On the method of reflections

TL;DR

This paper reviews and analyzes the method of reflections for solving boundary value problems in multiply connected domains, providing convergence proofs, modifications for unconditional convergence, and numerical tests.

Contribution

It offers a unified abstract formulation, convergence analysis, and improvements to the method of reflections for boundary value problems.

Findings

Proved unconditional convergence of the sequential method.

Modified the parallel method for unconditional convergence.

Presented numerical tests validating the methods.

Abstract

This paper aims at reviewing and analysing the method of reflections. The latter is an iterative procedure designed to linear boundary value problems set in multiply connected domains. Being based on a decomposition of the domain boundary, this method is particularly well-suited to numerical solvers relying on integral representation formulas. For the parallel and sequential forms of the method appearing in the literature, we propose a general abstract formulation in a given Hilbert setting and interpret the procedure in terms of subspace corrections. We then prove the unconditional convergence of the sequential form and propose a modification of the parallel one that makes it unconditionally converging. An alternative proof of convergence is provided in a case which does not fit into the previous framework. We finally present some numerical tests.