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.

Algorithms for functional approximation (msc 65D15), Numerical differentiation (msc 65D25), Finite difference methods (msc 65N06), Multigrid methods; domain decomposition (msc 65N55)
Life Sciences (theme 5), Energy (theme 4)
CWI
Modelling, Analysis and Simulation [MAS]
Scientific Computing

Hemker, P.W, & Sprengel, F. (1999). On the representation of functions and finite difference operators on adaptive sparse grids. Modelling, Analysis and Simulation [MAS]. CWI.