A Method for Computing Nearly Singular Integrals

Abstract

We develop a method for computing a nearly singular integral, such as a double layer potential due to sources on a curve in the plane, evaluated at a point near the curve. The approach is to regularize the singularity and obtain a preliminary value from a standard quadrature rule. Then we add corrections for the errors due to smoothing and discretization, which are found by asymptotic analysis. We prove an error estimate for the corrected value, uniform with respect to the point of evaluation. One application is a simple method for solving the Dirichlet problem for Laplace's equation on a grid covering an irregular region in the plane, similar to an earlier method of A. Mayo [SIAM J. Sci. Statist. Comput., 6 (1985), pp. 144-157]. This approach could also be used to compute the pressure gradient due to a force on a moving boundary in an incompressible fluid. Computational examples are given for the double layer potential and for the Dirichlet problem.

Quadrature by expansion: A new method for the evaluation of layer potentials

“…Three other powerful approaches are (a) to design special purpose quadratures that integrate a specific class of singular functions with high-order accuracy [Alpert, 1999, Bremer et al, 2010, Helsing, 2009, Kapur and Rokhlin, 1997, Kolm and Rokhlin, 2001, Kress, 1995, Sidi and Israeli, 1988, Strain, 1995, Yarvin and Rokhlin, 1998], (b) to find a change of variables that removes the principal singularity [Bruno and Kunyansky, 2001, Davis and Rabinowitz, 1984, Duffy, 1982, Graglia and Lombardi, 2008, Hackbusch and Sauter, 1994, Jarvenpää et al, 2003, Khayat and Wilton, 2005, Kress, 1991, Schwab and Wendland, 1992, Ying et al, 2006, and (c) to regularize the kernel so that smooth rules can be applied, followed by corrections through asymptotic analysis or Richardson extrapolation [Beale and Lai, 2001, Goodman et al, 1990, Haroldsen and Meiron, 1990, Lowengrub et al, 1993, Schwab and Wendland, 1992. By contrast with singularity subtraction, methods of type (b) are sometimes referred to as using singularity cancellation.…”

Partially implicit motion of a sharp interface in Navier–Stokes flow

“…For the solution of the Stokes problem in (2), it is natural to use the IIM, as applied to Stokes flow in [18], and this was done in [3]. In the present work, as in [15], we use the boundary integral representation of the free space Stokes pressure and velocity (e.g., see [29]).…”

A kernel-free boundary integral method for elliptic boundary value problems

“…Here, the coefficient matrix in the right hand side of (40) is understood as an operator of grid functions. Note that the assumption (38) indicates the approximation property of the discrete elliptic operator: (41) in the discrete maximum norm for .…”

“…This paper presents a class of kernel-free boundary integral (KFBI) methods for solving the elliptic BVPs. It is similar, in spirit, to Li's augmented strategy for constant coefficient problems [25], Wiegmann and Bube's explicit jump II method [26] and Calhoun's Cartesian grid method [52], and is a direct extension of Mayo's original approach [13,41,42]. The most obvious difference of the method from others is that it works with more general elliptic operators with possible anisotropy and inhomogeneity.…”

“…The grid-based boundary integral method has been generalized to different problems such as biharmonic equations, Stokes and Navier-Stokes equations. Recently, J. T. Beale proposed methods for evaluating the (nearly) singular integrals in both two and three space dimensions, which make the grid-based boundary integral method more accurate [41,42]. Mayo's approach is fast when combined with the fast multipole method [43] provided that the Green's functions are analytically known.…”