Nonconforming $hp$-FE/BE coupling on unstructured meshes based on Nitsche's method

TL;DR

This paper develops a stable $hp$-finite element/boundary element coupling method on unstructured meshes using Nitsche's approach, with explicit error bounds and numerical validation.

Contribution

It introduces a Nitsche-based $hp$ coupling technique that avoids Babuška-Brezzi conditions, providing stability, explicit thresholds, and extendable analysis for complex domains.

Findings

The method is stable if the stabilization exceeds a certain explicit threshold.

Explicit a priori error estimates are derived for the coupling.

Numerical examples confirm convergence for various discretizations and solutions.

Abstract

We construct and analyse a $hp$-FE/BE coupling on non-matching meshes, based on Nitsche's method. Both the mesh size and the polynomial degree are changed to improve accuracy. Nitsche's method leads to a positive definite formulation, thus, unlike the mortar method, it does not require the Babu\v{s}ka-Brezzi condition for stability. The method is stable provided the stabilization function is larger than a certain threshold. We obtain an explicit bound for the threshold and derive a priori error estimates. Our analysis can be easily extended to the pure FE or the pure BE decomposition as well as to the case of more than two subdomains. The problem in a bounded domain is considered in detail, but the case of an unbounded BE subdomain and a bounded FE subdomain follows with similar arguments. We develop convergence analysis and provide numerical examples for quasi-uniform as well as geometrically refined $hp$ discretisations in both subdomains with analytic and singular solutions.