Efficient Operator-Coarsening Multigrid Schemes for Local Discontinuous Galerkin Methods
Abstract
An efficient hp-multigrid scheme is presented for local discontinuous Galerkin (LDG) discretizations of elliptic problems, formulated around the idea of separately coarsening the underlying discrete gradient and divergence operators. We show that traditional multigrid coarsening of the primal formulation leads to poor and suboptimal multigrid performance, whereas coarsening of the flux formulation leads to essentially optimal convergence and is equivalent to a purely geometric multigrid method. The resulting operator-coarsening schemes do not require the entire mesh hierarchy to be explicitly built, thereby obviating the need to compute quadrature rules, lifting operators, and other mesh-related quantities on coarse meshes. We show that good multigrid convergence rates are achieved in a variety of numerical tests on two-dimensional and three-dimensional uniform and adaptive Cartesian grids, as well as for curved domains using implicitly defined meshes and for multiphase elliptic interface problems with complex geometry. Extensions, e.g., to non-LDG discretizations and fully unstructured meshes, are briefly discussed.
- Authors:
-
- Harvard Univ., Cambridge, MA (United States)
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
- Publication Date:
- Research Org.:
- Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC)
- OSTI Identifier:
- 1580384
- Grant/Contract Number:
- AC02-05CH11231
- Resource Type:
- Accepted Manuscript
- Journal Name:
- SIAM Journal on Scientific Computing
- Additional Journal Information:
- Journal Volume: 41; Journal Issue: 6; Journal ID: ISSN 1064-8275
- Publisher:
- SIAM
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING
Citation Formats
Fortunato, Daniel, Rycroft, Chris H., and Saye, Robert. Efficient Operator-Coarsening Multigrid Schemes for Local Discontinuous Galerkin Methods. United States: N. p., 2019.
Web. doi:10.1137/18m1206357.
Fortunato, Daniel, Rycroft, Chris H., & Saye, Robert. Efficient Operator-Coarsening Multigrid Schemes for Local Discontinuous Galerkin Methods. United States. https://doi.org/10.1137/18m1206357
Fortunato, Daniel, Rycroft, Chris H., and Saye, Robert. Tue .
"Efficient Operator-Coarsening Multigrid Schemes for Local Discontinuous Galerkin Methods". United States. https://doi.org/10.1137/18m1206357. https://www.osti.gov/servlets/purl/1580384.
@article{osti_1580384,
title = {Efficient Operator-Coarsening Multigrid Schemes for Local Discontinuous Galerkin Methods},
author = {Fortunato, Daniel and Rycroft, Chris H. and Saye, Robert},
abstractNote = {An efficient hp-multigrid scheme is presented for local discontinuous Galerkin (LDG) discretizations of elliptic problems, formulated around the idea of separately coarsening the underlying discrete gradient and divergence operators. We show that traditional multigrid coarsening of the primal formulation leads to poor and suboptimal multigrid performance, whereas coarsening of the flux formulation leads to essentially optimal convergence and is equivalent to a purely geometric multigrid method. The resulting operator-coarsening schemes do not require the entire mesh hierarchy to be explicitly built, thereby obviating the need to compute quadrature rules, lifting operators, and other mesh-related quantities on coarse meshes. We show that good multigrid convergence rates are achieved in a variety of numerical tests on two-dimensional and three-dimensional uniform and adaptive Cartesian grids, as well as for curved domains using implicitly defined meshes and for multiphase elliptic interface problems with complex geometry. Extensions, e.g., to non-LDG discretizations and fully unstructured meshes, are briefly discussed.},
doi = {10.1137/18m1206357},
journal = {SIAM Journal on Scientific Computing},
number = 6,
volume = 41,
place = {United States},
year = {Tue Dec 17 00:00:00 EST 2019},
month = {Tue Dec 17 00:00:00 EST 2019}
}
Free Publicly Available Full Text
Publisher's Version of Record
Other availability
Search WorldCat to find libraries that may hold this journal
Cited by: 11 works
Citation information provided by
Web of Science
Web of Science
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.
