Analysis of a high order unfitted finite element method for elliptic interface problems

TL;DR

This paper introduces a new high order unfitted finite element method for elliptic interface problems that accurately approximates geometry using a parametric mapping, with proven optimal error bounds and confirmed by numerical tests.

Contribution

The paper presents a novel high order unfitted finite element method with a parametric mapping for geometry approximation, along with an a priori error analysis and numerical validation.

Findings

Achieves optimal order error bounds for geometry and finite element approximation.

Provides a simple implementation for interface reconstruction and parametric mapping.

Numerical experiments confirm theoretical error estimates.

Abstract

In the context of unfitted finite element discretizations the realization of high order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. We consider a new unfitted finite element method which achieves a high order approximation of the geometry for domains which are implicitly described by smooth level set functions. The method is based on a parametric mapping which transforms a piecewise planar interface (or surface) reconstruction to a high order approximation. Both components, the piecewise planar interface reconstruction and the parametric mapping are easy to implement. In this paper we present an a priori error analysis of the method applied to an interface problem. The analysis reveals optimal order error bounds for the geometry approximation and for the finite element approximation, for arbitrary high order discretization. The theoretical results are confirmed in numerical experiments.