Elliptic partial differential equations are an important tool in the analysis of the behavior of physical systems. They arise in the study of temperature distribution, stress and deformation due to torsion, incompressible fluid flow, electric and magnetic fields, and in many other fields of science and engineering. Very often their solutions may be obtained with the aid of an electronic digital computer, and so they are of interest to applied mathematicians, computer scientists, and engineers alike. The first phase of the proposed research would formulate, implement, and then evaluate a finite difference method designed to be most effective on elliptic boundary-value problems with singularities of known form. The evaluation process would consider the method's accuracy over a population of problems, as well as determine its computational efficiency over a range of grid sizes. The second phase of the project would implement vector and parallel versions of the original scalar algorithm on a Cray X-MP/24 supercomputer. The performance tradeoffs between increased vector length and increased multitasking would then be investigated.
