Mortar coupling of $hp$-discontinuous Galerkin and boundary element methods for the Helmholtz equation

TL;DR

This paper introduces a novel mortar coupling method combining discontinuous Galerkin and boundary element techniques for solving the Helmholtz equation with variable coefficients in 3D, ensuring quasi-optimality under certain conditions.

Contribution

It develops a new mortar coupling approach with a block structure and proves quasi-optimality for the $h$- and $p$-versions of the scheme, including a novel reconstruction operator.

Findings

The method achieves quasi-optimality under specific approximability conditions.

The coupling results in a block-structured system with nonsingular subblocks.

A new discontinuous-to-continuous reconstruction operator is introduced.

Abstract

We design and analyze a coupling of a discontinuous Galerkin finite element method with a boundary element method to solve the Helmholtz equation with variable coefficients in three dimensions. The coupling is realized with a mortar variable that is related to an impedance trace on a smooth interface. The method obtained has a block structure with nonsingular subblocks. We prove quasi-optimality of the $h$- and $p$-versions of the scheme, under a threshold condition on the approximability properties of the discrete spaces. Amongst others, an essential tool in the analysis is a novel discontinuous-to-continuous reconstruction operator on tetrahedral meshes with curved faces.