An adaptive fast multipole accelerated Poisson solver for complex geometries

Askham, T.; Cerfon, A. J.

doi:10.1016/j.jcp.2017.04.063

Title: An adaptive fast multipole accelerated Poisson solver for complex geometries

Abstract

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.

Authors:

^[1];

^[2]

Univ. of Washington, Seattle, WA (United States)
New York Univ. (NYU), NY (United States)

Publication Date:: Tue May 02 00:00:00 EDT 2017

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

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

OSTI Identifier:: 1533962

Alternate Identifier(s):: OSTI ID: 1398114

Grant/Contract Number:: FG02-86ER53223; FG02-88ER25053; SC0012398; FA9550-10-1-0180; FA9550-15-1-0385; DEFG0288ER25053

Resource Type:: Accepted Manuscript

Journal Name:: Journal of Computational Physics

Additional Journal Information:: Journal Volume: 344; Journal ID: ISSN 0021-9991

Publisher:: Elsevier

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; Poisson equation; Fast multipole method; Quadrature by expansion; Integral equations

Citation Formats


                    Askham, T., and Cerfon, A. J. An adaptive fast multipole accelerated Poisson solver for complex geometries.  United States: N. p., 2017. 
Web.  doi:10.1016/j.jcp.2017.04.063.

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                    Askham, T., & Cerfon, A. J. An adaptive fast multipole accelerated Poisson solver for complex geometries.  United States.  https://doi.org/10.1016/j.jcp.2017.04.063

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                    Askham, T., and Cerfon, A. J. Tue .  
"An adaptive fast multipole accelerated Poisson solver for complex geometries".  United States.  https://doi.org/10.1016/j.jcp.2017.04.063.  https://www.osti.gov/servlets/purl/1533962.

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@article{osti_1533962,

  title        = {An adaptive fast multipole accelerated Poisson solver for complex geometries},

  author       = {Askham, T. and Cerfon, A. J.},

  abstractNote = {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.},

  doi          = {10.1016/j.jcp.2017.04.063},

  journal      = {Journal of Computational Physics},

  number       = ,

  volume       = 344,

  place        = {United States},

  year         = {Tue May 02 00:00:00 EDT 2017},

  month        = {Tue May 02 00:00:00 EDT 2017}

}

Copy to clipboard

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https://doi.org/10.1016/j.jcp.2017.04.063

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Works referenced in this record:

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Dragomir, Sever S.
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The use of the ghost fluid method for Poisson's equation to simulate streamer propagation in point-to-plane and point-to-point geometries
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Works referencing / citing this record:

Integral Equation Formulation of the Biharmonic Dirichlet Problem
journal, September 2017

Rachh, M.; Askham, T.
Journal of Scientific Computing, Vol. 75, Issue 2
DOI: 10.1007/s10915-017-0559-8

Adaptive quadrature by expansion for layer potential evaluation in two dimensions
text, January 2017

Klinteberg, Ludvig af; Tornberg, Anna-Karin
arXiv
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Distributed and Adaptive Fast Multipole Method In Three Dimensions
preprint, January 2020

Bull, Jonathan; Engblom, Stefan
arXiv
DOI: 10.48550/arxiv.2002.04894

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Title: An adaptive fast multipole accelerated Poisson solver for complex geometries

Abstract

Citation Formats

Nonlinear, three-dimensional magnetohydrodynamics of noncircular tokamaks journal, January 1976

A Fast Poisson Solver for Complex Geometries journal, May 1995

ECOM: A fast and accurate solver for toroidal axisymmetric MHD equilibria journal, May 2015

Fast Parallel Iterative Solution of Poisson’s and the Biharmonic Equations on Irregular Regions journal, January 1992

Laplace's Equation and the Dirichlet-Neumann Map in Multiply Connected Domains journal, April 1993

A Fast Adaptive Multipole Algorithm for Particle Simulations journal, July 1988

A Fast Direct Solver for Structured Linear Systems by Recursive Skeletonization journal, January 2012

High-Order Corrected Trapezoidal Quadrature Rules for Singular Functions journal, August 1997

A Direct Adaptive Poisson Solver of Arbitrary Order Accuracy journal, May 1996

On the evaluation of layer potentials close to their sources journal, February 2008

On the Convergence of Local Expansions of Layer Potentials journal, January 2013

An Implementation of the Fast Multipole Method without Multipoles journal, July 1992

Hybrid Gauss-Trapezoidal Quadrature Rules journal, January 1999

Remarks on algorithm 006: An adaptive algorithm for numerical integration over an N-dimensional rectangular region journal, December 1980

Quadrature by expansion: A new method for the evaluation of layer potentials journal, November 2013

Laplace’s equation and the Dirichlet–Neumann map: a new mode for Mikhlin’s method journal, January 2005

A fast, high-order solver for the Grad–Shafranov equation journal, June 2013

Universal quadratures for boundary integral equations on two-dimensional domains with corners journal, November 2010

An Improved Fast Multipole Algorithm for Potential Fields journal, November 1998

On the solution of elliptic partial differential equations on regions with corners journal, January 2016

Immersed boundary smooth extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods journal, January 2016

Corner singularities for elliptic problems: Integral equations, graded meshes, quadrature, and compressed inverse preconditioning journal, October 2008

A Robust and Accurate Solver of Laplace's Equation with General Boundary Conditions on General Domains in the Plane journal, June 2012

