Compact approximation of a two-dimensional boundary value problem for elliptic equations of the 2nd order with a discontinuous coefficient

For an elliptic equation of the 2nd order with variable discontinuous coefficients and the right side, a scheme of the 4th order of accuracy is constructed. On the jump line, the conditions of docking (Kirchhoff) are assumed to be fulfilled. The use of Richardson extrapolation, as numerical experiments have shown, increases the order of accuracy to about the 6th. It is shown that relaxation methods, including multigrid methods, are applicable to solving such systems linear algebraic equations (SLAE) corresponding to a compact finite-difference approximation of the problem. In comparison with the classical approximation, the accuracy increases by about 100 times with the same labor intensity. Various variants of the equation and boundary conditions are considered, as well as the problem of determining eigenvalues and functions for a piecewise constant coefficient of the equation.