Optimality of multilevel preconditioners for local mesh refinement in three dimensions

TL;DR

This paper proves the optimality of multilevel preconditioners, specifically BPX and WHB, for local mesh refinement in three dimensions, extending previous 2D results and ensuring linear computational complexity.

Contribution

It extends the 2D BPX and WHB optimality results to 3D local red-green mesh refinement, establishing their linear complexity and stability in higher dimensions.

Findings

BPX norm equivalence is optimal for 3D local refinement.

WHB preconditioner is optimal in 3D local mesh refinement.

Practical BPX implementation has linear computational complexity.

Abstract

In this article, we establish optimality of the Bramble-Pasciak-Xu (BPX) norm equivalence and optimality of the wavelet modified (or stabilized) hierarchical basis (WHB) preconditioner in the setting of local 3D mesh refinement. In the analysis of WHB methods, a critical first step is to establish the optimality of BPX norm equivalence for the refinement procedures under consideration. While the available optimality results for the BPX norm have been constructed primarily in the setting of uniformly refined meshes, a notable exception is the local 2D red-green result due to Dahmen and Kunoth. The purpose of this article is to extend this original 2D optimality result to the local 3D red-green refinement procedure introduced by Bornemann-Erdmann-Kornhuber (BEK), and then to use this result to extend the WHB optimality results from the quasiuniform setting to local 2D and 3D red-green refinement scenarios. The BPX extension is reduced to establishing that locally enriched finite element subspaces allow for the construction of a scaled basis which is formally Riesz stable. It is possible to show that the number of degrees of freedom used for smoothing is bounded by a constant times the number of degrees of freedom introduced at that level of refinement, indicating that a practical implementable version of the resulting BPX preconditioner for the BEK refinement setting has provably optimal (linear) computational complexity per iteration. An interesting implication of the optimality of the WHB preconditioner is the a priori H1-stability of the L2-projection. The theoretical framework employed supports arbitrary spatial dimension d >= 1 and requires no coefficient smoothness assumptions beyond those required for well-posedness in H1.