A high-order unfitted finite element method for moving interface problems

TL;DR

This paper introduces a high-order unfitted finite element method for moving interface problems in fluid dynamics, providing thorough error analysis and demonstrating optimal convergence in numerical experiments.

Contribution

The paper develops a new high-order unfitted finite element method for moving interface problems and derives error estimates, including for pressure in the $H^1$-norm, which is novel.

Findings

Optimal convergence for $k=3$ and $4$ in numerical tests.

Error estimates for pressure in $H^1$-norm are established.

Method effectively handles severely deforming interfaces.

Abstract

We propose a $k^{\rm th}$-order unfitted finite element method ($2\le k\le 4$) to solve the moving interface problem of the Oseen equations. Thorough error estimates for the discrete solutions are presented by considering errors from interface-tracking, time integration, and spatial discretization. In literatures on time-dependent Stokes interface problems, error estimates for the discrete pressure are usually sub-optimal, namely, $(k-1)^{\rm th}$-order, under the $L^2$-norm. We have obtained a $(k-1)^{\rm th}$-order error estimate for the discrete pressure under the $H^1$-norm. Numerical experiments for a severely deforming interface show that optimal convergence orders are obtained for $k = 3$ and $4$.