An unfitted finite element method for elliptic interface problem with low regularity estimates

TL;DR

This paper introduces and analyzes an unfitted finite element method for elliptic interface problems that effectively handles solutions with low regularity, ensuring stability and accurate error estimates through local polynomial extensions and ghost penalty techniques.

Contribution

The paper develops a novel unfitted finite element method with stability and error analysis for low-regularity solutions in elliptic interface problems, using local polynomial extensions and ghost penalties.

Findings

Method achieves robust stability near interfaces.

Error estimates are valid for solutions with low regularity.

Numerical tests confirm the method's accuracy in 2D and 3D.

Abstract

In this paper, we present and analyze an unfitted finite element method for the elliptic interface problem. We consider the case that the interface is $C^2$-smooth or polygonal, and the exact solution $u \in H^{1+s}(\Omega_0 \cup \Omega_1)$ for any $s > 0$. The stability near the interface is guaranteed by a local polynomial extension technique combined with ghost penalty bilinear forms, from which the robust condition number estimates and the error estimates are derived. Furthermore, the jump penalty term for weakly enforcing the jump condition in our method is also defined based on the local polynomial extension, which enables us to establish the error estimation particularly for solutions with low regularity. We perform a series of numerical tests in two and three dimensions to illustrate the accuracy of the proposed method.