Estimates for generalized sparse grid hierarchical basis preconditioners

TL;DR

This paper improves estimates for hierarchical basis preconditioners in sparse grid discretizations, extending results to arbitrary dimensions and generalized spaces, with bounds that are nearly sharp.

Contribution

It provides new bounds for generalized sparse grid spaces in arbitrary dimensions, extending previous estimates and demonstrating near-sharpness for various subclasses.

Findings

Bounds are extended to arbitrary dimensions d>1.

Results apply to generalized sparse grid spaces with monotone index sets.

Bounds are shown to be sharp up to dimension-dependent constants.

Abstract

We reconsider some estimates the paper "M. Griebel, P. Oswald, On additive Schwarz preconditioners for sparse grid discretizations. Numer. Math. 66 (1994), 449-463" concerning the hierarchical basis preconditioner for sparse grid discretizations. The improvement is in three directions: We consider arbitrary space dimensions d>1, give bounds for generalized sparse grid spaces with arbitrary monotone index set, and show that the bounds are sharp up to constants depending only on d, at least for a subclass of generalized sparse grid spaces containing full grid, standard sparse grid spaces, and energy-norm optimized sparse grid spaces.