Minimizing computer cost for the solution of certain scientific problems

Many scientific problems require solution of the Laplace, Poission or Fourier equation. These equations occur in heat flow, fluid flow, diffusion and structural problems. It is well known that these types of problems lead to large sets of simultaneous equations that frequently require a number of iterations consuming a lot of computer dollars before a solution is obtained. Frequently one must solve a few hundred to a few thousand simultaneous equations. Numerical methods likely to be used for solution include: (1) Liebmann, an explicit method, (2) alternating direction implicit procedure and (3) banded matrix inversion technique.