TY - JOUR

T1 - Ubiquitous evaluation of layer potentials using Quadrature by Kernel-Independent Expansion

AU - Rahimian, Abtin

AU - Barnett, Alex

AU - Zorin, Denis

N1 - Funding Information:
Acknowledgements We extend our thanks to Manas Rachh, Andreas Klöckner, Michael O’Neil, and Leslie Greengard for stimulating conversations about various aspects of this work. A.R. and D.Z. acknowledge the support of the US National Science Foundation (NSF) through Grant DMS-1320621; A.B. acknowledges the support of the NSF through Grant DMS-1216656.

PY - 2018/6/1

Y1 - 2018/6/1

N2 - We introduce a quadrature scheme—QBKIX —for the ubiquitous high-order 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 to match 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 (allowing simple generalization to 3D), and compatibility with fast algorithms such as the kernel-independent FMM.

AB - We introduce a quadrature scheme—QBKIX —for the ubiquitous high-order 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 to match 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 (allowing simple generalization to 3D), and compatibility with fast algorithms such as the kernel-independent FMM.

KW - Boundary integral equations

KW - Elliptic Boundary value problem

KW - High order quadrature

KW - Kernel-independent

KW - Near-singular integrals

UR - http://www.scopus.com/inward/record.url?scp=85033437536&partnerID=8YFLogxK

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U2 - 10.1007/s10543-017-0689-2

DO - 10.1007/s10543-017-0689-2

M3 - Article

AN - SCOPUS:85033437536

VL - 58

SP - 423

EP - 456

JO - BIT Numerical Mathematics

JF - BIT Numerical Mathematics

SN - 0006-3835

IS - 2

ER -