Ubiquitous evaluation of layer potentials using Quadrature by Kernel-Independent Expansion

We introduce a quadrature scheme—QBKIX—for the ubiquitous highorder accurate evaluation of singular layer potentials associated with general elliptic PDEs, i.e., a scheme that yields high accuracy at all distances to the domain boundary as well as on the boundary itself. Relying solely on point evaluations of the underlying kernel, our scheme is essentially PDE-independent; in particular, no analytic expansion nor addition theorem is required. Moreover, it applies to boundary integrals with singular, weakly singular, and hypersingular kernels. Our work builds upon quadrature by expansion, which approximates the potential by an analytic expansion in the neighborhood of each expansion center. In contrast, we use a sum of fundamental solutions lying on a ring enclosing the neighborhood, and solve a small dense linear system for their coefficients tomatch the potential on a smaller concentric ring.We test the new method with Laplace, Helmholtz, Yukawa, Stokes, and Navier (elastostatic) kernels in two dimensions (2D) using adaptive, panel-based boundary quadratures on smooth and corner domains. Advantages of the algorithm include its relative simplicity of implementation, immediate extension to new kernels, dimension-independence Communicated by Ralf Hiptmair. B Abtin Rahimian [email protected] Alex Barnett [email protected] Denis Zorin [email protected] 1 Department of Computer Science, University of Colorado, Boulder, CO 80302, USA 2 Department of Mathematics, Dartmouth College, Hanover, NH 03755, USA 3 Courant Institute of Mathematical Sciences, New York University, New York, NY 10003, USA