Works referenced in this record:
Preconditioning Methods for Local Discontinuous Galerkin Discretizations
journal, January 2003
- Kanschat, Guido
- SIAM Journal on Scientific Computing, Vol. 25, Issue 3
Superconvergence of the Local Discontinuous Galerkin Method for Elliptic Problems on Cartesian Grids
journal, January 2001
- Cockburn, Bernardo; Kanschat, Guido; Perugia, Ilaria
- SIAM Journal on Numerical Analysis, Vol. 39, Issue 1
Manycore parallel computing for a hybridizable discontinuous Galerkin nested multigrid method
preprint, January 2017
- Fabien, M. S.; Knepley, M. G.; Mills, R. T.
- arXiv
Multilevel methods for discontinuous Galerkin FEM on locally refined meshes
journal, November 2004
- Kanschat, Guido
- Computers & Structures, Vol. 82, Issue 28
A multilevel discontinuous Galerkin method
journal, September 2003
- Gopalakrishnan, J.; Kanschat, G.
- Numerische Mathematik, Vol. 95, Issue 3
Implicit mesh discontinuous Galerkin methods and interfacial gauge methods for high-order accurate interface dynamics, with applications to surface tension dynamics, rigid body fluid–structure interaction, and free surface flow: Part II
journal, September 2017
- Saye, Robert
- Journal of Computational Physics, Vol. 344
A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier–Stokes Equations
journal, March 1997
- Bassi, F.; Rebay, S.
- Journal of Computational Physics, Vol. 131, Issue 2
Schwarz Methods for a Preconditioned WOPSIP Method for Elliptic Problems
journal, January 2012
- Antonietti, Paola F.; Ayuso de Dios, Blanca; Brenner, Susanne C.
- Computational Methods in Applied Mathematics, Vol. 12, Issue 3
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
Multigrid algorithms for $$\varvec{hp}$$ h p -version interior penalty discontinuous Galerkin methods on polygonal and polyhedral meshes
journal, May 2017
- Antonietti, P. F.; Houston, P.; Hu, X.
- Calcolo, Vol. 54, Issue 4
Fast Numerical Integration on Polytopic Meshes with Applications to Discontinuous Galerkin Finite Element Methods
journal, August 2018
- Antonietti, Paola F.; Houston, Paul; Pennesi, Giorgio
- Journal of Scientific Computing, Vol. 77, Issue 3
V-cycle Multigrid Algorithms for Discontinuous Galerkin Methods on Non-nested Polytopic Meshes
journal, July 2018
- Antonietti, P. F.; Pennesi, G.
- Journal of Scientific Computing, Vol. 78, Issue 1
An Interior Penalty Finite Element Method with Discontinuous Elements
journal, August 1982
- Arnold, Douglas N.
- SIAM Journal on Numerical Analysis, Vol. 19, Issue 4
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
High‐order accuratep‐multigrid discontinuous Galerkin solution of the Euler equations
journal, July 2009
- Bassi, F.; Ghidoni, A.; Rebay, S.
- International Journal for Numerical Methods in Fluids, Vol. 60, Issue 8
A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier–Stokes Equations
journal, March 1997
- Bassi, F.; Rebay, S.
- Journal of Computational Physics, Vol. 131, Issue 2
Algebraic multigrid for discontinuous Galerkin discretizations of heterogeneous elliptic problems: AMG4DG
journal, February 2012
- Bastian, Peter; Blatt, Markus; Scheichl, Robert
- Numerical Linear Algebra with Applications, Vol. 19, Issue 2
Convergence of Multigrid Algorithms for Interior Penalty Methods
journal, May 2005
- Brenner, Susanne C.; Zhao, Jie
- Applied Numerical Analysis & Computational Mathematics, Vol. 2, Issue 1
Mixed Discontinuous Galerkin Methods for Darcy Flow
journal, June 2005
- Brezzi, F.; Hughes, T. J. R.; Marini, L. D.
- Journal of Scientific Computing, Vol. 22-23, Issue 1-3
Discontinuous Galerkin approximations for elliptic problems
journal, January 2000
- Brezzi, F.; Manzini, G.; Marini, D.
