On the representation of functions and finite difference operators on adaptive sparse grids

In this paper we describe methods to approximate functions and differential operators on adaptive sparse 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, FD operators allow a more efficient operator evaluation than finite element operators. However, the structure of the FD operators is more complex. In order to examine the possibility to construct efficient solution methods, we analyze the discrete FD (Laplace) operator and compare its hierarchical representation on sparse and on full grids. The analysis gives a motivation for a MG solution algorithm.