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.