Optimal adaptivity for non-symmetric FEM/BEM coupling

TL;DR

This paper establishes a framework for proving optimal convergence rates of adaptive algorithms in non-symmetric FEM/BEM coupling problems, using a novel connection to LU-factorization of infinite matrices.

Contribution

It introduces a new approach to prove general quasi-orthogonality for non-symmetric FEM/BEM coupling, enabling proof of optimal adaptive algorithm convergence.

Findings

Proves general quasi-orthogonality for Johnson-Nedelec coupling.

Shows standard adaptive algorithm converges optimally.

Framework potentially applicable to other non-symmetric problems like Stokes.

Abstract

We develop a framework which allows us to prove the essential general quasi-orthogonality for the non-symmetric Johnson-Nedelec finite element/boundary element coupling. General quasi-orthogonality was first proposed in [Axioms of Adaptivity, 2014] as a necessary ingredient of optimality proofs and is the major difficulty on the way to prove rate optimal convergence of adaptive algorithms for many strongly non-symmetric problems. The proof exploits a new connection between the general quasi-orthogonality and LU-factorization of infinite matrices. We then derive that a standard adaptive algorithm for the Johnson-Nedelec coupling converges with optimal rates. The developed techniques are fairly general and can most likely be applied to other problems like Stokes equation.