  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.
