A FAST DIRECT ALGORITHM FOR THE SOLUTION OF THE LAPLACE EQUATION ON REGIONS WITH FRACTAL BOUNDARIES

An algorithm is presented for the rapid direct solution of the Laplace equation on regions with fractal boundaries. In a typical application, the numerical simulation has to be on a very large scale involving at least tens of thousands of equations with as many unknowns, in order to obtain any meaningful results. Attempts to use conventional techniques have encountered insurmountable difficulties, due to excessive CPU time requirements of the computations involved. Indeed, conventional direct algorithms for the solution of linear systems require order O(N3) operations for the solution of an N x N-problem, while classical iterative methods require order O(N2) operations, with the constant strongly dependent on the problem in question. In either case, the computational expense is prohibitive for large-scale problems. The direct algorithm of the present paper requires O(N) operations with a constant dependent only on the geometry of the boundary, making it considerably more practical for large-scale problems encountered in the computation of harmonic measure of fractals, complex iteration theory, potential theory, and growth phenomena such as crystallization, electrodeposition, viscous fingering, and diffusion-limited aggregation. (C) 1994 Academic Press, Inc.