A MultiMesh Finite Element Method for the Stokes Problem

TL;DR

This paper introduces a multimesh finite element method for solving the Stokes problem, handling overlapping meshes with stabilization techniques to ensure stability and optimal convergence in flow simulations.

Contribution

It develops a novel multimesh finite element formulation for the Stokes problem, incorporating stabilization for cut elements and verifying optimal convergence.

Findings

Stable and coercive discretization achieved.

Optimal convergence demonstrated.

Effective stabilization for cut elements.

Abstract

The multimesh finite element method enables the solution of partial differential equations on a computational mesh composed by multiple arbitrarily overlapping meshes. The discretization is based on a continuous--discontinuous function space with interface conditions enforced by means of Nitsche's method. In this contribution, we consider the Stokes problem as a first step towards flow applications. The multimesh formulation leads to so called cut elements in the underlying meshes close to overlaps. These demand stabilization to ensure coercivity and stability of the stiffness matrix. We employ a consistent least-squares term on the overlap to ensure that the inf-sup condition holds. We here present the method for the Stokes problem, discuss the implementation, and verify that we have optimal convergence.