On the Representation of Functions and Finite Difference Operators on Adaptive Dyadic Grids

Abstract

Abstract In this paper we describe methods to approximate functions and dif- ferential operators on adaptive sparse (dyadic) grids. We distinguish between several representations of a function on the sparse grid and we describe how finite difference (FD) operators can be applied to these representations. For general variable coefficient equations on sparse grids, genuine finite element (FE) discretizations are not feasible and FD operators allow an easier operator evaluation than the adapted FE operators. However, the structure of the FD operators is complex. With the aim to construct an efficient multigrid procedure, we analyze the structure of the discrete Laplacian in its hierarchical representation and show the relation between the full and the sparse grid case. The rather complex relations, that are expressed by scaling matrices for each separate coordinate direction, make us doubt about the pos- sibility of constructing efficient preconditioners that show spectral equivalence. Hence, we question the possibility of constructing a natural multigrid algorithm with optimal O(N) efficiency. We conjecture that for the efficient solution of a general class of adaptive grid problems it is better to accept an additional condition for the dyadic grids (condition L) and to apply adaptive hp-discretization.