Solving the equation -u(xx) -epsilon u(yy) = f(x, y, u) by an O(h(4)) finite difference method

The semi-linear equation -u(xx) - epsilon u(yy) = f(x, y, u) with Dirichlet boundary conditions is solved by an O(h(4)) finite difference method, whic h has local truncation error O(h(2)) at the mesh points neighboring the bou ndary and O(h(4)) at most interior mesh points. It is proved that the finit e difference method is O(h(4)) uniformly convergent as h --> 0. The method is considered in the form of a system of algebraic equations with a nine di agonal sparse matrix. The system of algebraic equations is solved by an imp licit iterative method combined with Gauss elimination. A Mathematica modul e is designed for the purpose of testing and using the method. To illustrat e the method, the equation of twisting a springy rod is solved. (C) 2000 Jo hn Wiley & Sons, Inc.