A Scalable Preconditioner for a Primal Discontinuous Petrov--Galerkin Method

Barker, A. T.; Dobrev, V.; Gopalakrishnan, J.; Kolev, T.

doi:10.1137/16m1104780

A Scalable Preconditioner for a Primal Discontinuous Petrov--Galerkin Method

Journal Article · Tue Apr 24 00:00:00 EDT 2018 · SIAM Journal on Scientific Computing

DOI:https://doi.org/10.1137/16m1104780· OSTI ID:1734606

Barker, A. T. ^[1]; Dobrev, V. ^[1]; ^[2]; Kolev, T. ^[1]

Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Center for Applied Scientific Computing
Portland State Univ., OR (United States)

In this work, we show how a scalable preconditioner for the primal discontinuous Petrov--Galerkin (DPG) method can be developed using existing algebraic multigrid (AMG) preconditioning techniques. The stability of the DPG method gives a norm equivalence which allows us to exploit existing AMG algorithms and software. We show how these algebraic preconditioners can be applied directly to a Schur complement system arising from the DPG method. One of our intermediate results shows that a generic stable decomposition implies a stable decomposition for the Schur complement. This justifies the application of algebraic solvers directly to the interface degrees of freedom. Combining such results, we obtain the first massively scalable algebraic preconditioner for the DPG system.

View Accepted Manuscript (DOE)

Research Organization:: Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA); US Air Force Office of Scientific Research (AFOSR); US Army Research Office (ARO)

Grant/Contract Number:: AC52-07NA27344

OSTI ID:: 1734606

Report Number(s):: LLNL-JRNL--710378; 848304

Journal Information:: SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 2 Vol. 40; ISSN 1064-8275

Publisher:: SIAMCopyright Statement

Country of Publication:: United States

Language:: English

References (22)

