Finite volume schemes for the electrical impedance tomography problem

In electrical impedance tomography, the currents are applied on electrodes placed on the surface of an object. The electrical conductivity is reconstructed in the interior of the object using voltage measurements on its surface. Knowing the conductivity distribution provides information about the internal object's structure which can be useful for medical and industry applications. Instability of the EIT problem causes difficulties challenging a successful reconstruction. Since a static EIT imaging is sensitive to the measurement noise and approximation errors, this paper addresses the problem of reducing the latter. The finite volume method is presented for solving the EIT forward problem, which is a significant part of the inverse problem. Finite volume schemes were obtained for unstructured grids. The schemes were derived for three kinds of finite volumes, which can be considered relative to triangulation of the domain. The approaches are based on vertex-centered and cell-centered discretization, where the numerical solution is associated with mesh vertices or mesh elements, respectively. In the first case, a finite volume approximation was introduced on barycentric volumes and Dirichlet–Voronoi volumes on the assumption of a linear distribution of electric potential. Triangular finite volumes were utilized for approximation based on cell-centered discretization. Both cases suggested a piecewise constant conductivity distribution over grid cells. Numerical comparison for the obtained finite volume schemes was carried out on a test problem that can be solved analytically. The results were compared to a solution by the finite element method.