Multilevel decompositions and norms for negative order Sobolev spaces

TL;DR

This paper develops multilevel decompositions and norms for negative order Sobolev spaces, enabling efficient computation and stable preconditioning on complex meshes.

Contribution

It introduces stable multilevel decompositions for $H^{-s}$ spaces on simplicial meshes, with applications to preconditioning and norm evaluation.

Findings

Stable multilevel decompositions for $H^{-s}$ spaces are constructed.

Preconditioners with mesh-independent condition numbers are developed.

Efficient evaluation methods for negative order Sobolev norms are proposed.

Abstract

We consider multilevel decompositions of piecewise constants on simplicial meshes that are stable in $H^{-s}$ for $s\in (0,1)$. Proofs are given in the case of uniformly and locally refined meshes. Our findings can be applied to define local multilevel diagonal preconditioners that lead to bounded condition numbers (independent of the mesh-sizes and levels) and have optimal computational complexity. Furthermore, we discuss multilevel norms based on local (quasi-)projection operators that allow the efficient evaluation of negative order Sobolev norms. Numerical examples and a discussion on several extensions and applications conclude this article.