A multimesh finite element method for the Navier–Stokes equations based on projection methods

The multimesh finite element method is a technique for solving partial differential equations on multiple non-matching meshes by enforcing interface conditions using Nitsche’s method. Since the non-matching meshes can result in arbitrarily cut cells, additional stabilization terms are needed to obtain a stable method. In this contribution we extend the multimesh finite element method to the Navier–Stokes equations based on the incremental pressure-correction scheme. For each step in the pressure-correction scheme, we derive a multimesh finite element formulation with suitable stabilization terms. The proposed scheme is implemented for arbitrary many overlapping two dimensional domains, yielding expected spatial and temporal convergence rates for the Taylor–Green problem, and demonstrates good agreement for the drag and lift coefficients for the Turek–Schäfer benchmark (DFG benchmark 2D-3). Finally, we illustrate the capabilities of the proposed scheme by optimizing the layout of obstacles in a two dimensional channel. •We propose a multimesh finite element method for the Navier–Stokes equations.•The method is based on the incremental pressure correction scheme.•Stabilization terms ensure a robust scheme for arbitrary mesh intersections.•The scheme shows good performance on several numerical examples.