Afternote to Coupling at a distance: convergence analysis and a priori error estimates

TL;DR

This paper proves convergence and provides error estimates for a novel HDG-BEM coupling method that uses a non-coincident boundary for the computational domain, addressing previous theoretical gaps.

Contribution

It offers the first rigorous convergence proof and a priori error estimates for the HDG-BEM coupling with a non-coincident boundary.

Findings

Proved convergence of the relaxed HDG-BEM coupling algorithm.

Derived a priori error estimates for the numerical solution.

Validated theoretical results with numerical experiments.

Abstract

In their article "Coupling at a distance HDG and BEM", Cockburn, Sayas and Solano proposed an iterative coupling of the hybridizable discontinuous Galerkin method (HDG) and the boundary element method (BEM) to solve an exterior Dirichlet problem. The novelty of the numerical scheme consisted of using a computational domain for the HDG discretization whose boundary did not coincide with the coupling interface. In their article, the authors provided extensive numerical evidence for convergence, but the proof of convergence and the error analysis remained elusive at that time. In this article we fill the gap by proving the convergence of a relaxation of the algorithm and providing a priori error estimates for the numerical solution.