Nonconforming $hp$-FE/BE coupling on unstructured meshes based on Nitsche's method
For computational scientists solving PDEs with coupled finite and boundary elements, this work offers a stable and flexible coupling technique that handles non-matching meshes and varying polynomial degrees, though it is an incremental improvement over existing mortar methods.
This paper develops and analyzes an hp-FE/BE coupling method on non-matching meshes using Nitsche's method, which avoids the Babuška-Brezzi condition and provides a positive definite formulation. The method achieves stable and accurate results for both quasi-uniform and geometrically refined hp discretizations, with explicit error bounds and numerical validation.
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 Babuška-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.