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Title: Efficient Low-Order Refined Preconditioners for High-Order Matrix-Free Continuous and Discontinuous Galerkin Methods

Abstract

In this paper, we design preconditioners for the matrix-free solution of high-order continuous and discontinuous Galerkin discretizations of elliptic problems based on finite element method--spectral element method (FEM-SEM) equivalence and additive Schwarz methods. The high-order operators are applied without forming the system matrix, making use of sum factorization for efficient evaluation. The system is preconditioned using a spectrally equivalent low-order (p = 1) finite element operator discretization on a refined mesh. The low-order refined mesh is anisotropic and not shape regular in the polynomial degree of the high-order operator, requiring specialized solvers to treat the anisotropy. We make use of an element-structured, geometric multigrid V-cycle with ordered ILU(0) smoothing. The preconditioner is parallelized through an overlapping additive Schwarz method that is robust in h and p. The method is extended to interior penalty and Bassi and Rebay (BR2) discontinuous Galerkin discretizations, for which it is also robust in the size of the penalty parameter. Finally, numerical results are presented on a variety of examples, verifying the uniformity of the preconditioner.

Authors:
ORCiD logo [1]
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  1. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Publication Date:
Research Org.:
Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1768376
Alternate Identifier(s):
OSTI ID: 1769114
Report Number(s):
LLNL-JRNL-787240
Journal ID: ISSN 1064-8275; 984351
Grant/Contract Number:  
AC52-07NA27344
Resource Type:
Accepted Manuscript
Journal Name:
SIAM Journal on Scientific Computing
Additional Journal Information:
Journal Volume: 42; Journal Issue: 5; Journal ID: ISSN 1064-8275
Publisher:
SIAM
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; preconditioning; matrix-free; discontinuous Galerkin

Citation Formats

Pazner, Will E. Efficient Low-Order Refined Preconditioners for High-Order Matrix-Free Continuous and Discontinuous Galerkin Methods. United States: N. p., 2020. Web. doi:10.1137/19m1282052.
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Pazner, Will E. Efficient Low-Order Refined Preconditioners for High-Order Matrix-Free Continuous and Discontinuous Galerkin Methods. United States. https://doi.org/10.1137/19m1282052
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Pazner, Will E. Mon . "Efficient Low-Order Refined Preconditioners for High-Order Matrix-Free Continuous and Discontinuous Galerkin Methods". United States. https://doi.org/10.1137/19m1282052. https://www.osti.gov/servlets/purl/1768376.
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@article{osti_1768376,
title = {Efficient Low-Order Refined Preconditioners for High-Order Matrix-Free Continuous and Discontinuous Galerkin Methods},
author = {Pazner, Will E.},
abstractNote = {In this paper, we design preconditioners for the matrix-free solution of high-order continuous and discontinuous Galerkin discretizations of elliptic problems based on finite element method--spectral element method (FEM-SEM) equivalence and additive Schwarz methods. The high-order operators are applied without forming the system matrix, making use of sum factorization for efficient evaluation. The system is preconditioned using a spectrally equivalent low-order (p = 1) finite element operator discretization on a refined mesh. The low-order refined mesh is anisotropic and not shape regular in the polynomial degree of the high-order operator, requiring specialized solvers to treat the anisotropy. We make use of an element-structured, geometric multigrid V-cycle with ordered ILU(0) smoothing. The preconditioner is parallelized through an overlapping additive Schwarz method that is robust in h and p. The method is extended to interior penalty and Bassi and Rebay (BR2) discontinuous Galerkin discretizations, for which it is also robust in the size of the penalty parameter. Finally, numerical results are presented on a variety of examples, verifying the uniformity of the preconditioner.},
doi = {10.1137/19m1282052},
journal = {SIAM Journal on Scientific Computing},
number = 5,
volume = 42,
place = {United States},
year = {Mon Oct 05 00:00:00 EDT 2020},
month = {Mon Oct 05 00:00:00 EDT 2020}
}
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Works referenced in this record:

Spectral methods for problems in complex geometries
journal, August 1980

Efficient Operator-Coarsening Multigrid Schemes for Local Discontinuous Galerkin Methods
journal, January 2019

  • Fortunato, Daniel; Rycroft, Chris H.; Saye, Robert
  • SIAM Journal on Scientific Computing, Vol. 41, Issue 6
  • DOI: 10.1137/18M1206357

Fully discrete hp-finite elements: fast quadrature
journal, May 2001

  • Melenk, J. M.; Gerdes, K.; Schwab, C.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 190, Issue 32-33
  • DOI: 10.1016/S0045-7825(00)00322-4

p-Multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier–Stokes equations
journal, July 2005

