Numerical Approximation and Analysis of the Inverse Robin Problem Using the Kohn-Vogelius Method

TL;DR

This paper develops a numerical method using the Kohn-Vogelius approach and finite element discretization to solve the inverse Robin problem, providing error estimates and numerical validation for recovering Robin coefficients.

Contribution

It introduces a regularized variational framework with error analysis for the inverse Robin problem using finite element methods.

Findings

Error estimates relate mesh size, time step, and noise level.

Numerical experiments validate the theoretical error bounds.

The method effectively recovers Robin coefficients in practical scenarios.

Abstract

In this work, we numerically investigate the inverse Robin problem of recovering a piecewise constant Robin coefficient in an elliptic or parabolic problem from the Cauchy data on a part of the boundary, a problem that commonly arises in applications such as non-destructive corrosion detection. We employ a Kohn-Vogelius type variational functional for the regularized reconstruction, and discretize the resulting optimization problem using the Galerkin finite element method on a graded mesh. We establish rigorous error estimates on the recovered Robin coefficient in terms of the mesh size, temporal step size and noise level. This is achieved by combining the approximation error of the direct problem, a priori estimates on the functional, and suitable conditional stability estimates of the continuous inverse problem. We present several numerical experiments to illustrate the approach and to complement the theoretical findings.