**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.

Last modified: Fri Feb 29 10:10:00 EST 2008