- Numerical Methods for Partial Differential Equations, Vol. 16, Issue 4
An A Priori Error Analysis of the Local Discontinuous Galerkin Method for Elliptic Problems
journal, January 2000
- Castillo, Paul; Cockburn, Bernardo; Perugia, Ilaria
- SIAM Journal on Numerical Analysis, Vol. 38, Issue 5
An Analysis of the Minimal Dissipation Local Discontinuous Galerkin Method for Convection–Diffusion Problems
journal, March 2007
- Cockburn, Bernardo; Dong, Bo
- Journal of Scientific Computing, Vol. 32, Issue 2
Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems
journal, January 2009
- Cockburn, Bernardo; Gopalakrishnan, Jayadeep; Lazarov, Raytcho
- SIAM Journal on Numerical Analysis, Vol. 47, Issue 2
The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
journal, December 1998
- Cockburn, Bernardo; Shu, Chi-Wang
- SIAM Journal on Numerical Analysis, Vol. 35, Issue 6
Manycore Parallel Computing for a Hybridizable Discontinuous Galerkin Nested Multigrid Method
journal, January 2019
- Fabien, Maurice S.; Knepley, Matthew G.; Mills, Richard T.
- SIAM Journal on Scientific Computing, Vol. 41, Issue 2
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
A multilevel discontinuous Galerkin method
journal, September 2003
- Gopalakrishnan, J.; Kanschat, G.
- Numerische Mathematik, Vol. 95, Issue 3
Multilevel methods for discontinuous Galerkin FEM on locally refined meshes
journal, November 2004
- Kanschat, Guido
- Computers & Structures, Vol. 82, Issue 28
A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids
journal, January 2006
- Luo, Hong; Baum, Joseph D.; Löhner, Rainald
- Journal of Computational Physics, Vol. 211, Issue 2
Smoothed aggregation multigrid solvers for high-order discontinuous Galerkin methods for elliptic problems
journal, August 2011
- Olson, Luke N.; Schroder, Jacob B.
- Journal of Computational Physics, Vol. 230, Issue 18
The Compact Discontinuous Galerkin (CDG) Method for Elliptic Problems
journal, January 2008
- Peraire, J.; Persson, P. -O.
- SIAM Journal on Scientific Computing, Vol. 30, Issue 4
A sparse and high-order accurate line-based discontinuous Galerkin method for unstructured meshes
journal, January 2013
- Persson, Per-Olof
- Journal of Computational Physics, Vol. 233
Smoothed Aggregation Multigrid for the Discontinuous Galerkin Method
journal, January 2009
- Prill, F.; Lukáčová-Medviďová, M.; Hartmann, R.
- SIAM Journal on Scientific Computing, Vol. 31, Issue 5
High-Order Quadrature Methods for Implicitly Defined Surfaces and Volumes in Hyperrectangles
journal, January 2015
- Saye, R. I.
- SIAM Journal on Scientific Computing, Vol. 37, Issue 2
Implicit mesh discontinuous Galerkin methods and interfacial gauge methods for high-order accurate interface dynamics, with applications to surface tension dynamics, rigid body fluid–structure interaction, and free surface flow: Part I
journal, September 2017
- Saye, Robert
- Journal of Computational Physics, Vol. 344
Implicit mesh discontinuous Galerkin methods and interfacial gauge methods for high-order accurate interface dynamics, with applications to surface tension dynamics, rigid body fluid–structure interaction, and free surface flow: Part II
journal, September 2017
- Saye, Robert
- Journal of Computational Physics, Vol. 344
Algebraic multigrid techniques for discontinuous Galerkin methods with varying polynomial order
journal, May 2014
- Siefert, C.; Tuminaro, R.; Gerstenberger, A.
- Computational Geosciences, Vol. 18, Issue 5
Robust multigrid for high-order discontinuous Galerkin methods: A fast Poisson solver suitable for high-aspect ratio Cartesian grids
journal, December 2016
- Stiller, Jörg
- Journal of Computational Physics, Vol. 327
An Adaptive Level Set Approach for Incompressible Two-Phase Flows
journal, January 1999
- Sussman, Mark; Almgren, Ann S.; Bell, John B.
- Journal of Computational Physics, Vol. 148, Issue 1
Iterative Methods by Space Decomposition and Subspace Correction
journal, December 1992
- Xu, Jinchao
- SIAM Review, Vol. 34, Issue 4
Works referencing / citing this record:
The ultraspherical spectral element method
text, January 2020
- Fortunato, Daniel; Hale, Nicholas; Townsend, Alex
- arXiv