An adaptive fast multipole accelerated Poisson solver for complex geometries

Askham, T.; Cerfon, A. J.

doi:10.1016/j.jcp.2017.04.063

An adaptive fast multipole accelerated Poisson solver for complex geometries

Journal Article · Tue May 02 00:00:00 EDT 2017 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2017.04.063· OSTI ID:1533962

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Univ. of Washington, Seattle, WA (United States); DOE/OSTI
New York Univ. (NYU), NY (United States)

Here, we present a fast, direct and adaptive Poisson solver for complex two-dimensional geometries based on potential theory and fast multipole acceleration. More precisely, the solver relies on the standard decomposition of the solution as the sum of a volume integral to account for the source distribution and a layer potential to enforce the desired boundary condition. The volume integral is computed by applying the FMM on a square box that encloses the domain of interest. For the sake of efficiency and convergence acceleration, we first extend the source distribution (the right-hand side in the Poisson equation) to the enclosing box as a $$C$$⁰ function using a fast, boundary integral-based method. We demonstrate on multiply connected domains with irregular boundaries that this continuous extension leads to high accuracy without excessive adaptive refinement near the boundary and, as a result, to an extremely efficient “black box” fast solver.

View Accepted Manuscript (DOE)

Research Organization:: New York Univ. (NYU), NY (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Fusion Energy Sciences (FES); US Air Force Office of Scientific Research (AFOSR)

Grant/Contract Number:: FG02-86ER53223; FG02-88ER25053; SC0012398

OSTI ID:: 1533962

Alternate ID(s):: OSTI ID: 1398114

Journal Information:: Journal of Computational Physics, Journal Name: Journal of Computational Physics Vol. 344; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (37)

