Inverse problems and invisibility cloaking for FEM models and resistor networks

In this paper we consider inverse problems for resistor networks and for models obtained via the Finite Element Method (FEM) for the conductivity equation. These correspond to discrete versions of the inverse conductivity problem of Calder\'on. We characterize FEM models corresponding to a given triangulation of the domain that are equivalent to certain resistor networks, and apply the results to study nonuniqueness of the discrete inverse problem. It turns out that the degree of nonuniqueness for the discrete problem is larger than the one for the partial differential equation. We also study invisibility cloaking for FEM models, and show how an arbitrary body can be surrounded with a layer so that the cloaked body has the same boundary measurements as a given background medium.