In view of the Dirichlet boundary value problem of elliptic partial differential equations with variable coefficients, Taylor expansion is firstly applied for an establishment of a five point difference scheme, thus proving the existence and uniqueness of the difference scheme solution. Secondly, a prior estimation formula for the difference scheme solution can be obtained by applying the extremum principle, with its convergence and stability further proved. Thirdly, Richardson extrapolation method is applied to establish an extrapolation format with fourth-order accuracy. Finally, the Gauss-Seidel iterative method is applied to solve the numerical example, with the numerical results showing that the Richardson extrapolation method greatly improves the accuracy of the numerical solution.