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Rapid evaluation of Newtonian potentials on planar domains

Zewen Shen, Kirill Serkh

TL;DR

The paper develops a fast, high-order method for evaluating Newtonian potentials on planar domains by transforming volume integrals into boundary layer potentials via Green's third identity and evaluating them with the Helsing-Ojala approach. A key advance is the use of high-order (up to $N\approx 20$) bivariate polynomial interpolation in the monomial basis, together with an efficient per-element anti-Laplacian mapping, enabling near- and self-interactions to be computed with machine precision. The resulting algorithm achieves linear-time complexity in the number of discretization nodes, with a small, localized precomputation cost, and is validated through extensive numerical experiments on Poisson's equation and volume integral equations. The work also provides theoretical justification for monomial-basis interpolation in 2-D, including stability and error bounds, and outlines clear paths to extensions to Helmholtz-type problems and 3-D domains.

Abstract

The accurate and efficient evaluation of Newtonian potentials over general 2-D domains is important for the numerical solution of Poisson's equation and volume integral equations. In this paper, we present a simple and efficient high-order algorithm for computing the Newtonian potential over a planar domain discretized by an unstructured mesh. The algorithm is based on the use of Green's third identity for transforming the Newtonian potential into a collection of layer potentials over the boundaries of the mesh elements, which can be easily evaluated by the Helsing-Ojala method. One important component of our algorithm is the use of high-order (up to order 20) bivariate polynomial interpolation in the monomial basis, for which we provide extensive justification. The performance of our algorithm is illustrated through several numerical experiments.

Rapid evaluation of Newtonian potentials on planar domains

TL;DR

The paper develops a fast, high-order method for evaluating Newtonian potentials on planar domains by transforming volume integrals into boundary layer potentials via Green's third identity and evaluating them with the Helsing-Ojala approach. A key advance is the use of high-order (up to ) bivariate polynomial interpolation in the monomial basis, together with an efficient per-element anti-Laplacian mapping, enabling near- and self-interactions to be computed with machine precision. The resulting algorithm achieves linear-time complexity in the number of discretization nodes, with a small, localized precomputation cost, and is validated through extensive numerical experiments on Poisson's equation and volume integral equations. The work also provides theoretical justification for monomial-basis interpolation in 2-D, including stability and error bounds, and outlines clear paths to extensions to Helmholtz-type problems and 3-D domains.

Abstract

The accurate and efficient evaluation of Newtonian potentials over general 2-D domains is important for the numerical solution of Poisson's equation and volume integral equations. In this paper, we present a simple and efficient high-order algorithm for computing the Newtonian potential over a planar domain discretized by an unstructured mesh. The algorithm is based on the use of Green's third identity for transforming the Newtonian potential into a collection of layer potentials over the boundaries of the mesh elements, which can be easily evaluated by the Helsing-Ojala method. One important component of our algorithm is the use of high-order (up to order 20) bivariate polynomial interpolation in the monomial basis, for which we provide extensive justification. The performance of our algorithm is illustrated through several numerical experiments.
Paper Structure (18 sections, 2 theorems, 28 equations, 11 figures, 5 tables)

This paper contains 18 sections, 2 theorems, 28 equations, 11 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Mathematical and numerical preliminaries
  3. Newtonian potential
  4. The Helsing-Ojala method for the close evaluation of 1-D layer potentials
  5. Bivariate polynomial interpolation in the monomial basis
  6. Numerical algorithm
  7. Construction of the anti-Laplacian mapping
  8. Close and self-evaluation of Newtonian potential over a mesh element
  9. Generalization to an arbitrary domain
  10. Time complexity analysis
  11. Precomputation for each mesh element:
  12. Close and self-evaluation of the Newtonian potential over a mesh element at a single target location:
  13. Numerical experiments
  14. Bivariate polynomial interpolation in the monomial basis
  15. Newtonian potential generated over a mesh element
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let $\Omega$ be a 2-D planar domain and $f$ be an integrable function on $\Omega$. Suppose that $\varphi:\Omega\to\mathbbm{R}$ satisfies $\nabla^2 \varphi=f$. Then, for $x\in \mathbbm{R}^2\setminus \partial\Omega$, where $\mathbbm{1}_\Omega$ denotes the indicator function for the domain $\Omega$, and $n_y$ denotes the outward pointing unit normal vector at the point $y$.

Figures (11)

  • Figure 1: The 20th order Vioreanu-Rokhlin nodes over a triangle, and the associated Lebesgue constants for various orders of approximation.. The $x$-axis label $N$ denotes the order of approximation. One may observe in Figure \ref{['fig:tri:leb']} that the Lebesgue constant for the Vioreanu-Rokhlin nodes does not exhibit monotonic growth, which is due to the heuristic nature of the algorithm used to construct these nodes.
  • Figure 2: The growth of $\kappa(V^{(N)})$ for triangles with different aspect ratios. The colors of the triangles in Figure \ref{['fig:cond:1']} correspond to the line colors depicted in Figure \ref{['fig:cond:2']}. The boxes in Figure \ref{['fig:cond:1']} define the local coordinate for each triangle.
  • Figure 3: Bivariate polynomial interpolation over the equilateral triangle $\Delta_1$ by monomials and Koornwinder polynomials.
  • Figure 4: Bivariate polynomial interpolation over the flattened triangle $\Delta_2$ by monomials and Koornwinder polynomials.
  • Figure 5: Bivariate polynomial interpolation over the curved triangle element $\widetilde{\Delta}$ by monomials.
  • ...and 6 more figures

Theorems & Definitions (8)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • Theorem 3.1
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Remark 5.1