In this article, we consider the two-dimensional mathematical statement for the direct problem of electroimpedance tomography (EIT). The peculiarity of this formulation is the use of an elliptic equation with piecewise constant coefficients and a special integro-differential boundary condition at the contact boundary of the electrodes: partial derivative/partial derivative x (sigma(x, y)partial derivative u/partial derivative x) + partial derivative/partial derivative y (sigma(x, y)partial derivative u/partial derivative y) = 0, (x, y) is an element of D; sigma partial derivative u/partial derivative n = 0, (x, y) is not an element of boolean OR(L)(l=1) E-l; (7) sigma partial derivative u/partial derivative n = 1/z(l)E(l)(integral(El) uds + z(l)I(l)) - u/z(l,) (x, y) is an element of E-l, l = 1, 2, ..., L. In [10], it is proved that for the considered mathematical setting, the solution exists and is unique if the following conditions are met: Sigma(L)(l=1) U-l = 0, Sigma(L)(l=1) I-l = 0. To solve problem (7), due to the complexity of the geometric shape of the area of the object under study, we use the finite volume method (FVM) [6] on unstructured grids. The solution of the finite-difference problem yields an approximate solution of the differential problem in the nodes of an unstructured grid. In the numerical solution of differential equations of the FVM, an important element is the computational grid. A well-constructed computational grid significantly simplifies the solution of differential equations and makes it closer to the exact solution. The accuracy and efficiency of the numerical study of the problem depend on properties of the computational grid used. In this paper, the grid was built using the Gambit program, which is a part of the ANSYS Fluent software package. The numerical calculations carried out for the test problem - a round disk with two electrodes inhomogeneous in electrical conductivity - are compared with an approximate analytical solution in the form of Fourier series [14]. The numerical solution of the difference scheme was performed by the Gauss method with partial selection of the main element. To obtain estimates of the comparison of numerical and analytical solutions, the value of the root mean square error (RMSE) was used. Table 1 shows the calculated RMSE values for various values of the electrical conductivity ratio sigma 2/sigma 1, various grids: with nodes thickening only on the electrodes, to the entire boundary of the region, with special placement of nodes on the border of the circular insert, and without it. As a result, a good agreement has been obtained for various ratios of electrical conductivity coefficients. In addition, with a fourfold increase in the number of triangles in the grid, a decrease in RMSE is observed, which indicates the convergence of the resulting difference scheme.
