An FMM Accelerated Poisson Solver for Complicated Geometries in the Plane using Function Extension

TL;DR

This paper introduces an efficient, high-order accurate Poisson solver for complex 2D geometries that uses function extension and FMM acceleration, improving speed and robustness over traditional methods.

Contribution

The novel aspect is the scheme for creating source extensions on adaptive quad-trees, enabling universal stencils and precomputed interpolation matrices for complex geometries.

Findings

Demonstrates high-order convergence in complex geometries

Shows significant speed improvements with FMM acceleration

Robustly handles piecewise smooth boundaries

Abstract

We describe a new, adaptive solver for the two-dimensional Poisson equation in complicated geometries. Using classical potential theory, we represent the solution as the sum of a volume potential and a double layer potential. Rather than evaluating the volume potential over the given domain, we first extend the source data to a geometrically simpler region with high order accuracy. This allows us to accelerate the evaluation of the volume potential using an efficient, geometry-unaware fast multipole-based algorithm. To impose the desired boundary condition, it remains only to solve the Laplace equation with suitably modified boundary data. This is accomplished with existing fast and accurate boundary integral methods. The novelty of our solver is the scheme used for creating the source extension, assuming it is provided on an adaptive quad-tree. For leaf boxes intersected by the boundary, we construct a universal "stencil" and require that the data be provided at the subset of those points that lie within the domain interior. This universality permits us to precompute and store an interpolation matrix which is used to extrapolate the source data to an extended set of leaf nodes with full tensor-product grids on each. We demonstrate the method's speed, robustness and high-order convergence with several examples, including domains with piecewise smooth boundaries.