Convergence of the compact finite difference method for second-order elliptic equations

In this paper, we give a fourth-order compact finite difference scheme for the general forms of two point boundary value problems and two dimensional elliptic partial differential equations (PDE’s). By decomposing the coefficient matrix into a sum of several matrixes after we discretize the original problem, we can obtain a lower bound for the smallest eigenvalue of the coefficient matrix. Thus we prove the compact finite difference scheme converges with the fourth order of accuracy. To solve the discretized block tri-diagonal matrix equations of the two dimensional elliptic PDE’s, we develop an efficient iterative method: Full Multigrid Method. In our numerical experiments, we compare the compact finite difference method with the finite element method and Crank-Nicolson finite difference method. The results show that the compact finite difference scheme is a highly efficient and accurate method.