A New Fast-Multipole Accelerated Poisson Solver in Two Dimensions

Abstract

Abstract. We present an adaptive fast multipole method for solving the Poisson equation in two dimensions. The algorithm is direct, assumes that the source distribution is discretized using an adaptive quad-tree, and allows for Dirichlet, Neumann, periodic, and free-space conditions to be imposed on the boundary of a square. The amount of work per grid point is comparable to that of classical fast solvers, even for highly nonuniform grids.

Stress computations on perforated polygonal domains

“…We refrain from giving more details since our code is not entirely optimal with respect to the latest development in the field. More efficient and consequent implementations for some kernels in 2D do exist [32,15]. We speculate that speedups with a factor of 5 -10 are possible for the problem sizes treated in this paper.…”

“…A more radical way to go is to abandon the idea of solving integral equations completely, and instead compute the solution to the stress problem directly, as a sum of reflected layer potentials. Promising work on this type of fast direct solvers have been reported by Ethridge and Greengard for Laplace's and Poisson's equations on the unit square, and extensions to complex geometry are underway [15].…”

“…These undesirable effects could, in theory, be offset by preconditioners using techniques such as multigrid [24] or domain decomposition [4]. Still, for many realistic problems and geometries it is often difficult to achieve optimal performance [15]. The total work tends to grow faster than linearly with the number of discretization points N. The achievable accuracy decreases as the system size increases.…”

“…Their advantages over methods based on partial differential equations are most pronounced for large and illconditioned problems. Therefore, we argue, unless even more efficient numerical methods emerge [15], integral equation methods will become increasingly appreciated as computers get more powerful and more difficult problems are attacked. New integral equations need to be developed for solid mechanics problems.…”

A Comparison of Splittings and Integral Equation Solvers for a Nonseparable Elliptic Equation

“…Two such fast direct solver classes are fast Fourier transform (FFT) or cyclic reduction accelerated finite difference methods and fast multipole accelerated integral methods. While FFT accelerated finite difference solvers is the classic choice, Ethridge and Greengard [12] and Huang, Cheng, and Leiterman [23] recently showed that these can be outperformed by integral method solvers. The work per grid point for the solver classes is comparable, at least for moderate precision arithmetic.…”