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A fast, high-order scheme for evaluating volume potentials on complex 2D geometries via area-to-line integral conversion and domain mappings
Numerical Analysis · Submitted 11 Mar 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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 Poincar\'e 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 1.00 | 0.90 | 0.27 | asserted 1.00 |
| s1 | 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). | method 0.99 | 0.80 | 0.04 | asserted 0.82 |
| s2 | 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 Poincar\'e lemma, followed by leveraging existing one-dimensional near-singular and singular quadratures appropriate to the singular nature of the kernel. | method 0.99 | 0.84 | 0.07 | asserted 1.00 |
| s3 | 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. | method 0.83 | 0.83 | 0.20 | asserted 1.00 |
| s4 | 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. | method 0.93 | 0.54 | 0.03 | asserted 0.99 |
| s5 | 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. | result 0.99 | 0.97 | 0.41 | asserted 1.00 |
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The paper
Original abstract
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 Poincar\'e 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.
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How the literature uses it
4 arXiv papers cite this in 21 sentences
Papers citing it · 16 sentences without a confident label
Uses its method or tool 1 paper
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When $Δ$ is a curved element, the blending function method [ref] provides an elegant solution to the problem, as is mentioned in [ref] .
Accelerating potential evaluation over unstructured meshes in two dimensions · Integration over a mesh element · 0.91
Compares with it 1 paper
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The scheme proposed in the present work differs from that introduced in reference [ref] (see also [ref] ), where the volume potential is evaluated over irregular domain regions using a direct numerical quadrature of the singular and near-singular integrals.
Fast, high-order numerical evaluation of volume potentials via polynomial density interpolation · Introduction · 1.00
Cites it as background 2 papers
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We refer the readers to [ref] for a thorough literature review of methods belonging to these two categories.
Accelerating potential evaluation over unstructured meshes in two dimensions · Introduction · 0.99
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Restricting attention to work in this vein where the resulting homogeneous problem is solved with integral equation techniques, there have been a variety of techniques proposed for producing a valid particular solution, generally on uniform grids using finite difference, finite element, or Fourier methods—see [ref] for more details.
Fast, high-order numerical evaluation of volume potentials via polynomial density interpolation · Introduction · 0.91
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