Speaker: Harper Langston, New York University
Abstract: Many problems in scientific computing call for a fast and efficient solution to elliptic partial differential equations. Three particular problems of this sort are the Poisson equation, the Modified Helmholtz equation, and the Stokes equations. The solutions to these equations are often required inside or outside of a complex domain subject to Dirichlet boundary conditions. For simple rectangular, circular or spherical domains, well-established fast methods for solving such problems exist. For more complex domain discretizations, many solvers rely on adaptive meshes, domain decomposition strategies and multigrid acceleration. When geometries become complex, grid-generation methods can become unstructured and computationally expensive. Another approach is to incorporate fast direct solvers, which are efficient, scalable, and do not require a hierarchy of grids. Fast-Multipole-Method based solvers in particular allow for the handling of very non-uniform force distributions, allow for complex geometries and provide a high degree of accuracy. For solving elliptic PDEs in complex geometries, we therefore propose an embedded boundary solver approach which decouples an inhomogeneous interior/exterior Dirichlet elliptic PDE into a problem involving a simple domain with distributed forces and problem absent forces with complex geometry. For the first problem, we introduce an FMM-based volume solver using precomputed interactions and 4th, 6th or 8th order polynomial approximations to forces. For the second problem, a pre-existing FMM-based boundary-integral solver is used. Both FMM implementations are additionally designed to be kernel-independent such that the extension from the Poisson equation to more complex PDEs such as the Stokes equations is straightforward. Examples of the volume solvers and coupled embedded boundary solvers will be discussed.