A PROOF OF CONVERGENCE FOR THE COMBINATION TECHNIQUE FOR THE LAPLACE EQUATION USING TOOLS OF SYMBOLIC COMPUTATION

For a simple model problem - the Laplace equation on the unit square w ith a Dirichlet boundary function vanishing for x = 0, x = 1 and y = 1 , and equaling some suitable g(x) for y = 0 - we present a proof of co nvergence for the so-called combination technique, a modern, efficient and easily parallelizable sparse grid solver for elliptic partial dif ferential equations that recently gained importance in fields of appli cations like computational fluid dynamics. For full square grids with meshwidth h and O(h(-)2) grid points, the order O(h(2)) of the discret ization error was shown in (Hofman, 1967), if g(x) is an element of C- 2[0, 1]. In this paper, we show that the error of the solution produce d by the combination technique on a sparse grid with only O((h(-1) log (2)(h(-1))) grid points is of the order O(h(2) log(2)(h(-1))), if g is an element of C-4[0, 1], and g(0) = g(1) = g ''(0) = g ''(1) = 0. The crucial task of the proof, i.e. the determination of the discretizati on error on rectangular grids with arbitrary meshwidths in each coordi nate direction, is supported by an extensive and interactive use of Ma ple.