$L^2$-error analysis of an isoparametric unfitted finite element method for elliptic interface problems

TL;DR

This paper extends previous analysis of a high-order unfitted finite element method for elliptic interface problems by deriving optimal $L^2$-error bounds, complementing existing $H^1$-norm error estimates.

Contribution

It provides the first $L^2$-error analysis for the method, demonstrating optimal convergence rates in the $L^2$-norm.

Findings

Achieves optimal $L^2$-error bounds for the method.

Extends previous $H^1$-error analysis to $L^2$-norm.

Confirms high-order accuracy of the unfitted finite element approach.

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. Recently a new unfitted finite element method was introduced which achieves a high order approximation of the geometry for domains which are implicitly described by smooth level set functions. This method is based on a parametric mapping which transforms a piecewise planar interface (or surface) reconstruction to a high order approximation. In the paper [C. Lehrenfeld, A. Reusken, \emph{Analysis of a High Order Finite Element Method for Elliptic Interface Problems}, arXiv 1602.02970, Accepted for publication in IMA J. Numer. Anal.] an a priori error analysis of the method applied to an interface problem is presented. The analysis reveals optimal order discretization error bounds in the $H^1$-norm. In this paper we extend this analysis and derive optimal $L^2$-error bounds.