Corrected trapezoidal rules for singular implicit boundary integrals

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.

High order corrected trapezoidal rules for a class of singular integrals

“…The closest grid nodes to each surface normal line certainly are expected to exhibit jumps as one refines the grids (decreases h). Thus, as noted in [6], the errors (4.5) as functions of h are generally not smooth. Consequently we cannot see a clear slope.…”

“…By construction of the quadrature rule (3.12) we expect it to be third order accurate in h. However from the plots we observe order of accuracy ≥ 3.5. We conjecture that an additional cancellation of errors occurs when adding the results from each plane (see [6] for a related discussion regarding V 2 h ). A rigorous analysis of this behavior is beyond the scope of this article.…”

“…This was the approach used in [6], although there (2.11) was used also on the second term instead of the punctured trapezoidal rule.…”

“…We tested our third order rule V 3 h (3.12). Moreover we compared with the previously developed second order rule, denoted by V 2 h , from [6]. In the presented simulations, we take the component n z of n to be dominant if | tan θ| < √ 2, where n/|n| = (sin θ cos φ, sin θ sin φ, cos θ).…”

“…In [6], we derived a second order accurate method. In this paper, we generalize our approach systematically to derive higher order methods.…”