Equation in Complex Geometry

We are working on fast, adaptive methods for the solution of the Poisson equation

in complex geometry, subject to linear boundary conditions in interior or exterior domains. In simple geometries (circular or rectangular domains) with regular grids, there are well-known fast direct solvers based on the fast Fourier transform (FFT) that are well-suited to the task. In practical problems, however, involving complex geometries, highly inhomogeneous source distributions (

- A. McKenney, L. Greengard and A. Mayo,
*A Fast Poisson Solver for Complex Geometries*, J. Comput. Phys. 118, 348 (1995). - L. Greengard and J.-Y. Lee,
*A Direct Adaptive Poisson Solver of Arbitrary Order Accuracy*, J. Comput. Phys. 125, 415 (1996). - F. Ethridge and L. Greengard,
*A New Fast-Multipole Accelerated Poisson Solver inTwo Dimensions*, SIAM J. Sci. Comput. 23, 741 (2001). - H. Langston, L. Greengard, and D. Zorin
*A Free-Space Adaptive FMM-Based PDE Solver in Three Dimensions*, Comm. Appl. Math. and Comp. Sci. 6, 79 (2011)