On the Numerical Solution of Three-Dimensional Boundary Value Problems by Separation of Variables
E. K. Blum, G. J. Reid · SIAM Journal on Numerical Analysis · 1988
The classical method of separation of variables is subject to certain deficiencies as a numerical method for the solution of the Dirichlet problem for the Laplace equation. The method is only applicable directly to a region that is a curvilinear box of a separable coordinate system and the series expansion for the solution does not converge uniformly in the box except for special “zero-edge” boundary value functions. The series also converges rather slowly in the mean. In this paper, we present a modification of the classical method that yields a more rapidly convergent series expansion for boundary value functions which need not be zero on the edges of the box and we apply the method to three-dimensional boxes. We also give some ideas for extension to more general three-dimensional regions and for parallelization. The modification proceeds by first representing the given boundary value function as a sum of a “zero-edge” function and at most twelve “separable” boundary value functions. Each of these functions defines an auxiliary Dirichlet problem and the solution of the given problem is the sum of the solutions of the auxiliary problems. The “zero-edge” problem is solved by the classical method. However, for each of the “separable” boundary value problems, a uniformly convergent series solution is found in terms of eigenfunctions for a corresponding “Robin” boundary value problem. A three-dimensional example illustrates the improvement in accuracy achieved by the modified separation of variables method.