A stabilized Nitsche overlapping mesh method for the Stokes problem

TL;DR

This paper introduces a stabilized Nitsche overlapping mesh method for the Stokes problem, ensuring stability, optimal convergence, and condition number independence from interface location, validated through 3D numerical examples.

Contribution

The paper extends Nitsche's method with stabilization for overlapping meshes in Stokes problems, providing stability, convergence, and condition number bounds.

Findings

Method is stable and optimally convergent.

Condition number estimate is independent of interface position.

Numerical examples confirm theoretical results in 3D.

Abstract

We develop a Nitsche-based formulation for a general class of stabilized finite element methods for the Stokes problem posed on a pair of overlapping, non-matching meshes. By ex- tending the least-squares stabilization to the overlap region, we prove that the method is stable, consistent, and optimally convergent. To avoid an ill-conditioned linear algebra system, the scheme is augmented by a least-squares term measuring the discontinuity of the solution in the overlap region of the two meshes. As a consequence, we may prove an estimate for the condition number of the resulting stiffness matrix that is independent of the location of the interface. Finally, we present numerical examples in three spatial dimensions illustrating and confirming the theoretical results.