Pointwise Convergence Of The Combination Technique For Laplace's Equation

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 combination technique, a modern, eecient, and easily parallelizable sparse grid solver for elliptic partial diierential equations that recently gained importance in elds of applications like computational uid dynamics. For full square grids with meshwidth h and O(h ?2) grid points, the order O(h 2) of the discretization error using nite diierences was shown in 12], if g(x) 2 C 2 0; 1]. In this paper, we show that the nite diierence discretization 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 the Fourier coeecients b k of ~ g, the 2-periodic and 0-symmetric extension of g, fullll jb k j c g k ?3?" for some arbitrary small positive ". If 0 < " 1, this is vaild for g 2 C 4 0; 1] and g(0) = g(1) = g 00 (0) = g 00 (1) = 0, e.g.. A simple transformation even shows that g 2 C 4 0; 1] is suucient. Furthermore, we present results of numerical experiments with functions g of varying smoothness.