New developments on the coupling of mixed-FEM and BEM for the three-dimensional Stokes problem

TL;DR

This paper advances the coupling of mixed finite element and boundary element methods for the 3D exterior Stokes problem, establishing well-posedness, stability, and extending classical procedures without restrictive smoothness assumptions.

Contribution

It extends Johnson & Nédélec and Costabel & Han coupling procedures to 3D Stokes problems, providing new stability and coerciveness proofs under minimal regularity conditions.

Findings

Proved well-posedness of coupled formulations.

Extended classical coupling procedures to 3D Stokes.

Provided explicit conditions for stable finite and boundary element discretizations.

Abstract

In this paper we consider the three dimensional exterior Stokes problem and study the solvability of the corresponding continuous and discrete formulations that arise from the coupling of a dual-mixed variational formulation with the boundary integral equation method. More precisely, after employing the incompressibility condition to eliminate the pressure, we consider the resulting velocity-stress-vorticity approach with different kind of boundary conditions on an annular bounded domain, and couple the underlying equations with either one or two boundary integral equations arising from the application of the usual and normal traces to the Green representation formula in the exterior unbounded region. As a result, we obtain saddle point operator equations, which are then analyzed by the well-known Babuska-Brezzi theory. We prove the well-posedness of the continuous formulations, identifying previously the space of solutions of the associated homogeneous problem, and specify explicit hypotheses to be satisfied by the finite element and boundary element subspaces in order to guarantee the stability of the respective Galerkin schemes. In particular, following a similar analysis given recently for the Laplacian, we are able to extend the classical Johnson & N\'ed\'elec procedure to the present case, without assuming any restrictive smoothness requirement on the coupling boundary, but only Lipschitz-continuity. In addition, and differently from known approaches for the elasticity problem, we are also able to extend the Costabel & Han coupling procedure to the 3D Stokes problem by providing a direct proof of the required coerciveness property, that is without argueing by contradiction, and by using the natural norm of each space instead of mesh-dependent norms. Finally, we briefly describe concrete examples of discrete spaces satisfying the aforementioned hypotheses.