An adaptive finite element method for solving 3D electromagnetic volume integral equation with applications in microwave thermometry

Abstract

An adaptive finite element method (AFEM) for the numerical solution of an electromagnetic volume integral equation (VIE) is presented. To solve the model VIE, the problem is formulated as an optimal control problem for minimization of Tikhonov's regularization functional. A posteriori error estimates in the obtained finite element reconstruction and in the underlying Tikhonov's functional are derived. Based on these estimates, adaptive finite element algorithms are formulated and numerically tested on the problem of microwave hyperthermia in cancer treatment. In this problem, the temperature change of a target in the computational domain results in the change of its dielectric properties. Numerical examples of monitoring this change show robust and qualitative three dimensional reconstructions of the target using the proposed adaptive algorithms.