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Title: An algebraic multigrid method for Q2-Q1 mixed discretizations of the Navier-Stokes equations

Abstract

Algebraic multigrid (AMG) preconditioners are considered for discretized systems of partial differential equations (PDEs) where unknowns associated with different physical quantities are not necessarily colocated at mesh points. Specifically, we investigate a Q2-Q1 mixed finite element discretization of the incompressible Navier–Stokes equations where the number of velocity nodes is much greater than the number of pressure nodes. Consequently, some velocity degrees of freedom (DOFs) are defined at spatial locations where there are no corresponding pressure DOFs. Thus, AMG approaches leveraging this colocated structure are not applicable. This article instead proposes an automatic AMG coarsening that mimics certain pressure/velocity DOF relationships of the Q2-Q1 discretization. The main idea is to first automatically define coarse pressures in a somewhat standard AMG fashion and then to carefully (but automatically) choose coarse velocity unknowns so that the spatial location relationship between pressure and velocity DOFs resembles that on the finest grid. To define coefficients within the intergrid transfers, an energy minimization AMG (EMIN-AMG) is utilized. EMIN-AMG is not tied to specific coarsening schemes and grid transfer sparsity patterns, and so it is applicable to the proposed coarsening. Numerical results highlighting solver performance are given on Stokes and incompressible Navier–Stokes problems.

Authors:
 [1];  [2]
+ Show Author Affiliations
  1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). Center for Computing Research
  2. Sandia National Lab. (SNL-CA), Livermore, CA (United States). Center for Computing Research
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Sandia National Lab. (SNL-CA), Livermore, CA (United States); Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
OSTI Identifier:
1399874
Alternate Identifier(s):
OSTI ID: 1394531
Report Number(s):
SAND-2016-6518J
Journal ID: ISSN 1099-1506; 644837; TRN: US1703220
Grant/Contract Number:  
AC04-94AL85000; AC05-00OR22725
Resource Type:
Accepted Manuscript
Journal Name:
Numerical Linear Algebra with Applications (Online)
Additional Journal Information:
Journal Name: Numerical Linear Algebra with Applications (Online); Journal Volume: 24; Journal Issue: 6; Journal ID: ISSN 1099-1506
Publisher:
Wiley
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; algebraic multigrid; mixed finite element discretizations; Navier–Stokes equations; preconditioning

Citation Formats

Prokopenko, Andrey, and Tuminaro, Raymond S. An algebraic multigrid method for Q2-Q1 mixed discretizations of the Navier-Stokes equations. United States: N. p., 2016. Web. doi:10.1002/nla.2109.
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Prokopenko, Andrey, & Tuminaro, Raymond S. An algebraic multigrid method for Q2-Q1 mixed discretizations of the Navier-Stokes equations. United States. https://doi.org/10.1002/nla.2109
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Prokopenko, Andrey, and Tuminaro, Raymond S. Fri . "An algebraic multigrid method for Q2-Q1 mixed discretizations of the Navier-Stokes equations". United States. https://doi.org/10.1002/nla.2109. https://www.osti.gov/servlets/purl/1399874.
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@article{osti_1399874,
title = {An algebraic multigrid method for Q2-Q1 mixed discretizations of the Navier-Stokes equations},
author = {Prokopenko, Andrey and Tuminaro, Raymond S.},
abstractNote = {Algebraic multigrid (AMG) preconditioners are considered for discretized systems of partial differential equations (PDEs) where unknowns associated with different physical quantities are not necessarily colocated at mesh points. Specifically, we investigate a Q2-Q1 mixed finite element discretization of the incompressible Navier–Stokes equations where the number of velocity nodes is much greater than the number of pressure nodes. Consequently, some velocity degrees of freedom (DOFs) are defined at spatial locations where there are no corresponding pressure DOFs. Thus, AMG approaches leveraging this colocated structure are not applicable. This article instead proposes an automatic AMG coarsening that mimics certain pressure/velocity DOF relationships of the Q2-Q1 discretization. The main idea is to first automatically define coarse pressures in a somewhat standard AMG fashion and then to carefully (but automatically) choose coarse velocity unknowns so that the spatial location relationship between pressure and velocity DOFs resembles that on the finest grid. To define coefficients within the intergrid transfers, an energy minimization AMG (EMIN-AMG) is utilized. EMIN-AMG is not tied to specific coarsening schemes and grid transfer sparsity patterns, and so it is applicable to the proposed coarsening. Numerical results highlighting solver performance are given on Stokes and incompressible Navier–Stokes problems.},
doi = {10.1002/nla.2109},
journal = {Numerical Linear Algebra with Applications (Online)},
number = 6,
volume = 24,
place = {United States},
year = {Fri Jul 01 00:00:00 EDT 2016},
month = {Fri Jul 01 00:00:00 EDT 2016}
}
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https://doi.org/10.1002/nla.2109
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Figures / Tables:

