Asymptotic Approximations for the Close Evaluation of Double-Layer Potentials

Abstract

When using the boundary integral equation method to solve a boundary value problem, the evaluation of the solution near the boundary is challenging to compute because the layer potentials that represent the solution are nearly-singular integrals. To address this close evaluation problem, we apply an asymptotic analysis of these nearly singular integrals and obtain an asymptotic approximation. We derive the asymptotic approximation for the case of the double-layer potential in two and three dimensions, representing the solution of the interior Dirichlet problem for Laplace's equation. By doing so, we obtain an asymptotic approximation given by the Dirichlet data at the boundary point nearest to the interior evaluation point plus a nonlocal correction. We present numerical methods to compute this asymptotic approximation, and we demonstrate the efficiency and accuracy of the asymptotic approximation through several examples. These examples show that the asymptotic approximation is useful as it accurately approximates the close evaluation of the double-layer potential while requiring only modest computational resources.

Efficient high-order singular quadrature schemes in magnetic fusion

“…Such a scheme would, for example, be necessary if we wanted to evaluate the off-surface vacuum field potential Φ given by (2.8), or to compute the off-surface magnetic field near a bounding surface in the integral equation formulation of the calculation of Taylor states as implemented in BIEST [33]. Both extensions reflect questions which have not yet been addressed in a definitive way by the applied mathematics community [1,4,5,8,9,12,21,23,24,26,27,[44][45][46]48,51,[53][54][55]. These problems are the subject of ongoing work, with results to be reported in the future.…”

“…(2.8), have not been proposed in the magnetic fusion community. In fact, designing efficient and accurate numerical methods for off-surface evaluations near arbitrary surfaces in three dimensions remains an open problem in applied mathematics, although the pioneering work of [1,8,9,12,27,44,48,51,[53][54][55] must be highlighted. Fortunately, for many applications, including free-boundary MHD equilibrium calculations, the vacuum magnetic field is only needed on the magnetic surface, so that only Eq.…”

Highly accurate special quadrature methods for Stokesian particle suspensions in confined geometries

“…As in the free-space setting in Section 3.3, the evaluation point x is classified into one of three regions (see Figure 8). The potential is evaluated using (52) in the direct quadrature region and ( 53)- (54) in the other two regions. The reader may wonder why we in the upsampled quadrature region do not simply evaluate (52) using upsampled quadrature.…”

“…The extrapolatory method used by Lu et al 2 falls into the same category. Other types of methods are based on regularizing the kernel and adding corrections, [48][49][50][51] density interpolation techniques, 52 coordinate rotations and a subtraction method, 53 asymptotic approximations, 54 analytical expressions available only for spheres 55 or floating partitions of unity. [56][57][58] Many of these methods are target-specific, and their cost grows rapidly if there are many nearly singular target points.…”

Modified Representations for the Close Evaluation Problem

“…There exists a plethora of manners to address the close evaluation problem: using extraction methods based on Taylor series expansions [10], regularizing the nearly singular behavior of the integrand and adding corrections [11,12], compensating quadrature rules via interpolation [13], using Quadrature By Expansion related techniques (QBX) [9,[14][15][16][17][18][19], using adaptive methods [20], using singularity subtraction techniques and interpolation [21][22][23], or using asymptotic approximations [24][25][26], to name a few. Most techniques rely on either providing corrections to the kernel (related to the fundamental solution of the PDE at stake), or to the density (solution of the boundary integral equation).…”