Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations

TL;DR

This paper presents a non-iterative quasi-reversibility method for numerically solving a data completion problem in Maxwell's equations, effectively recovering missing boundary data with proven convergence and demonstrated efficiency.

Contribution

It introduces and analyzes a new quasi-reversibility approach with mixed variational formulations for Maxwell's equations data completion problem.

Findings

Proved well-posedness and convergence of the method

Validated effectiveness through numerical simulations in 2D and 3D

Demonstrated robustness with noisy data

Abstract

This paper concerns the numerical resolution of a data completion problem for the time-harmonic Maxwell equations in the electric field. The aim is to recover the missing data on the inaccessible part of the boundary of a bounded domain from measured data on the accessible part. The non-iterative quasi-reversibility method is studied and different mixed variational formulations are proposed. Well-posedness, convergence and regularity results are proved. Discretization is performed by means of edge finite elements. Various two- and three-dimensional numerical simulations attest the efficiency of the method, in particular for noisy data.