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

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.