Optimal additive Schwarz preconditioning for adaptive 2D IGA boundary element methods

TL;DR

This paper develops and analyzes multilevel diagonal preconditioners for 2D isogeometric boundary element methods, ensuring mesh-independent condition numbers and optimal computational complexity for solving integral equations.

Contribution

It introduces a new class of multilevel diagonal preconditioners for adaptive 2D IGA boundary element methods, with proven mesh-independent condition numbers.

Findings

Condition number is independent of mesh size and refinement level.

Preconditioners lead to optimal iterative solver complexity.

Numerical results confirm theoretical predictions.

Abstract

We define and analyze (local) multilevel diagonal preconditioners for isogeometric boundary elements on locally refined meshes in two dimensions. Hypersingular and weakly-singular integral equations are considered. We prove that the condition number of the preconditioned systems of linear equations is independent of the mesh-size and the refinement level. Therefore, the computational complexity, when using appropriate iterative solvers, is optimal. Our analysis is carried out for closed and open boundaries and numerical examples confirm our theoretical results.