Explicit polynomial preserving trace liftings on a triangle: Explicit trace liftings Ainsworth, Mark; Demkowicz, Leszek Mathematische Nachrichten, Vol. 282, Issue 5 https://doi.org/10.1002/mana.200610762	journal	April 2009
A class of discontinuous Petrov-Galerkin methods. II. Optimal test functions Demkowicz, L.; Gopalakrishnan, J. Numerical Methods for Partial Differential Equations, Vol. 27, Issue 1 https://doi.org/10.1002/num.20640	journal	October 2010
BoomerAMG: A parallel algebraic multigrid solver and preconditioner Henson, Van Emden; Yang, Ulrike Meier Applied Numerical Mathematics, Vol. 41, Issue 1 https://doi.org/10.1016/S0168-9274(01)00115-5	journal	April 2002
A class of discontinuous Petrov–Galerkin methods. Part III: Adaptivity Demkowicz, Leszek; Gopalakrishnan, Jay; Niemi, Antti H. Applied Numerical Mathematics, Vol. 62, Issue 4 https://doi.org/10.1016/j.apnum.2011.09.002	journal	April 2012
A robust DPG method for convection-dominated diffusion problems II: Adjoint boundary conditions and mesh-dependent test norms Chan, Jesse; Heuer, Norbert; Bui-Thanh, Tan Computers & Mathematics with Applications, Vol. 67, Issue 4 https://doi.org/10.1016/j.camwa.2013.06.010	journal	March 2014
A primal DPG method without a first-order reformulation Demkowicz, L.; Gopalakrishnan, J. Computers & Mathematics with Applications, Vol. 66, Issue 6 https://doi.org/10.1016/j.camwa.2013.06.029	journal	October 2013
Analysis of the discontinuous Petrov–Galerkin method with optimal test functions for the Reissner–Mindlin plate bending model Calo, Victor M.; Collier, Nathaniel O.; Niemi, Antti H. Computers & Mathematics with Applications, Vol. 66, Issue 12 https://doi.org/10.1016/j.camwa.2013.07.012	journal	January 2014
Convergence rates of the DPG method with reduced test space degree Bouma, Timaeus; Gopalakrishnan, Jay; Harb, Ammar Computers & Mathematics with Applications, Vol. 68, Issue 11 https://doi.org/10.1016/j.camwa.2014.08.004	journal	December 2014
Breaking spaces and forms for the DPG method and applications including Maxwell equations Carstensen, C.; Demkowicz, L.; Gopalakrishnan, J. Computers & Mathematics with Applications, Vol. 72, Issue 3 https://doi.org/10.1016/j.camwa.2016.05.004	journal	August 2016
A geometric multigrid preconditioning strategy for DPG system matrices Roberts, Nathan V.; Chan, Jesse Computers & Mathematics with Applications, Vol. 74, Issue 8 https://doi.org/10.1016/j.camwa.2017.06.055	journal	October 2017
A class of discontinuous Petrov–Galerkin methods. Part I: The transport equation Demkowicz, L.; Gopalakrishnan, J. Computer Methods in Applied Mechanics and Engineering, Vol. 199, Issue 23-24 https://doi.org/10.1016/j.cma.2010.01.003	journal	April 2010
A class of discontinuous Petrov–Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1D Zitelli, J.; Muga, I.; Demkowicz, L. Journal of Computational Physics, Vol. 230, Issue 7 https://doi.org/10.1016/j.jcp.2010.12.001	journal	April 2011
Robust operator estimates and the application to substructuring methods for first-order systems Wieners, Christian; Wohlmuth, Barbara ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 48, Issue 5 https://doi.org/10.1051/m2an/2014006	journal	August 2014
Domain decomposition preconditioners for the discontinuous Petrov–Galerkin method Li, Xiang; Xu, Xuejun ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 51, Issue 3 https://doi.org/10.1051/m2an/2016050	journal	May 2017
Polynomial extension operators for $H^1$, $\boldsymbol {\mathit {H}}(\mathbf {curl})$ and $\boldsymbol {\mathit {H}}(\mathbf {div})$-spaces on a cube Costabel, M.; Dauge, M.; Demkowicz, L. Mathematics of Computation, Vol. 77, Issue 264 https://doi.org/10.1090/S0025-5718-08-02108-X	journal	April 2008
Polynomial extension operators. Part III Demkowicz, L.; Gopalakrishnan, J.; Schöberl, J. Mathematics of Computation, Vol. 81, Issue 279 https://doi.org/10.1090/S0025-5718-2011-02536-6	journal	September 2012
An analysis of the practical DPG method Gopalakrishnan, J.; Qiu, W. Mathematics of Computation, Vol. 83, Issue 286 https://doi.org/10.1090/S0025-5718-2013-02721-4	journal	May 2013
Nodal Auxiliary Space Preconditioning in H(curl) and H(div) Spaces Hiptmair, Ralf; Xu, Jinchao SIAM Journal on Numerical Analysis, Vol. 45, Issue 6 https://doi.org/10.1137/060660588	journal	January 2007
Optimal Finite-Element Interpolation on Curved Domains Bernardi, Christine SIAM Journal on Numerical Analysis, Vol. 26, Issue 5 https://doi.org/10.1137/0726068	journal	October 1989
Algebraic Multigrid for Linear Systems Obtained by Explicit Element Reduction Brunner, Thomas A.; Kolev, Tzanio V. SIAM Journal on Scientific Computing, Vol. 33, Issue 5 https://doi.org/10.1137/100801640	journal	January 2011
Parallel Auxiliary Space AMG Solver for $H(div)$ Problems Kolev, Tzanio V.; Vassilevski, Panayot S. SIAM Journal on Scientific Computing, Vol. 34, Issue 6 https://doi.org/10.1137/110859361	journal	January 2012
Parallel Auxiliary Space AMG for H(Curl) Problems Vassilevski, Tzanio V. Kolev Panayot S. Journal of Computational Mathematics, Vol. 27, Issue 5 https://doi.org/10.4208/jcm.2009.27.5.013	journal	June 2009

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A Scalable Preconditioner for a Primal Discontinuous Petrov--Galerkin Method

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