Stable decompositions of $hp$-BEM spaces and an optimal Schwarz preconditioner for the hypersingular integral operator in 3D

TL;DR

This paper develops stable $hp$-boundary element space decompositions and introduces an optimal Schwarz preconditioner for the hypersingular integral operator in 3D, ensuring uniform condition number bounds across polynomial degrees.

Contribution

It provides a new stable decomposition framework for fractional Sobolev spaces on surfaces and applies it to design an optimal Schwarz preconditioner for hypersingular BEM discretizations.

Findings

Stable decompositions of Sobolev spaces are established.

Preconditioner achieves uniform condition number bounds.

Application to $p$-version boundary element methods.

Abstract

We consider fractional Sobolev spaces $H^\theta(\Gamma)$, $\theta \in [0,1]$, on a 2D surface $\Gamma$. We show that functions in $H^\theta(\Gamma)$ can be decomposed into contributions with local support in a stable way. Stability of the decomposition is inherited by piecewise polynomial subspaces. Applications include the analysis of additive Schwarz preconditioners for discretizations of the hypersingular integral operator by the $p$-version of the boundary element method with condition number bounds that are uniform in the polynomial degree $p$.