Error Estimation Methods for the Finite-Difference Solution for Poisson’s Equation

The finite-difference method is universally used for the approximation of differential equations. In this thesis two different approaches are reviewed for the error estimation of the approximation of the Dirichlet problem for elliptic equations, specifically Poisson’s and Laplace’s equations using various finite-difference schemes. The first approach is based on the difference analogue of the maximum principle. Applying Gerschgorin’s majorant method to the analysis , also the order of accuracy of the proposed scheme is obtained. The second approach uses the difference analogue of Green’s function and Green’s third identity. In order to obtain an order of approximation, Gerschgorin’s majorant method is applied in this approach also. Both methods gave similar approximations. Keywords: Finite-difference, maximum principle, Gerschgorin’s majorant method, Green’s function, Green’s third identity.