Discrete potential theory for two-dimensional Laplace and Poisson difference equations

Charles Saltzer
Jan 1958

A method is given for solving problems associated with Laplace and Poisson equations which, in general, requires considerably fewer equations than the usual method and which gives a convergent solution by the method of successive approximations. For infinite regions, by this method, the exact solution for the Dirichlet and Neumann problems can be found by solving a system of equations with as many variables as there are boundary points of the region. In addition, at each stage of the iteration a best possible estimate of the error of the approximate solution with respect to the exact solution of the difference equation for the Dirichlet problem is furnished, and for the Neumann problem, a bound for the error of the normal difference of the approximate solution is given.

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