An adaptive fast multipole accelerated Poisson solver for complex geometries
This work provides an efficient black-box solver for Poisson equations in complex geometries, benefiting computational scientists who need accurate solutions without manual tuning.
The authors present a fast, direct, and adaptive Poisson solver for complex 2D geometries using potential theory and fast multipole acceleration, achieving high accuracy without excessive adaptive refinement near boundaries. The solver demonstrates efficiency on multiply connected domains with irregular boundaries.
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.