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

TL;DR

This paper introduces a multimesh finite element method for Navier-Stokes equations that handles multiple non-matching meshes, using stabilization and projection techniques to achieve accurate and stable solutions for complex fluid flow problems.

Contribution

It extends the multimesh finite element method to Navier-Stokes equations with stabilization, enabling efficient simulation on complex geometries with non-matching meshes.

Findings

Achieves expected convergence rates on Taylor-Green problem

Demonstrates good agreement with benchmark drag and lift coefficients

Successfully optimizes obstacle layout in a channel flow

Abstract

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 variational formulation. 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 overall scheme yields expected spatial and temporal convergence rates on the Taylor-Green problem, and demonstrates good agreement for the drag and lift coefficients on the Turek-Schafer benchmark (DFG benchmark 2D-3). Finally, we illustrate the capabilities of the proposed scheme by optimizing the layout of obstacles in a channel.