On convergence of grid boundary value problems for functions defined in cells and faces of a grid
For stationary diffusion type equations convergence of grid nonhomogeneous boundary value problems of the variant of the Mimetic Finite Difference method (MFD method) is studied, in which grid scalars are defined in grid cells and the grid vectors are given by their normal local coordinates on the plane faces of the grid cells. Grid equations and boundary conditions are formulated in the operator form with the use of consistence grid analogs of first-order invariant differential and boundary operators. Convergence is investigated on the basis of the theory of operator difference schemes: a priori estimates of the error rate of the solution in terms of the approximation error are obtained guaranteeing first-order convergence under inhomogeneous boundary conditions of the first, second, and third kind in a domain with a curvilinear boundary. The grid analogues of the imbedding inequalities and approximation relations obtained earlier are used.