A Nonlinear Optimization Procedure for Generalized Gaussian Quadratures journal, January 2010

Spectrally Accurate Quadratures for Evaluation of Layer Potentials Close to the Boundary for the 2D Stokes and Laplace Equations journal, January 2015

GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems journal, July 1986

Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries journal, September 2003

On Direct Methods for Solving Poisson’s Equations journal, December 1970

Approximating real functions which possess nth derivatives of bounded variation and applications journal, November 2008

A free-space adaptive FMM-Based PDE solver in three dimensions journal, January 2011

Evaluation of Layer Potentials Close to the Boundary for Laplace and Helmholtz Problems on Analytic Planar Domains journal, January 2014

The Fast Solution of Poisson’s and the Biharmonic Equations on Irregular Regions journal, April 1984

Accurate derivative evaluation for any Grad–Shafranov solver journal, January 2016

The use of the ghost fluid method for Poisson's equation to simulate streamer propagation in point-to-plane and point-to-point geometries journal, February 2009

Fast convolution with free-space Green's functions journal, October 2016

Approximating real functions which possess nth derivatives of bounded variation and applications journal, November 2008

Laplace’s equation and the Dirichlet–Neumann map: a new mode for Mikhlin’s method 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 journal, February 2009

A New Fast-Multipole Accelerated Poisson Solver in Two Dimensions journal, January 2001

Integral Equation Formulation of the Biharmonic Dirichlet Problem journal, September 2017

Adaptive quadrature by expansion for layer potential evaluation in two dimensions text, January 2017

Distributed and Adaptive Fast Multipole Method In Three Dimensions preprint, January 2020

Nonlinear, three-dimensional magnetohydrodynamics of noncircular tokamaks
journal, January 1976

A Fast Poisson Solver for Complex Geometries
journal, May 1995

ECOM: A fast and accurate solver for toroidal axisymmetric MHD equilibria
journal, May 2015

Fast Parallel Iterative Solution of Poisson’s and the Biharmonic Equations on Irregular Regions
journal, January 1992

Laplace's Equation and the Dirichlet-Neumann Map in Multiply Connected Domains
journal, April 1993

A Fast Adaptive Multipole Algorithm for Particle Simulations
journal, July 1988

A Fast Direct Solver for Structured Linear Systems by Recursive Skeletonization
journal, January 2012

High-Order Corrected Trapezoidal Quadrature Rules for Singular Functions
journal, August 1997

A Direct Adaptive Poisson Solver of Arbitrary Order Accuracy
journal, May 1996

On the evaluation of layer potentials close to their sources
journal, February 2008

On the Convergence of Local Expansions of Layer Potentials
journal, January 2013

An Implementation of the Fast Multipole Method without Multipoles
journal, July 1992

Hybrid Gauss-Trapezoidal Quadrature Rules
journal, January 1999

Remarks on algorithm 006: An adaptive algorithm for numerical integration over an N-dimensional rectangular region
journal, December 1980

Quadrature by expansion: A new method for the evaluation of layer potentials
journal, November 2013

Laplace’s equation and the Dirichlet–Neumann map: a new mode for Mikhlin’s method
journal, January 2005

A fast, high-order solver for the Grad–Shafranov equation
journal, June 2013

Universal quadratures for boundary integral equations on two-dimensional domains with corners
journal, November 2010

An Improved Fast Multipole Algorithm for Potential Fields
journal, November 1998

On the solution of elliptic partial differential equations on regions with corners
journal, January 2016

Immersed boundary smooth extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods
journal, January 2016

Corner singularities for elliptic problems: Integral equations, graded meshes, quadrature, and compressed inverse preconditioning
journal, October 2008

A Robust and Accurate Solver of Laplace's Equation with General Boundary Conditions on General Domains in the Plane
journal, June 2012

A Nonlinear Optimization Procedure for Generalized Gaussian Quadratures
journal, January 2010

Spectrally Accurate Quadratures for Evaluation of Layer Potentials Close to the Boundary for the 2D Stokes and Laplace Equations
journal, January 2015

GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
journal, July 1986

Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries
journal, September 2003

On Direct Methods for Solving Poisson’s Equations
journal, December 1970

Approximating real functions which possess nth derivatives of bounded variation and applications
journal, November 2008

A free-space adaptive FMM-Based PDE solver in three dimensions
journal, January 2011

Evaluation of Layer Potentials Close to the Boundary for Laplace and Helmholtz Problems on Analytic Planar Domains
journal, January 2014

The Fast Solution of Poisson’s and the Biharmonic Equations on Irregular Regions
journal, April 1984

Accurate derivative evaluation for any Grad–Shafranov solver
journal, January 2016

The use of the ghost fluid method for Poisson's equation to simulate streamer propagation in point-to-plane and point-to-point geometries
journal, February 2009

Fast convolution with free-space Green's functions
journal, October 2016

Approximating real functions which possess nth derivatives of bounded variation and applications
journal, November 2008

Laplace’s equation and the Dirichlet–Neumann map: a new mode for Mikhlin’s method
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
journal, February 2009

A New Fast-Multipole Accelerated Poisson Solver in Two Dimensions
journal, January 2001

Integral Equation Formulation of the Biharmonic Dirichlet Problem
journal, September 2017

Adaptive quadrature by expansion for layer potential evaluation in two dimensions
text, January 2017

Distributed and Adaptive Fast Multipole Method In Three Dimensions
preprint, January 2020