An adaptive finite element method in reconstruction of coefficients in Maxwell's equations from limited observations

TL;DR

This paper introduces an adaptive finite element method for reconstructing dielectric permittivity and magnetic permeability in Maxwell's equations from limited boundary data, improving accuracy through a posteriori error estimates and adaptive mesh refinement.

Contribution

The paper develops a novel adaptive finite element approach with a posteriori error estimates for coefficient reconstruction in Maxwell's equations, enhancing solution accuracy with local mesh refinement.

Findings

A posteriori error estimates effectively guide adaptive refinement.

Adaptive meshes significantly improve reconstruction accuracy.

Numerical experiments confirm the efficiency of the proposed method.

Abstract

We propose an adaptive finite element method for the solution of a coefficient inverse problem of simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions in the Maxwell's system using limited boundary observations of the electric field in 3D. We derive a posteriori error estimates in the Tikhonov functional to be minimized and in the regularized solution of this functional, as well as formulate corresponding adaptive algorithm. Our numerical experiments justify the efficiency of our a posteriori estimates and show significant improvement of the reconstructions obtained on locally adaptively refined meshes.