The Numerical Solution of Laplace's Equation

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Abstract
This paper considers in detail numerical methods of solving Laplace's equation in an arbitrary two-dimensional region with given boundary values. The methods involve the solution of approximating difference equations by iterative procedures. Modifications of the standard Liebmann procedure are developed which lead to a great increase in the convenience and rapidity of obtaining such a numerical solution. These modifications involve the use of formulas which simultaneously improve a block of points in place of a single point; methods of operating on the differences of trial functions in place of the functions themselves; and also a method of extrapolating to the final solution of the difference equations. The theory underlying these procedures is considered in detail by a new method which involves the expansion of the error and difference functions in terms of eigenfunctions. This permits definite comparison of rates of convergence of various procedures. The techniques of handling practical problems are considered in detail.

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