Most scientific problems are formulated as partial differential equations which must be solved numerically. Sometimes the solution can be expressed as a boundary integral involving an unknown density function which can also be computed numerically by solving an integral equation. Thanks to recently developed algorithms such as iterative linear system solvers and the fast multipole method it is sometimes more efficient to solve the integral equation. Recent work has dealt with the numerical solution of integral equations and stable and efficient methods for evaluating the solution at various points once the integral representation in known.

A typical problem is to compute the magnetic field induced by a wire wrapped around a U-shaped recording head and carrying a current. The procedure for solving the problem is to discretize the surface of the region using triangles, solve a boundary integral equation to determine an integral expression for the solution, use this expression to evaluate the discrete Laplacian at the mesh points of an embedding cube, and finally apply a fast Poisson solver to determine the value of the field at all mesh points, both inside and outside the recording head.

It is not unusual to require a 1000 by 1000 by 1000 regular grid; i.e., one billion mesh points. These cannot be stored on a single processor, so a multiprocessor must be used. The 3-D fast Poisson solver has been parallelized and run on the IBM SP2. This enables the solution of a problem that is p times as large on p processors, in almost the same amount of time that the original problem requires on one processor.

For more details, see the references:

Fast Parallel Iterative Solution of Poisson's and the Biharmonic Equations on Irregular Regions
SIAM J. Sci. Stat. Comput., 13 (1992), pp. 101-117 ,
by A. Mayo and A. Greenbaum.

Laplace's Equation and the Dirichlet-Neumann Map in Multiply Connected Domains
J. Comput. Phys., 105 (1993), pp. 267-278 ,
by A. Greenbaum, L. Greengard, and G. McFadden.