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

Michael Karkulik, Jens Markus Melenk, Alexander Rieder

Abstract

We consider fractional Sobolev spaces $H^θ(Γ)$, $θ\in [0,1]$, on a 2D surface $Γ$. We show that functions in $H^θ(Γ)$ 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 $p$.

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

Abstract

We consider fractional Sobolev spaces , , on a 2D surface . We show that functions in 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 -version of the boundary element method with condition number bounds that are uniform in the polynomial degree .

Paper Structure

This paper contains 23 sections, 29 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Notation, assumptions, and main results
  3. Geometric and functional setting
  4. Meshes and polynomial spaces
  5. Main results
  6. Applications
  7. Interpolation of discrete $L^2$ spaces with weights
  8. Additive Schwarz preconditioning for the $p$-BEM
  9. Interpolation of Sobolev norms
  10. Interpolation of Sobolev norms and seminorms
  11. Interpolation of weighted spaces
  12. Trace theorems and local estimates
  13. Interpolation of weighted spaces
  14. Averaging operators
  15. Preliminaries
  16. ...and 8 more sections

Key Result

Theorem 2.4

For $\theta \in (0,1)$, there exists a constant $C_{\theta}$ that depends only on $\Gamma$, $\Omega$, the $\gamma$-shape regularity of the triangulation ${\mathcal{T}}$, and $\theta$, such that for all $u \in \widetilde{H}^{\theta}(\Gamma)$, and for all decompositions with $\operatorname{supp}u_V \subseteq \omega_V$, $\operatorname{supp}u_e \subseteq \omega_e$ and $\operatorname{supp}u_K\subseteq

Theorems & Definitions (61)

  • Definition 2.2
  • Theorem 2.4
  • proof
  • Theorem 2.5: stable decomposition of $\widetilde{H}^\theta(\Gamma)$---continuous and discrete
  • Theorem 2.6: stable localization of $\widetilde{{\mathcal{S}}}^{\mathbf{p},1}({\mathcal{T}})$
  • Theorem 2.7: stable decomposition of $H^\theta(\Gamma)$ --- continuous and discrete
  • Remark 2.8
  • Proposition 3.1: tartar07
  • Definition 3.2: locally comparable
  • Theorem 3.3
  • ...and 51 more