Figure 1 Figure 1: Q2 –Q1 pressure (left) and velocity (right) elements.
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Works referenced in this record:

Monolithic Multigrid Methods for Two-Dimensional Resistive Magnetohydrodynamics
journal, January 2016

  • Adler, James H.; Benson, Thomas R.; Cyr, Eric C.
  • SIAM Journal on Scientific Computing, Vol. 38, Issue 1
  • DOI: 10.1137/151006135

A Multigrid Tutorial, Second Edition
book, January 2000

  • Briggs, William L.; Henson, Van Emden; McCormick, Steve F.
  • Other Titles in Applied Mathematics
  • DOI: 10.1137/1.9780898719505

Local Fourier analysis for multigrid with overlapping smoothers applied to systems of PDEs
journal, January 2011

  • MacLachlan, Scott P.; Oosterlee, C. W.
  • Numerical Linear Algebra with Applications, Vol. 18, Issue 4
  • DOI: 10.1002/nla.762

Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems
journal, September 1996

A comparative study of efficient iterative solvers for generalized Stokes equations
journal, January 2008

  • Larin, Maxim; Reusken, Arnold
  • Numerical Linear Algebra with Applications, Vol. 15, Issue 1
  • DOI: 10.1002/nla.561

The Construction of the Coarse de Rham Complexes with Improved Approximation Properties
journal, January 2014

  • Lashuk, Ilya V.; Vassilevski, Panayot S.
  • Computational Methods in Applied Mathematics, Vol. 14, Issue 2
  • DOI: 10.1515/cmam-2014-0004

A New Approximate Block Factorization Preconditioner for Two-Dimensional Incompressible (Reduced) Resistive MHD
journal, January 2013

  • Cyr, Eric C.; Shadid, John N.; Tuminaro, Raymond S.
  • SIAM Journal on Scientific Computing, Vol. 35, Issue 3
  • DOI: 10.1137/12088879X

Bootstrap AMG
journal, January 2011

  • Brandt, A.; Brannick, J.; Kahl, K.
  • SIAM Journal on Scientific Computing, Vol. 33, Issue 2
  • DOI: 10.1137/090752973

Truly monolithic algebraic multigrid for fluid-structure interaction
journal, September 2010

  • Gee, M. W.; Küttler, U.; Wall, W. A.
  • International Journal for Numerical Methods in Engineering, Vol. 85, Issue 8
  • DOI: 10.1002/nme.3001

Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations
journal, September 1982

  • Brooks, Alexander N.; Hughes, Thomas J. R.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 32, Issue 1-3
  • DOI: 10.1016/0045-7825(82)90071-8

Energy Optimization of Algebraic Multigrid Bases
journal, June 1999

Element agglomeration coarse Raviart-Thomas spaces with improved approximation properties: COARSE RAVIART-THOMAS SPACES WITH IMPROVED APPROXIMATION PROPERTIES
journal, January 2012

  • Lashuk, I. V.; Vassilevski, P. S.
  • Numerical Linear Algebra with Applications, Vol. 19, Issue 2
  • DOI: 10.1002/nla.1819

Algebraic Multigrid Methods Based on Compatible Relaxation and Energy Minimization
book, January 2007

Block-implicit multigrid solution of Navier-Stokes equations in primitive variables
journal, July 1986

Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow
journal, June 2007

  • Elman, Howard C.; Ramage, Alison; Silvester, David J.
  • ACM Transactions on Mathematical Software, Vol. 33, Issue 2
  • DOI: 10.1145/1236463.1236469

Mixed Finite Element Methods and Applications
book, January 2013

A numerical solution of the Navier-Stokes equations using the finite element technique
journal, January 1973

On an energy minimizing basis for algebraic multigrid methods
journal, October 2004

An Energy-minimizing Interpolation for Robust Multigrid Methods
journal, January 1999

  • Wan, W. L.; Chan, Tony F.; Smith, Barry
  • SIAM Journal on Scientific Computing, Vol. 21, Issue 4
  • DOI: 10.1137/S1064827598334277

Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface
journal, January 1965

  • Harlow, Francis H.; Welch, J. Eddie
  • Physics of Fluids, Vol. 8, Issue 12
  • DOI: 10.1063/1.1761178

A General Interpolation Strategy for Algebraic Multigrid Using Energy Minimization
journal, January 2011

  • Olson, Luke N.; Schroder, Jacob B.; Tuminaro, Raymond S.
  • SIAM Journal on Scientific Computing, Vol. 33, Issue 2
  • DOI: 10.1137/100803031

AMGe---Coarsening Strategies and Application to the Oseen Equations
journal, January 2006

  • Wabro, Markus
  • SIAM Journal on Scientific Computing, Vol. 27, Issue 6
  • DOI: 10.1137/040610350

Algebraic multigrid methods for constrained linear systems with applications to contact problems in solid mechanics
journal, March 2004

  • Adams, Mark F.
  • Numerical Linear Algebra with Applications, Vol. 11, Issue 23
  • DOI: 10.1002/nla.374

AMG by element agglomeration and constrained energy minimization interpolation
journal, January 2006

  • Kolev, Tzanio V.; Vassilevski, Panayot S.
  • Numerical Linear Algebra with Applications, Vol. 13, Issue 9
  • DOI: 10.1002/nla.494

An efficient smoother for the Stokes problem
journal, February 1997

A comparison of preconditioners for incompressible Navier-Stokes solvers
journal, August 2008

  • ur Rehman, M.; Vuik, C.; Segal, G.
  • International Journal for Numerical Methods in Fluids, Vol. 57, Issue 12
  • DOI: 10.1002/fld.1684

Numerical simulation of fluid–structure interaction problems on hybrid meshes with algebraic multigrid methods
journal, July 2011

  • Yang, Huidong; Zulehner, Walter
  • Journal of Computational and Applied Mathematics, Vol. 235, Issue 18
  • DOI: 10.1016/j.cam.2011.05.046

Numerical performance of smoothers in coupled multigrid methods for the parallel solution of the incompressible Navier-Stokes equations
journal, January 2000

Multigrid transfers for nonsymmetric systems based on Schur complements and Galerkin projections: SCHUR COMPLEMENT-BASED MULTIGRID TRANSFERS
journal, June 2013

  • Wiesner, T. A.; Tuminaro, R. S.; Wall, W. A.
  • Numerical Linear Algebra with Applications, Vol. 21, Issue 3
  • DOI: 10.1002/nla.1889

An ILU preconditioner with coupled node fill-in for iterative solution of the mixed finite element formulation of the 2D and 3D Navier-Stokes equations
journal, September 1992

  • Dahl, O.; Wille, S. Ø
  • International Journal for Numerical Methods in Fluids, Vol. 15, Issue 5
  • DOI: 10.1002/fld.1650150503

General Constrained Energy Minimization Interpolation Mappings for AMG
journal, January 2010

  • Vassilevski, Panayot S.
  • SIAM Journal on Scientific Computing, Vol. 32, Issue 1
  • DOI: 10.1137/080726252

Coupled algebraic multigrid methods for the Oseen problem
journal, October 2004

Algebraic Multigrid for Selected PDE Systems
conference, August 2002

On the Algebraic Construction of Multilevel Transfer Operators
journal, July 2000

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    Figure 7 (p. 16)figure
    Table 1 (p. 17)table
    Figure 8 (p. 18)figure
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