A posteriori error estimation in a finite element method for reconstruction of dielectric permittivity

TL;DR

This paper develops a posteriori error estimates for finite element methods used in reconstructing dielectric permittivity from boundary electric field measurements, aiding in more accurate inverse problem solutions.

Contribution

It introduces a novel a posteriori error estimation technique for finite element approximations in a coefficient inverse problem involving Maxwell's equations.

Findings

Derived an error estimate for the difference between computed and true permittivity.

Provided a bound for the error based on computed electric field and Lagrangian multipliers.

Enhanced the reliability of finite element reconstructions in electromagnetic inverse problems.

Abstract

We present a posteriori error estimates for finite element approximations in a minimization approach to a coefficient inverse problem. The problem is that of reconstructing the dielectric permittivity $\varepsilon = \varepsilon(\mathbf{x})$, $\mathbf{x}\in\Omega\subset\mathbb{R}^3$, from boundary measurements of the electric field. The electric field is related to the permittivity via Maxwell's equations. The reconstruction procedure is based on minimization of a Tikhonov functional where the permittivity, the electric field and a Lagrangian multiplier function are approximated by peicewise polynomials. Our main result is an estimate for the difference between the computed coefficient $\varepsilon_h$ and the true minimizer $\varepsilon$, in terms of the computed functions.