y> = ,o(n ul U2)n54-(a’+a2) (Q, (n afll,au2 ’ ’

Note that p(n, (~1, ~2) E](2/v)(ln 2)2, &P”(ln 2)2[. Finally, let us look at Q 1 and Q2. For a proof that

see [7]. Now, we define

t

I

s := Itw E]2/n, 2a’ [, h :=

Y, ffl? a:

PI 1 and

I + Q2b Y, ~1,~2)) .

(41)

(42)

(43)

we can

(44)

P2 ] are bounded from above by some P(y),

Again, we use that T(t, y) < eCf’ry for I > 0 and e-’ 6 k!/tk for each integer k > 0 and each t > 0. Furthermore, a short calculation verifies that pn = (arsinh II)/ h and u = sh. Hence, with k = 3, we get

H. Bungartz et al. /Mathematics and Computers in Simulation 42 (1996) 595-605 603

b3Ql(n,y,m,~2)l <n3P(y)eppnT~(1 +lz)9/2 s3 = gPb)e -((arsinhu)/h)rry 1

(1 + u2p

Tr3 +p3P(Y) ((

3! 1 3P(Y) (W3 1

arsinhv)/h)ny)3 (1 + 1?)3/2 = 7 (arsinhv)3 (1 + v2)3/2

3PCY) II3 1

= 4y3 (arsinhv)3 (1 + v2)3/2’

Since

V3 1

(arsinhv)3 (1 + 1_?)3/2 <l

for v > 0, we have shown

3PCY) b3Q,,,m Y, (31,02>1 < 4y3.

In a similar way, we show that

choosing k = 4 this time:

s3 arsinhv ln3Q2(n, y, ~1, cr2)l < zu3 P(y)e-((arsinhu)~h)~~ v2Jr+ v2 (~ -

V >

4! s

(

arsinhv

((arsinhv)h)ny4 v2dm ~ - v >

3P(Y) (SW4 (

arsinhv 1 =-

TY4 (arsinhv)4v2dm v p-dG7 >

3P(Y) V2 ~- TY4 (arsinhv)4dw >

< 3P(Y) 1 P(Y) -‘Fj=ny4’

since V2

(arsinhv)4Jm

for all v > 0. Altogether, starting from (41), we have shown

1 <-

3

604 H. Bungartz, et al. /Mathematics and Computers in Simulation 42 (1996) 595-605

So, the left-hand side of (36) is bounded for all positive at, ~2, and N 6 2ar by

“% ]bn]]@(n, y)] < “% 3n’ (F + $) $(ln2)2 n=l l7=l

< 3P(Y) + P(Y) G-r4 \ ( --) qg(ln2)?‘rZ f. 4y3 ny4

Accordingly, we show (37) - (39). This completes the proof of the error splitting (27)-(29) leading to i --Lii,j = O(/z;ht) +0(/z:) +0(/z:).

Choosing now N := 2’ = hi*, we receive u - li = O(hZ) from (25) (k = 2) and Gi,j - ii,j = O(hz) from (26). Hence, we learn from (24) that the error u - ui,j of the solution on a rectangular grid with meshwidths h, = 2-’ and h, = 2-j has the form

u - ui,j = O(h;h;) + O(h;) + O(h$). (45)

if the smoothness requirements (5) hold, which is the desired result and the link to [5] and, thus, to (2).

6. Concluding remarks

The discretization error u - u~,~ of the solution of our Dirichlet problem obtained on the full square grid G,,, is of the order 0(4-n), if g E C2[0, 1] [7]; thus, the error u - U: n of the solution obtained by

the combination technique on the sparse grid c,.,, is only slightly deteriorated to 0(4-‘n), if g E C4[0, l] and g(0) = g( 1) = g”(0) = g”( 1) = 0. These more severe smoothness requirements were to be expected because of the interpolation properties of sparse grids [I].

In this paper, we presented a proof for the combination technique. We did not focus on numerical aspects of or experiences with the combination technique, since a lot of numerical results can be found in other reports [3-61.

Our proof of the combination technique is an example for the strong support computer algebra programs like Maple can provide even in a more theoretical context of numerical analysis like convergence theory of PDE solvers. Of course, a task like finding bounds for complicated functions of four unknowns, as it has to be done here, generally still lies beyond the posibilities of symbolic computation tools, because this usually requires the computation of maxima and minima, i.e. the computation of zeros. Nevertheless, the interactive use of Maple helped us, e.g., to try out and find substitutions suitable for simplifying large expressions, to factorize these expressions, and, finally, to reformulate our problem, enabling us to find the presupposed order terms. To some extent, Maple played a similar role for our problem as calculators did for many problems a few decades ago.

References

[l] H. Bungartz, Diinne Gitter und deren Anwendung bei der adaptiven Liisung der dreidimensionalen Poisson-Gleichung, Dissertation, Technische Universitst Miinchen, 1992.

[2] B. Char, K. Geddes, G. Gonnet, B. Leong, M. Monagan and S. Watt, Maple V Library Reference Manual (Springer, New York, 1991).

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[3] M. Griebel, The combination technique for the sparse grid solution of PDEs on multiprocessor machines, Parallel Processing Letters 2 (1) (1992) 6 l-70.

[4] M. Griebel, W. Huber and C. Zenger, A fast poisson solver for turbulence simulation on parallel computers using sparse grids, in: E. Hirschel, ed., Notes on Numerical Fluid Mechanics, Vol. 38 (Vieweg, Braunschweig, 1993).

[5] M. Griebel, M. Schneider and C. Zenger, A combination technique for the solution of sparse grid problem, in: P. de Green and R. Beauwens, eds., Proc. IMACS Intemat. Symp. on Iterative Methods in Linear Algebra (Brussels, April 1991) (Elsevier, Amsterdam, 1992).

[6] M. Griebel and V. Thumer, The efficient solution of fluid dynamics problems by the combination technique, Institut fur Informatik der Technischen Universitlt Miinchen, SFB-Report 342/l/93 A, 1993.

[7] P. Hofmann, Asymptotic expansions of the discretization error of boundary value problems in rectangular domains, Numer. Math. 9 (1967) 302-322.

[8] J. Walsh and D. Young, On the degree of convergence of solutions of difference equations to the solution of the Dirichlet problem, J. Math and Phys. 33 (1954) 8&83.

[9] C. Zenger, Sparse grids, in: W. Hackbusch, ed., Parallel Algorithms for Partial Differential Equation, Proc. Sixth GAMM- Seminar (Kiel, January 1990), Notes on Numerical Fluid Mechanics, Vol. 31 (Vieweg, Braunschweig, 1991).

[lo] C. Zenger, Representation of functions of several variables by sparse grids, in: V. Ganzha, V. Rudenko and E. Vorozhtsov, eds., Computer Algebra and its Applications to Mechanics (Nova, New York, 1992).