This research is concerned with the development of efficient, versatile, and accurate algorithms to solve partial differential equations using the boundary integral method in three space dimensions. These schemes will be based on recently developed techniques for efficiently approximating the boundaries of three-dimensional bodies using linear approximants arising from complementary pivoting methods. The new techniques are ideally suited for use in the numerical approximation of integrals over surfaces and manifolds inherent in the boundary integral method. Both interior and exterior problems for the Laplace, heat, wave, and biharmonic equations will be treated. Time dependent and time independent problems will be studied.
