Fast algorithms for Quadrature by Expansion I: Globally valid expansions

Abstract The use of integral equation methods for the efficient numerical solution of PDE boundary value problems requires two main tools: quadrature rules for the evaluation of layer potential integral operators with singular kernels, and fast algorithms for solving the resulting dense linear systems. Classically, these tools were developed separately. In this work, we present a unified numerical scheme based on coupling Quadrature by Expansion , a recent quadrature method, to a customized Fast Multipole Method (FMM) for the Helmholtz equation in two dimensions. The method allows the evaluation of layer potentials in linear-time complexity, anywhere in space, with a uniform, user-chosen level of accuracy as a black-box computational method. Providing this capability requires geometric and algorithmic considerations beyond the needs of standard FMMs as well as careful consideration of the accuracy of multipole translations. We illustrate the speed and accuracy of our method with various numerical examples.

[1]  Leslie Greengard,et al.  On the efficient representation of the half-space impedance Green’s function for the Helmholtz equation , 2011, 1109.6708.

[2]  Anna-Karin Tornberg,et al.  Estimation of quadrature errors in layer potential evaluation using quadrature by expansion , 2016, 1603.08366.

[3]  R. Kress,et al.  Integral equation methods in scattering theory , 1983 .

[4]  Peter Werner,et al.  Über das Dirichletsche Außenraumproblem für die Helmholtzsche Schwingungsgleichung , 1965 .

[5]  Anna-Karin Tornberg,et al.  Error estimation for quadrature by expansion in layer potential evaluation , 2017, Adv. Comput. Math..

[6]  Jean-François Richard,et al.  Methods of Numerical Integration , 2000 .

[7]  Charles L. Epstein,et al.  On the Convergence of Local Expansions of Layer Potentials , 2012, SIAM J. Numer. Anal..

[8]  Per-Gunnar Martinsson,et al.  High-order accurate methods for Nyström discretization of integral equations on smooth curves in the plane , 2014, Adv. Comput. Math..

[9]  Leslie Greengard,et al.  Quadrature by expansion: A new method for the evaluation of layer potentials , 2012, J. Comput. Phys..

[10]  J. CARRIERt,et al.  A FAST ADAPTIVE MULTIPOLE ALGORITHM FOR PARTICLE SIMULATIONS * , 2022 .

[11]  Seppo Järvenpää,et al.  Singularity extraction technique for integral equation methods with higher order basis functions on plane triangles and tetrahedra , 2003 .

[12]  R. Kanwal.  Linear Integral Equations , 1925, Nature.

[13]  E. Nyström.  Über Die Praktische Auflösung von Integralgleichungen mit Anwendungen auf Randwertaufgaben , 1930 .

[14]  Norman Yarvin,et al.  Generalized Gaussian Quadratures and Singular Value Decompositions of Integral Operators , 1998, SIAM J. Sci. Comput..

[15]  Philip Rabinowitz,et al.  Methods of Numerical Integration , 1985 .

[16]  P. Martinsson,et al.  High-order accurate Nystrom discretization of integral equations with weakly singular kernels on smooth curves in the plane , 2011, 1112.6262.

[17]  V. Rokhlin.  Rapid Solution of Integral Equations of Scattering Theory , 1990 .

[18]  Leslie Greengard,et al.  Fast algorithms for volume integrals in potential theory , 2000 .

[19]  Alex H. Barnett,et al.  Evaluation of Layer Potentials Close to the Boundary for Laplace and Helmholtz Problems on Analytic Planar Domains , 2013, SIAM J. Sci. Comput..

[20]  Ronald F. Boisvert,et al.  NIST Handbook of Mathematical Functions , 2010 .

[21]  Johan Helsing.  A Fast and Stable Solver for Singular Integral Equations on Piecewise Smooth Curves , 2011, SIAM J. Sci. Comput..

[22]  Lloyd N. Trefethen,et al.  Is Gauss Quadrature Better than Clenshaw-Curtis? , 2008, SIAM Rev..

[23]  D. Zorin,et al.  Ubiquitous evaluation of layer potentials using Quadrature by Kernel-Independent Expansion , 2016, 1612.00977.

[24]  L. Greengard,et al.  Accelerating fast multipole methods for the Helmholtz equation at low frequencies , 1998 .

[25]  James Bremer,et al.  A Nonlinear Optimization Procedure for Generalized Gaussian Quadratures , 2010, SIAM J. Sci. Comput..

[26]  Thorsten Gerber,et al.  Handbook Of Mathematical Functions , 2016 .