IMPROVED NUMERICAL-SOLUTIONS OF LAPLACE EQUATION

Three different methods for generating numerical solutions of Laplace's equation are derived using a differential operator formalism for a Taylor series expansion of the electric potential about a point. The expansions are evaluated on three different two-dimensional grids in the x-y plane to yield algorithms good up to, but not including, fourth order. The accuracy and convergence of the algorithms are compared by applying them to a rectangular boundary value problem. Second, it is shown that the convergence criterion for numerical solutions can be interpreted as a charge density; and, therefore, the numerical solutions are, in fact, solutions of Poisson's equation. Further, it is demonstrated that the numerical solutions of Laplace's equation are bounded above and below by the solutions of Poisson's equation corresponding to a maximum uniform charge density derived from the convergence criterion. For this reason, it is recommended that numerical solutions of Laplace's equation should be accompanied by a statement of the maximum uniform charge density.