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

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 $p$.

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