Numerical Solution of Symmetric Positive Differential Equations

A finite-difference method for the solution of symmetric positive linear differential equations is developed. The method is applicable to any region with piecewise smooth boundaries. Methods for solution of the finite-difference equations are discussed. The finite-difference solutions are shown to converge at essentially the rate $O({h^{1/2}})$ as $h \to 0,h$, being the maximum distance between adjacent mesh-points. An alternate finite-difference method is given with the advantage that the finite-difference equations can be solved iteratively. However, there are strong limitations on the mesh arrangements which can be used with this method.