An adaptive fast multipole accelerated Poisson solver for complex geometries

TL;DR

This paper introduces an adaptive, fast multipole accelerated Poisson solver for complex 2D geometries that combines potential theory, boundary integral methods, and efficient volume integral computation for high accuracy and efficiency.

Contribution

It develops a novel, adaptive direct Poisson solver that efficiently handles complex geometries using potential theory and fast multipole methods, with a boundary integral-based extension for improved convergence.

Findings

Achieves high accuracy without excessive adaptive refinement.

Demonstrates efficiency on multiply connected domains with irregular boundaries.

Provides a black-box solver suitable for complex geometries.

Abstract

We present a fast, direct and adaptive Poisson solver for complex two-dimensional geometries based on potential theory and fast multipole acceleration. More precisely, the solver relies on the standard decomposition of the solution as the sum of a volume integral to account for the source distribution and a layer potential to enforce the desired boundary condition. The volume integral is computed by applying the FMM on a square box that encloses the domain of interest. For the sake of efficiency and convergence acceleration, we first extend the source distribution (the right-hand side in the Poisson equation) to the enclosing box as a $C^0$ function using a fast, boundary integral-based method. We demonstrate on multiply connected domains with irregular boundaries that this continuous extension leads to high accuracy without excessive adaptive refinement near the boundary and, as a result, to an extremely efficient "black box" fast solver.