Summary: This work is related to one of the methods of medical imaging – electrical impedance tomography (EIT). A feature of the considered two-dimensional mathematical formulation for the direct EIT problem is the use of an elliptic type equation with piecewise constant coefficients and a special integro-differential boundary condition at the contact boundary of the electrodes. For an approximate solution of the problem under consideration, a numerical method was developed using unstructured grids, the finite volume method for barycentric cells, and the Gaussian elimination method with the choice of the main element. The validation of the method was performed on a known analytical solution.