Approximating real functions which possess nth derivatives of bounded variation and applications Dragomir, Sever S. Computers & Mathematics with Applications, Vol. 56, Issue 9 https://doi.org/10.1016/j.camwa.2008.03.049	journal	November 2008
Laplace’s equation and the Dirichlet–Neumann map: a new mode for Mikhlin’s method Helsing, Johan; Wadbro, Eddie Journal of Computational Physics, Vol. 202, Issue 2 https://doi.org/10.1016/j.jcp.2004.06.024	journal	January 2005
The use of the ghost fluid method for Poisson's equation to simulate streamer propagation in point-to-plane and point-to-point geometries Celestin, Sebastien; Bonaventura, Zdenek; Zeghondy, Barbar Journal of Physics D: Applied Physics, Vol. 42, Issue 6 https://doi.org/10.1088/0022-3727/42/6/065203	journal	February 2009
A New Fast-Multipole Accelerated Poisson Solver in Two Dimensions Ethridge, Frank; Greengard, Leslie SIAM Journal on Scientific Computing, Vol. 23, Issue 3 https://doi.org/10.1137/s1064827500369967	journal	January 2001
Laplace's Equation and the Dirichlet-Neumann Map in Multiply Connected Domains Greenbaum, A.; Greengard, L.; McFadden, G. B. Journal of Computational Physics, Vol. 105, Issue 2 https://doi.org/10.1006/jcph.1993.1073	journal	April 1993
A Fast Poisson Solver for Complex Geometries McKenney, A.; Greengard, L.; Mayo, A. Journal of Computational Physics, Vol. 118, Issue 2 https://doi.org/10.1006/jcph.1995.1104	journal	May 1995
A Direct Adaptive Poisson Solver of Arbitrary Order Accuracy Greengard, Leslie; Lee, June-Yub Journal of Computational Physics, Vol. 125, Issue 2 https://doi.org/10.1006/jcph.1996.0103	journal	May 1996
Remarks on algorithm 006: An adaptive algorithm for numerical integration over an N-dimensional rectangular region Genz, A. C.; Malik, A. A. Journal of Computational and Applied Mathematics, Vol. 6, Issue 4 https://doi.org/10.1016/0771-050X(80)90039-X	journal	December 1980
Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries Popinet, Stéphane Journal of Computational Physics, Vol. 190, Issue 2 https://doi.org/10.1016/S0021-9991(03)00298-5	journal	September 2003
ECOM: A fast and accurate solver for toroidal axisymmetric MHD equilibria Lee, Jungpyo; Cerfon, Antoine Computer Physics Communications, Vol. 190 https://doi.org/10.1016/j.cpc.2015.01.015	journal	May 2015
On the evaluation of layer potentials close to their sources Helsing, Johan; Ojala, Rikard Journal of Computational Physics, Vol. 227, Issue 5 https://doi.org/10.1016/j.jcp.2007.11.024	journal	February 2008
Corner singularities for elliptic problems: Integral equations, graded meshes, quadrature, and compressed inverse preconditioning Helsing, Johan; Ojala, Rikard Journal of Computational Physics, Vol. 227, Issue 20 https://doi.org/10.1016/j.jcp.2008.06.022	journal	October 2008
Universal quadratures for boundary integral equations on two-dimensional domains with corners Bremer, James; Rokhlin, Vladimir; Sammis, Ian Journal of Computational Physics, Vol. 229, Issue 22 https://doi.org/10.1016/j.jcp.2010.06.040	journal	November 2010
A fast, high-order solver for the Grad–Shafranov equation Pataki, Andras; Cerfon, Antoine J.; Freidberg, Jeffrey P. Journal of Computational Physics, Vol. 243 https://doi.org/10.1016/j.jcp.2013.02.045	journal	June 2013
Quadrature by expansion: A new method for the evaluation of layer potentials Klöckner, Andreas; Barnett, Alexander; Greengard, Leslie Journal of Computational Physics, Vol. 252 https://doi.org/10.1016/j.jcp.2013.06.027	journal	November 2013
Immersed boundary smooth extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods Stein, David B.; Guy, Robert D.; Thomases, Becca Journal of Computational Physics, Vol. 304 https://doi.org/10.1016/j.jcp.2015.10.023	journal	January 2016
On the solution of elliptic partial differential equations on regions with corners Serkh, Kirill; Rokhlin, Vladimir Journal of Computational Physics, Vol. 305 https://doi.org/10.1016/j.jcp.2015.10.024	journal	January 2016
Accurate derivative evaluation for any Grad–Shafranov solver Ricketson, L. F.; Cerfon, A. J.; Rachh, M. Journal of Computational Physics, Vol. 305 https://doi.org/10.1016/j.jcp.2015.11.015	journal	January 2016
Fast convolution with free-space Green's functions Vico, Felipe; Greengard, Leslie; Ferrando, Miguel Journal of Computational Physics, Vol. 323 https://doi.org/10.1016/j.jcp.2016.07.028	journal	October 2016
Nonlinear, three-dimensional magnetohydrodynamics of noncircular tokamaks Strauss, H. R. Physics of Fluids, Vol. 19, Issue 1 https://doi.org/10.1063/1.861310	journal	January 1976
