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

Abstract

We propose a $k^{\mathrm{th}}$-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, $(k-1{)}^{\mathrm{th}}$-order, under the $L^2$-norm. We have obtained a $(k-1{)}^{\mathrm{th}}$-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 $4$.