Adaptive BEM with inexact PCG solver yields almost optimal computational costs

TL;DR

This paper introduces an adaptive boundary element method combined with an inexact PCG solver that achieves near-optimal computational costs by using an optimal preconditioner and adaptive strategies for mesh refinement and solver termination.

Contribution

It develops an adaptive BEM algorithm with an inexact PCG solver, providing an additive Schwarz preconditioner with linear complexity and proven almost optimal computational complexity.

Findings

Achieves optimal algebraic convergence rates.

Preconditioner with uniformly bounded condition numbers.

Computational complexity is nearly optimal.

Abstract

We consider the preconditioned conjugate gradient method (PCG) with optimal preconditioner in the frame of the boundary element method (BEM) for elliptic first-kind integral equations. Our adaptive algorithm steers the termination of PCG as well as the local mesh-refinement. Besides convergence with optimal algebraic rates, we also prove almost optimal computational complexity. In particular, we provide an additive Schwarz preconditioner which can be computed in linear complexity and which is optimal in the sense that the condition numbers of the preconditioned systems are uniformly bounded. As model problem serves the 2D or 3D Laplace operator and the associated weakly-singular integral equation with energy space $\widetilde{H}^{-1/2}(\Gamma)$. The main results also hold for the hyper-singular integral equation with energy space $H^{1/2}(\Gamma)$.