Numerical convergence of discrete extensions in a space-time finite element, fictitious domain method for the Navier-Stokes equations

TL;DR

This paper investigates the numerical convergence of a space-time cut finite element method with a discrete extension approach for solving incompressible Navier-Stokes equations on evolving domains, focusing on stability and accuracy.

Contribution

It introduces and analyzes the convergence properties of a novel discrete extension technique within a higher order space-time CutFEM framework for Navier-Stokes equations.

Findings

Numerical results demonstrate convergence of the scheme.

The extension stabilizes solutions on irregular small cuts.

The method effectively handles evolving domains.

Abstract

A key ingredient of our fictitious domain, higher order space-time cut finite element (CutFEM) approach for solving the incompressible Navier--Stokes equations on evolving domains (cf.\ \cite{Bause2021}) is the extension of the physical solution from the time-dependent flow domain $\Omega_f^t$ to the entire, time-independent computational domain $\Omega$. The extension is defined implicitly and, simultaneously, aims at stabilizing the discrete solution in the case of unavoidable irregular small cuts. Here, the convergence properties of the scheme are studied numerically for variations of the combined extension and stabilization.