Adaptive Methods for the Poisson
Equation in Complex Geometry
We are working on fast, adaptive methods for the
solution of the Poisson equation
Δ U = f
in complex geometry,
subject to linear boundary conditions in interior or exterior
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 (f
), or both,
there has been a lot of effort directed at developing
Most currently available solvers rely on iterative
techniques using multigrid, domain decomposition,
or some other preconditioning strategy.
Although there has been significant progress in this
direction, the available solvers
compare unfavorably with fast direct solvers in terms of work
We are developing methods which are direct,
high-order accurate, insensitive to the degree of adaptive mesh
refinement, and accelerated by the FMM.
Our goal is to produced solvers that are competitive (or nearly competitive)
with standard fast solvers in terms of work per gridpoint
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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)