Efficient additive Schwarz preconditioning for hypersingular integral equations on locally refined triangulations

TL;DR

This paper develops and analyzes a new local multilevel additive Schwarz preconditioner for hypersingular integral equations, achieving optimal condition number independence from mesh size and refinement levels on adaptive meshes.

Contribution

It introduces a novel local multilevel preconditioner that ensures the condition number remains bounded regardless of mesh refinement or element count.

Findings

Preconditioner achieves mesh-independent condition number growth.

Numerical experiments confirm effectiveness on 2D and 3D problems.

Applicable to both closed boundaries and open screens.

Abstract

For non-preconditioned Galerkin systems, the condition number grows with the number of elements as well as the quotient of the maximal and the minimal mesh-size. Therefore, reliable and effective numerical computations, in particular on adaptively refined meshes, require the development of appropriate preconditioners. We analyze and numerically compare multilevel additive Schwarz preconditioners for hypersingular integral equations, where 2D and 3D as well as closed boundaries and open screens are covered. The focus is on a new local multilevel preconditioner which is optimal in the sense that the condition number of the corresponding preconditioned system is independent of the number of elements, the local mesh-size, and the number of refinement levels.