Integral equation methods for elliptic problems with boundary conditions of mixed type

Abstract

Laplace's equation with mixed boundary conditions, that is, Dirichlet conditions on parts of the boundary and Neumann conditions on the remaining contiguous parts, is solved on an interior planar domain using an integral equation method. Rapid execution and high accuracy is obtained by combining equations which are of Fredholm's second kind with compact operators on almost the entire boundary with a recursive compressed inverse preconditioning technique. Then an elastic problem with mixed boundary conditions is formulated and solved in an analogous manner and with similar results. This opens up for the rapid and accurate solution of several elliptic problems of mixed type.

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.…”

“…It should be noted that in the two-dimensional case, several of these alternatives provide extremely effective schemes, especially [Bremer et al, 2010, Helsing, 2009, Helsing and Ojala, 2008a since they all permit local adaptivity and high order accuracy.…”

The effective conductivity of random checkerboards

“…In this section we give a brief review of how the method works with focus on issues important to its implementation. The machinery for four-wedge corners is very similar to the one developed for three-wedge corners in [15], but we shall use the simplified notation introduced in [16]. See both references for derivations and full definitions.…”

“…coa is a weighted prolongation operator, W is a matrix containing the quadrature weights on the diagonal, and subscript 'coa' indicates the coarse grid, see Section 5 of [16]. Eq.…”

“…• the use of recursive compressed inverse preconditioning [15,16] to resolve layer densities in the vicinity of quadruple-junctions…”