  • Fidkowski, Krzysztof J.; Oliver, Todd A.; Lu, James
  • Journal of Computational Physics, Vol. 207, Issue 1
  • DOI: 10.1016/j.jcp.2005.01.005

Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
journal, January 2002

  • Arnold, Douglas N.; Brezzi, Franco; Cockburn, Bernardo
  • SIAM Journal on Numerical Analysis, Vol. 39, Issue 5
  • DOI: 10.1137/S0036142901384162

Multigrid Strategies for Viscous Flow Solvers on Anisotropic Unstructured Meshes
journal, September 1998

Multigrid Algorithms for $hp$-Discontinuous Galerkin Discretizations of Elliptic Problems
journal, January 2015

  • Antonietti, Paola F.; Sarti, Marco; Verani, Marco
  • SIAM Journal on Numerical Analysis, Vol. 53, Issue 1
  • DOI: 10.1137/130947015

Semi-automatic sparse preconditioners for high-order finite element methods on non-uniform meshes
journal, May 2012

  • Austin, Travis M.; Brezina, Marian; Jamroz, Ben
  • Journal of Computational Physics, Vol. 231, Issue 14
  • DOI: 10.1016/j.jcp.2012.03.013

Towards a cost-effective ILU preconditioner with high level fill
journal, September 1992

  • D'Azevedo, E. F.; Forsyth, P. A.; Tang, Wei-Pai
  • BIT, Vol. 32, Issue 3
  • DOI: 10.1007/BF02074880

BoomerAMG: A parallel algebraic multigrid solver and preconditioner
journal, April 2002

Nodal discontinuous Galerkin methods on graphics processors
journal, November 2009

  • Klöckner, A.; Warburton, T.; Bridge, J.
  • Journal of Computational Physics, Vol. 228, Issue 21
  • DOI: 10.1016/j.jcp.2009.06.041

A Uniform Additive Schwarz Preconditioner for High-Order Discontinuous Galerkin Approximations of Elliptic Problems
journal, August 2016

  • Antonietti, Paola F.; Sarti, Marco; Verani, Marco
  • Journal of Scientific Computing, Vol. 70, Issue 2
  • DOI: 10.1007/s10915-016-0259-9

Scalable Low-Order Finite Element Preconditioners for High-Order Spectral Element Poisson Solvers
journal, January 2019

  • Bello-Maldonado, Pedro D.; Fischer, Paul F.
  • SIAM Journal on Scientific Computing, Vol. 41, Issue 5
  • DOI: 10.1137/18M1194997

Iterative Methods by Space Decomposition and Subspace Correction
journal, December 1992

On the utility of GPU accelerated high-order methods for unsteady flow simulations: A comparison with industry-standard tools
journal, April 2017

  • Vermeire, B. C.; Witherden, F. D.; Vincent, P. E.
  • Journal of Computational Physics, Vol. 334
  • DOI: 10.1016/j.jcp.2016.12.049

Multigrid for Matrix-Free High-Order Finite Element Computations on Graphics Processors
journal, May 2019

  • Kronbichler, Martin; Ljungkvist, Karl
  • ACM Transactions on Parallel Computing, Vol. 6, Issue 1
  • DOI: 10.1145/3322813

Smoothed aggregation multigrid solvers for high-order discontinuous Galerkin methods for elliptic problems
journal, August 2011

Efficient Nonlinear Solvers for Nodal High-Order Finite Elements in 3D
journal, July 2010

Additive Schwarz methods for thep-version finite element method
journal, December 1993

An Approximate Minimum Degree Ordering Algorithm
journal, October 1996

  • Amestoy, Patrick R.; Davis, Timothy A.; Duff, Iain S.
  • SIAM Journal on Matrix Analysis and Applications, Vol. 17, Issue 4
  • DOI: 10.1137/S0895479894278952

An explicit expression for the penalty parameter of the interior penalty method
journal, May 2005

A finite element multigrid preconditioner for Chebyshev–collocation methods
journal, May 2000

Finite-Element Preconditioning for Pseudospectral Solutions of Elliptic Problems
journal, March 1990

  • Deville, M. O.; Mund, E. H.
  • SIAM Journal on Scientific and Statistical Computing, Vol. 11, Issue 2
  • DOI: 10.1137/0911019

Ordering Methods for Preconditioned Conjugate Gradient Methods Applied to Unstructured Grid Problems
journal, July 1992

