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 with 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 convergence for the so-called combination technique, a modern, 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(2)) of the discretization 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 produced 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 discretization error on rectangular grids with arbitrary meshwidths in each coordinate direction, is supported by an extensive and interactive use of Maple.
combination technique symbolic computation sparse grids elliptic partial differential equations order of discretization error