Communications in Applied Mathematics and Computational Science

Computation of volume potentials on structured grids with the method of local corrections

Chris Kavouklis and Phillip Colella

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We present a new version of the method of local corrections (MLC) of McCorquodale, Colella, Balls, and Baden (2007), a multilevel, low-communication, noniterative domain decomposition algorithm for the numerical solution of the free space Poisson’s equation in three dimensions on locally structured grids. In this method, the field is computed as a linear superposition of local fields induced by charges on rectangular patches of size O(1) mesh points, with the global coupling represented by a coarse-grid solution using a right-hand side computed from the local solutions. In the present method, the local convolutions are further decomposed into a short-range contribution computed by convolution with the discrete Green’s function for a Q-th-order accurate finite difference approximation to the Laplacian with the full right-hand side on the patch, combined with a longer-range component that is the field induced by the terms up to order P1 of the Legendre expansion of the charge over the patch. This leads to a method with a solution error that has an asymptotic bound of O(hP)+O(hQ)+O(ϵh2)+O(ϵ), where h is the mesh spacing and ϵ is the max norm of the charge times a rapidly decaying function of the radius of the support of the local solutions scaled by h. The bound O(ϵ) is essentially the error of the global potential computed on the coarsest grid in the hierarchy. Thus, we have eliminated the low-order accuracy of the original method (which corresponds to P=1 in the present method) for smooth solutions, while keeping the computational cost per patch nearly the same as that of the original method. Specifically, in addition to the local solves of the original method we only have to compute and communicate the expansion coefficients of local expansions (that is, for instance, 20 scalars per patch for P=4). Several numerical examples are presented to illustrate the new method and demonstrate its convergence properties.

Article information

Commun. Appl. Math. Comput. Sci., Volume 14, Number 1 (2019), 1-32.

Received: 5 October 2016
Revised: 14 July 2018
Accepted: 16 July 2018
First available in Project Euclid: 25 July 2019

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Mathematical Reviews number (MathSciNet)

Primary: 65N06: Finite difference methods 65N12: Stability and convergence of numerical methods 65N15: Error bounds 68W10: Parallel algorithms

Poisson solver method of local corrections Mehrstellen stencils domain decomposition parallel solvers


Kavouklis, Chris; Colella, Phillip. Computation of volume potentials on structured grids with the method of local corrections. Commun. Appl. Math. Comput. Sci. 14 (2019), no. 1, 1--32. doi:10.2140/camcos.2019.14.1.

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  • A. S. Almgren, A fast adaptive vortex method using local corrections, Ph.D. thesis, University of California, Berkeley, 1991.
  • A. S. Almgren, T. Buttke, and P. Colella, A fast adaptive vortex method in three dimensions, J. Comput. Phys. 113 (1994), no. 2, 177–200.
  • C. R. Anderson, A method of local corrections for computing the velocity field due to a distribution of vortex blobs, J. Comput. Phys. 62 (1986), no. 1, 111–123.
  • G. T. Balls and P. Colella, A finite difference domain decomposition method using local corrections for the solution of Poisson's equation, J. Comput. Phys. 180 (2002), no. 1, 25–53.
  • G. T. Balls, A finite-difference domain decomposition method using local corrections for the solution of Poisson's equation, Ph.D. thesis, University of California, Berkeley, 1999.
  • J. Barnes and P. Hut, A hierarchical ${O(N \log N)}$ force-calculation algorithm, Nature 324 (1986), 446–449.
  • J. Carrier, L. Greengard, and V. Rokhlin, A fast adaptive multipole algorithm for particle simulations, SIAM J. Sci. Statist. Comput. 9 (1988), no. 4, 669–686.
  • L. Collatz, The numerical treatment of differential equations, 3rd ed., Die Grundlehren der mathematischen Wissenschaften, no. 60, Springer, 1960.
  • J. W. Eastwood and D. R. K. Brownrigg, Remarks on the solution of Poisson's equation for isolated systems, J. Comput. Phys. 32 (1979), no. 1, 24–38.
  • L. C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, no. 19, American Mathematical Society, 2010.
  • A. Gholami, D. Malhotra, H. Sundar, and G. Biros, FFT, FMM, or multigrid? A comparative study of state-of-the-art Poisson solvers for uniform and nonuniform grids in the unit cube, SIAM J. Sci. Comput. 38 (2016), no. 3, C280–C306.
  • D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer, 2001.
  • L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys. 73 (1987), no. 2, 325–348.
  • ––––, A new version of the fast multipole method for the Laplace equation in three dimensions, Acta Numer. 6 (1997), 229–269.
  • P. Henrici, Fast Fourier methods in computational complex analysis, SIAM Rev. 21 (1979), no. 4, 481–527.
  • R. Hockney, The potential calculation and some applications, Method. Comput. Phys. 9 (1970), 135–211.
  • R. A. James, The solution of Poisson's equation for isolated source distributions, J. Computational Phys. 25 (1977), no. 2, 71–93.
  • K. Lackner, Computation of ideal MHD equilibria, Comput. Phys. Commun. 12 (1976), no. 1, 33–44.
  • M. H. Langston, L. Greengard, and D. Zorin, A free-space adaptive FMM-based PDE solver in three dimensions, Commun. Appl. Math. Comput. Sci. 6 (2011), no. 1, 79–122.
  • S. Liska and T. Colonius, A parallel fast multipole method for elliptic difference equations, J. Comput. Phys. 278 (2014), 76–91.
  • D. Malhotra and G. Biros, PVFMM: a parallel kernel independent FMM for particle and volume potentials, Commun. Comput. Phys. 18 (2015), no. 3, 808–830.
  • ––––, Algorithm 967: a distributed-memory fast multipole method for volume potentials, ACM Trans. Math. Software 43 (2016), no. 2, 17.
  • A. Mayo, Fast high order accurate solution of Laplace's equation on irregular regions, SIAM J. Sci. Statist. Comput. 6 (1985), no. 1, 144–157.
  • P. McCorquodale, P. Colella, B. V. Straalen, and C. Kavouklis, High-performance implementations of the method of local corrections on parallel computers, preprint, 2018.
  • P. McCorquodale, P. Colella, G. T. Balls, and S. B. Baden, A local corrections algorithm for solving Poisson's equation in three dimensions, Commun. Appl. Math. Comput. Sci. 2 (2007), 57–81.
  • W. F. Spotz and G. F. Carey, A high-order compact formulation for the $3$D Poisson equation, Numer. Methods Partial Differential Equations 12 (1996), no. 2, 235–243.
  • F. Vico, L. Greengard, and M. Ferrando, Fast convolution with free-space Green's functions, J. Comput. Phys. 323 (2016), 191–203.