On Direct Methods for Solving Poisson’s Equations Buzbee, B. L.; Golub, G. H.; Nielson, C. W. SIAM Journal on Numerical Analysis, Vol. 7, Issue 4 https://doi.org/10.1137/0707049	journal	December 1970
The Fast Solution of Poisson’s and the Biharmonic Equations on Irregular Regions Mayo, Anita SIAM Journal on Numerical Analysis, Vol. 21, Issue 2 https://doi.org/10.1137/0721021	journal	April 1984
A Nonlinear Optimization Procedure for Generalized Gaussian Quadratures Bremer, James; Gimbutas, Zydrunas; Rokhlin, Vladimir SIAM Journal on Scientific Computing, Vol. 32, Issue 4 https://doi.org/10.1137/080737046	journal	January 2010
GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems Saad, Youcef; Schultz, Martin H. SIAM Journal on Scientific and Statistical Computing, Vol. 7, Issue 3 https://doi.org/10.1137/0907058	journal	July 1986
A Fast Adaptive Multipole Algorithm for Particle Simulations Carrier, J.; Greengard, L.; Rokhlin, V. SIAM Journal on Scientific and Statistical Computing, Vol. 9, Issue 4 https://doi.org/10.1137/0909044	journal	July 1988
Fast Parallel Iterative Solution of Poisson’s and the Biharmonic Equations on Irregular Regions Mayo, A.; Greenbaum, A. SIAM Journal on Scientific and Statistical Computing, Vol. 13, Issue 1 https://doi.org/10.1137/0913006	journal	January 1992
An Implementation of the Fast Multipole Method without Multipoles Anderson, Christopher R. SIAM Journal on Scientific and Statistical Computing, Vol. 13, Issue 4 https://doi.org/10.1137/0913055	journal	July 1992
A Fast Direct Solver for Structured Linear Systems by Recursive Skeletonization Ho, Kenneth L.; Greengard, Leslie SIAM Journal on Scientific Computing, Vol. 34, Issue 5 https://doi.org/10.1137/120866683	journal	January 2012
Evaluation of Layer Potentials Close to the Boundary for Laplace and Helmholtz Problems on Analytic Planar Domains Barnett, Alex H. SIAM Journal on Scientific Computing, Vol. 36, Issue 2 https://doi.org/10.1137/120900253	journal	January 2014
On the Convergence of Local Expansions of Layer Potentials Epstein, Charles L.; Greengard, Leslie; Klöckner, Andreas SIAM Journal on Numerical Analysis, Vol. 51, Issue 5 https://doi.org/10.1137/120902859	journal	January 2013
Spectrally Accurate Quadratures for Evaluation of Layer Potentials Close to the Boundary for the 2D Stokes and Laplace Equations Barnett, Alex; Wu, Bowei; Veerapaneni, Shravan SIAM Journal on Scientific Computing, Vol. 37, Issue 4 https://doi.org/10.1137/140990826	journal	January 2015
High-Order Corrected Trapezoidal Quadrature Rules for Singular Functions Kapur, Sharad; Rokhlin, Vladimir SIAM Journal on Numerical Analysis, Vol. 34, Issue 4 https://doi.org/10.1137/S0036142995287847	journal	August 1997
A New Fast-Multipole Accelerated Poisson Solver in Two Dimensions Ethridge, Frank; Greengard, Leslie SIAM Journal on Scientific Computing, Vol. 23, Issue 3 https://doi.org/10.1137/S1064827500369967	journal	January 2001
An Improved Fast Multipole Algorithm for Potential Fields Hrycak, Tomasz; Rokhlin, Vladimir SIAM Journal on Scientific Computing, Vol. 19, Issue 6 https://doi.org/10.1137/S106482759630989X	journal	November 1998
Hybrid Gauss-Trapezoidal Quadrature Rules Alpert, Bradley K. SIAM Journal on Scientific Computing, Vol. 20, Issue 5 https://doi.org/10.1137/S1064827597325141	journal	January 1999
A free-space adaptive FMM-Based PDE solver in three dimensions Langston, Harper; Greengard, Leslie; Zorin, Denis Communications in Applied Mathematics and Computational Science, Vol. 6, Issue 1 https://doi.org/10.2140/camcos.2011.6.79	journal	January 2011
A Robust and Accurate Solver of Laplace's Equation with General Boundary Conditions on General Domains in the Plane Ojala, Rikard Journal of Computational Mathematics, Vol. 30, Issue 4 https://doi.org/10.4208/jcm.1201-m3644	journal	June 2012

Cited By (3)

Integral Equation Formulation of the Biharmonic Dirichlet Problem Rachh, M.; Askham, T. Journal of Scientific Computing, Vol. 75, Issue 2 https://doi.org/10.1007/s10915-017-0559-8	journal	September 2017
Adaptive quadrature by expansion for layer potential evaluation in two dimensions Klinteberg, Ludvig af; Tornberg, Anna-Karin arXiv https://doi.org/10.48550/arxiv.1704.02219	text	January 2017
Distributed and Adaptive Fast Multipole Method In Three Dimensions Bull, Jonathan; Engblom, Stefan arXiv https://doi.org/10.48550/arxiv.2002.04894	preprint	January 2020

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97 MATHEMATICS AND COMPUTING
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An adaptive fast multipole accelerated Poisson solver for complex geometries

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