A fast, high-order scheme for evaluating volume potentials on complex 2D geometries via area-to-line integral conversion and domain mappings

by   Thomas G. Anderson, et al.

This article presents a new high-order accurate algorithm for finding a particular solution to a linear, constant-coefficient partial differential equation (PDE) by means of a convolution of the volumetric source function with the Green's function in complex geometries. Utilizing volumetric domain decomposition, the integral is computed over a union of regular boxes (lending the scheme compatibility with adaptive box codes) and triangular regions (which may be potentially curved near boundaries). Singular and near-singular quadrature is handled by converting integrals on volumetric regions to line integrals bounding a reference volume cell using cell mappings and elements of the Poincaré lemma, followed by leveraging existing one-dimensional near-singular and singular quadratures appropriate to the singular nature of the kernel. The scheme achieves compatibility with fast multipole methods (FMMs) and thereby optimal asymptotic complexity by coupling global rules for target-independent quadrature of smooth functions to local target-dependent singular quadrature corrections, and it relies on orthogonal polynomial systems on each cell for well-conditioned, high-order and efficient (with respect to number of required volume function evaluations) approximation of arbitrary volumetric sources. Our domain discretization scheme is naturally compatible with standard meshing software such as Gmsh, which are employed to discretize a narrow region surrounding the domain boundaries. We present 8th-order accurate results, demonstrate the success of the method with examples showing up to 12-digit accuracy on complex geometries, and, for static geometries, our numerical examples show well over 99% of evaluation time of the particular solution is spent in the FMM step.


page 3

page 7

page 18

page 24

page 25

page 26


Solution of Stokes flow in complex nonsmooth 2D geometries via a linear-scaling high-order adaptive integral equation scheme

We present a fast, high-order accurate and adaptive boundary integral sc...

A robust solver for elliptic PDEs in 3D complex geometries

We develop a boundary integral equation solver for elliptic partial diff...

General-purpose kernel regularization of boundary integral equations via density interpolation

This paper presents a general high-order kernel regularization technique...

Efficient high-order singular quadrature schemes in magnetic fusion

Several problems in magnetically confined fusion, such as the computatio...

Fast multipole methods for evaluation of layer potentials with locally-corrected quadratures

While fast multipole methods (FMMs) are in widespread use for the rapid ...

Efficient high-order accurate Fresnel diffraction via areal quadrature and the nonuniform FFT

We present a fast algorithm for computing the diffracted field from arbi...

Fast Computation of Electrostatic Potentials for Piecewise Constant Conductivities

We present a novel numerical method for solving the elliptic partial dif...