Optimal additive Schwarz methods for the $hp$-BEM: the hypersingular integral operator in 3D on locally refined meshes

TL;DR

This paper introduces a robust overlapping Schwarz preconditioner for the $hp$ boundary element method solving hypersingular integral equations in 3D, ensuring uniform condition numbers across mesh refinements and polynomial degrees.

Contribution

It develops a new additive Schwarz preconditioner that is effective for adaptively refined meshes and high polynomial orders in 3D boundary element methods.

Findings

Condition number bounded uniformly in mesh size and polynomial degree

Preconditioner effective on adaptively refined meshes

Numerical experiments confirm robustness across geometries

Abstract

We propose and analyze an overlapping Schwarz preconditioner for the $p$ and $hp$ boundary element method for the hypersingular integral equation in 3D. We consider surface triangulations consisting of triangles. The condition number is bounded uniformly in the mesh size $h$ and the polynomial order $p$. The preconditioner handles adaptively refined meshes and is based on a local multilevel preconditioner for the lowest order space. Numerical experiments on different geometries illustrate its robustness.