A stabilized nonconforming finite element method for the elliptic Cauchy problem

TL;DR

This paper introduces a stabilized nonconforming finite element method for solving the ill-posed elliptic Cauchy problem, providing error estimates and analyzing data perturbation effects, extending existing frameworks to nonconforming spaces.

Contribution

It extends the framework for finite element methods to include nonconforming spaces, reducing stabilization needs for ill-posed elliptic Cauchy problems.

Findings

Error estimates established using $L^2$-norm dependence.

Reduced stabilization required due to nonconforming spaces.

Analysis of data perturbation effects on solution stability.

Abstract

In this paper we propose a nonconforming finite element method for the solution of the ill-posed elliptic Cauchy problem. We prove error estimates using continuous dependence estimates in the $L^2$-norm. The effect of perturbations in data on the estimates is investigated. The recently derived framework from \cite{Bu13,Bu14} is extended to include the case of nonconforming approximation spaces and we show that the use of such spaces allows us to reduce the amount of stabilization necessary for convergence, even in the case of ill-posed problems.