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

Abstract

While potential theoretic techniques have received significant interest and found broad success in the solution of linear partial differential equations (PDEs) in mathematical physics, limited adoption is reported in the case of nonlinear and/or inhomogeneous problems (i.e. with distributed volumetric sources) owing to outstanding challenges in producing a particular solution on complex domains while simultaneously respecting the competing ideals of allowing complete geometric flexibility, enabling source adaptivity, and achieving optimal computational complexity. This article presents a new high-order accurate algorithm for finding a particular solution to the 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.