Corrected Trapezoidal Rules for Boundary Integral Equations in Three Dimensions

TL;DR

This paper introduces a simple correction to the trapezoidal rule for boundary integral equations in 3D, significantly improving convergence rates by adjusting only a few matrix elements, supported by rigorous error analysis.

Contribution

It proposes a minimal correction method for trapezoidal quadrature in 3D boundary integral equations, achieving higher convergence orders with simple diagonal modifications.

Findings

Diagonal correction yields $O(h^{3})$ convergence for Laplace and Helmholtz kernels.

Nine-point correction stencil achieves $O(h^{5})$ convergence.

Method extends 2D quadrature correction techniques to 3D boundary integral equations.

Abstract

The manuscript describes a quadrature rule that is designed for the high order discretization of boundary integral equations (BIEs) using the Nystr\"{o}m method. The technique is designed for surfaces that can naturally be parameterized using a uniform grid on a rectangle, such as deformed tori, or channels with periodic boundary conditions. When a BIE on such a geometry is discretized using the Nystr\"{o}m method based on the Trapezoidal quadrature rule, the resulting scheme tends to converge only slowly, due to the singularity in the kernel function. The key finding of the manuscript is that the convergence order can be greatly improved by modifying only a very small number of elements in the coefficient matrix. Specifically, it is demonstrated that by correcting only the diagonal entries in the coefficient matrix, $O(h^{3})$ convergence can be attained for the single and double layer potentials associated with both the Laplace and the Helmholtz kernels. A nine-point correction stencil leads to an $O(h^5)$ scheme. The method proposed can be viewed as a generalization of the quadrature rule of Duan and Rokhlin, which was designed for the 2D Lippmann-Schwinger equation in the plane. The techniques proposed are supported by a rigorous error analysis that relies on Wigner-type limits involving the Epstein zeta function and its parametric derivatives.