A fully adaptive, high-order, fast Poisson solver for complex two-dimensional geometries

TL;DR

This paper introduces a fully adaptive, high-order, fast Poisson solver for complex 2D geometries that combines boundary-conforming spectral methods with bulk box codes, achieving optimal complexity and high accuracy.

Contribution

It presents a novel hybrid framework that avoids boundary function extension, enabling efficient, high-accuracy solutions for elliptic problems in complex geometries.

Findings

Achieves $O(N)$ computational complexity with adaptive discretization.

Demonstrates 10-digit accuracy in 2D problems.

Efficiently handles volumetric and boundary data across multiple scales.

Abstract

We present a new framework for the fast solution of inhomogeneous elliptic boundary value problems in domains with smooth boundaries. High-order solvers based on adaptive box codes or the fast Fourier transform can efficiently treat the volumetric inhomogeneity, but require care to be taken near the boundary to ensure that the volume data is globally smooth. We avoid function extension or cut-cell quadratures near the boundary by dividing the domain into two regions: a bulk region away from the boundary that is efficiently treated with a truncated free-space box code, and a variable-width boundary-conforming strip region that is treated with a spectral collocation method and accompanying fast direct solver. Particular solutions in each region are then combined with Laplace layer potentials to yield the global solution. The resulting solver has an optimal computational complexity of $O(N)$ for an adaptive discretization with $N$ degrees of freedom. With an efficient two-dimensional (2D) implementation we demonstrate adaptive resolution of volumetric data, boundary data, and geometric features across a wide range of length scales, to typically 10-digit accuracy. The cost of all boundary corrections remains small relative to that of the bulk box code. The extension to 3D is expected to be straightforward in many cases because the strip ``thickens'' an existing boundary quadrature.