Corrected trapezoidal rules for singular implicit boundary integrals

TL;DR

This paper introduces higher-order quadrature corrections for boundary integral operators, enabling accurate numerical integration on surfaces using simple Cartesian grids by transforming surface integrals into volume integrals and correcting for singularities.

Contribution

It develops new higher-order correction methods for trapezoidal rule-based quadrature of boundary integrals, avoiding complex parameterizations and addressing singularities effectively.

Findings

Achieves higher-order accuracy in boundary integral computations.

Reduces complexity by transforming surface integrals into volume integrals.

Provides a correction method based on local singularity decomposition.

Abstract

We present new higher-order quadratures for a family of boundary integral operators re-derived using the approach introduced in [Kublik, Tanushev, and Tsai - J. Comp. Phys. 247: 279-311, 2013]. In this formulation, a boundary integral over a smooth, closed hypersurface is transformed into an equivalent volume integral defined in a sufficiently thin tubular neighborhood of the surface. The volumetric formulation makes it possible to use the simple trapezoidal rule on uniform Cartesian grids and relieves the need to use parameterization for developing quadrature. Consequently, typical point singularities in a layer potential extend along the surface's normal lines. We propose new higher-order corrections to the trapezoidal rule on the grid nodes around the singularities. This correction is based on local decompositions of the singularity and is dependent on the angle of approach to the singularity relative to the surface's principal curvature directions. The proposed decomposition, combined with the volumetric formulation, leads to a special quadrature error cancellation.