A Combination Technique for the Solution of Sparse Grid Problems

Two-dimensional sparse grids contain only O(h ?1 log(h ?1)) grid points, in contrast to the usually used full O(h ?2)-grids, whereas for a suuciently smooth function the accuracy of the representation is only slightly deteriorated from O(h 2) to O(h 2 log(h ?1)). The combination technique presented in this paper uses the solutions of O(log(h ?1)) diierent, on regular standard grids discretized problems with O(h ?1) grid points and e.g. diierent mesh sizes in the x-and y-direction to produce a sparse grid solution. Assuming a suuciently smooth solution and asymptotic error expansions for the diierent discrete solutions, it is shown that the leading error terms are cancelled by the combination technique, and a sparse grid solution with an accuracy of the order O(h 2 log(h ?1)) is created. The proof is extended to the three-dimensional case. Here, by the combination of O((log(h ?1)) 2) solutions on diierent grids with O(h ?1) grid points, an accuracy of the order O(h 2 (log(h ?1)) 2) is achieved. Additionally, the results of numerical experiments are reported. There, it can be seen that the combination approach works not only for suuciently smooth solutions of linear problems, but, to some extent, also for non-smooth solutions and even for non-linear problems. 1. Grids and Spaces. Let i;j be the equidistant rectangular grid on the unit square = 0; 1] 0; 1] with meshwidth h i = 2 ?i in x-and h j = 2 ?j in y-direction. Let moreover S i;j be the space of piecewise bilinear functions on grid i;j. A partial diierential equation Lu = f with appropriate boundary conditions can now be discretized on any grid i;j using the nite element method and standard basis functions. For reasons of simplicity we only consider homogeneous Dirichlet boundary conditions and restrict the space S i;j to the space S 0 i;j of functions that satisfy these boundary conditions.