  • D’Azevedo, E. F.; Forsyth, P. A.; Tang, Wei-Pai
  • SIAM Journal on Matrix Analysis and Applications, Vol. 13, Issue 3
  • DOI: 10.1137/0613057

Nested dyadic grids associated with Legendre–Gauss–Lobatto grids
journal, December 2014

A Class of Domain Decomposition Preconditioners for hp-Discontinuous Galerkin Finite Element Methods
journal, June 2010

An Overlapping Schwarz Method for Spectral Element Solution of the Incompressible Navier–Stokes Equations
journal, May 1997

Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods
journal, February 2018

An Interior Penalty Finite Element Method with Discontinuous Elements
journal, August 1982

  • Arnold, Douglas N.
  • SIAM Journal on Numerical Analysis, Vol. 19, Issue 4
  • DOI: 10.1137/0719052

Continuous interior penalty $hp$-finite element methods for advection and advection-diffusion equations
journal, January 2007

Discontinuous Galerkin approximations for elliptic problems
journal, January 2000

Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method
journal, July 2005

High-order CFD methods: current status and perspective: HIGH-ORDER CFD METHODS
journal, January 2013

  • Wang, Z. J.; Fidkowski, Krzysztof; Abgrall, Rémi
  • International Journal for Numerical Methods in Fluids, Vol. 72, Issue 8
  • DOI: 10.1002/fld.3767

Line relaxation for spectral multigrid methods
journal, July 1988

Newton-GMRES Preconditioning for Discontinuous Galerkin Discretizations of the Navier–Stokes Equations
journal, January 2008

  • Persson, P. -O.; Peraire, J.
  • SIAM Journal on Scientific Computing, Vol. 30, Issue 6
  • DOI: 10.1137/070692108

Approximation results for orthogonal polynomials in Sobolev spaces
journal, January 1982

Application of p-Multigrid to Discontinuous Galerkin Formulations of the Poisson Equation
journal, March 2006

  • Helenbrook, Brian T.; Atkins, H. L.
  • AIAA Journal, Vol. 44, Issue 3
  • DOI: 10.2514/1.15497

Finite-Element Preconditioning of G-NI Spectral Methods
journal, January 2010

  • Canuto, Claudio; Gervasio, Paola; Quarteroni, Alfio
  • SIAM Journal on Scientific Computing, Vol. 31, Issue 6
  • DOI: 10.1137/090746367

Preconditioned minimal residual methods for chebyshev spectral calculations
journal, September 1985

Parallel multigrid smoothing: polynomial versus Gauss–Seidel
journal, July 2003

On computing upper and lower bounds on the outputs of linear elasticity problems approximated by the smoothed finite element method
journal, January 2010

  • Xuan, Z. C.; Lassila, T.; Rozza, G.
  • International Journal for Numerical Methods in Engineering
  • DOI: 10.1002/nme.2825

Line relaxation for spectral multigrid methods
journal, July 1988

An explicit expression for the penalty parameter of the interior penalty method
journal, May 2005

Stabilization of spectral methods by finite element bubble functions
journal, January 1994

Finite-Element Preconditioning for Pseudospectral Solutions of Elliptic Problems
journal, March 1990

  • Deville, M. O.; Mund, E. H.
  • SIAM Journal on Scientific and Statistical Computing, Vol. 11, Issue 2
  • DOI: 10.1137/0911019

Low-Order Preconditioning of High-Order Triangular Finite Elements
journal, January 2018

  • Chalmers, Noel; Warburton, T.
  • SIAM Journal on Scientific Computing, Vol. 40, Issue 6
  • DOI: 10.1137/17m1149444

Interior penalty tensor-product preconditioners for high-order discontinuous Galerkin discretizations
conference, January 2018

  • Pazner, Will; Persson, Per-Olof
  • 2018 AIAA Aerospace Sciences Meeting
  • DOI: 10.2514/6.2018-1093

High-performance Implementation of Matrix-free High-order Discontinuous Galerkin Methods
preprint, January 2017

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    High-order matrix-free incompressible flow solvers with GPU acceleration and low-order refined preconditioners
    journal, May 2020

    Fast Solution of Fully Implicit Runge--Kutta and Discontinuous Galerkin in Time for Numerical PDEs, Part I: the Linear Setting
    journal, February 2022

    • Southworth, Ben S.; Krzysik, Oliver; Pazner, Will
    • SIAM Journal on Scientific Computing, Vol. 44, Issue 1
    • DOI: 10.1137/21m1389742

    The ultraspherical spectral element method
    text, January 2020

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