The Finite Difference Methods and Their Extrapolation for Solving Biharmonic Equations

In this paper, we research the standard 13-point difference scheme for solving the biharmonic equation. Existence and convergence of finite difference methods(FDM) solutions are obtained by estimating the lower bounds of the minimum eigenvalues of the discrete matrix and making use of the Taylor series expansions, respectively. The accuracy are proved to be O(h(2)). Moreover, the extrapolation techniques are used to improve the high accuracy of the solutions. For the given examples, numerical results show that the errors O(h(4)) is obtained from the first level extrapolation. The very accurate solutions with the errors O(10(-8)) are obtained by the third level extrapolation techniques under the homogeneous essential boundary conditions.