skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Block Preconditioners for Stable Mixed Nodal and Edge finite element Representations of Incompressible Resistive MHD

Abstract

The scalable iterative solution of strongly coupled three-dimensional incompressible resistive magnetohydrodynamics (MHD) equations is quite challenging because disparate time scales arise from the electromagnetics, the hydrodynamics, as well as the coupling between these systems. This study considers a mixed finite element discretization of a dual saddle point formulation of the incompressible resistive MHD equations using a stable nodal (Q2/Q1) discretization for the hydrodynamics and a stable edge-node discretization of a reduced form of the Maxwell equations. This paper introduces new approximate block factorization preconditioners for this system which reduce the system to approximate Schur complement systems that can be solved using algebraic multilevel methods. These preconditioners include a new augmentation-based approximation for the magnetic induction saddle point system as well as efficient approximations of the Schur complements that arise from the complex coupling between the Navier--Stokes equations and the Maxwell equations.

Authors:
 [1];  [1];  [1];  [2];  [1]
  1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
  2. Univ. of Maryland, College Park, MD (United States)
Publication Date:
Research Org.:
Univ. of Maryland, College Park, MD (United States); Lockheed Martin Corporation, Littleton, CO (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
OSTI Identifier:
1598347
Grant/Contract Number:  
SC0009301; AC04-94AL85000
Resource Type:
Accepted Manuscript
Journal Name:
SIAM Journal on Scientific Computing
Additional Journal Information:
Journal Volume: 38; Journal Issue: 6; Journal ID: ISSN 1064-8275
Publisher:
SIAM
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; magnetohydrodynamics; preconditioners; mixed finite elements

Citation Formats

Phillips, Edward G., Shadid, John N., Cyr, Eric C., Elman, Howard C., and Pawlowski, Roger P.. Block Preconditioners for Stable Mixed Nodal and Edge finite element Representations of Incompressible Resistive MHD. United States: N. p., 2016. Web. https://doi.org/10.1137/16M1074084.
Phillips, Edward G., Shadid, John N., Cyr, Eric C., Elman, Howard C., & Pawlowski, Roger P.. Block Preconditioners for Stable Mixed Nodal and Edge finite element Representations of Incompressible Resistive MHD. United States. https://doi.org/10.1137/16M1074084
Phillips, Edward G., Shadid, John N., Cyr, Eric C., Elman, Howard C., and Pawlowski, Roger P.. Thu . "Block Preconditioners for Stable Mixed Nodal and Edge finite element Representations of Incompressible Resistive MHD". United States. https://doi.org/10.1137/16M1074084. https://www.osti.gov/servlets/purl/1598347.
@article{osti_1598347,
title = {Block Preconditioners for Stable Mixed Nodal and Edge finite element Representations of Incompressible Resistive MHD},
author = {Phillips, Edward G. and Shadid, John N. and Cyr, Eric C. and Elman, Howard C. and Pawlowski, Roger P.},
abstractNote = {The scalable iterative solution of strongly coupled three-dimensional incompressible resistive magnetohydrodynamics (MHD) equations is quite challenging because disparate time scales arise from the electromagnetics, the hydrodynamics, as well as the coupling between these systems. This study considers a mixed finite element discretization of a dual saddle point formulation of the incompressible resistive MHD equations using a stable nodal (Q2/Q1) discretization for the hydrodynamics and a stable edge-node discretization of a reduced form of the Maxwell equations. This paper introduces new approximate block factorization preconditioners for this system which reduce the system to approximate Schur complement systems that can be solved using algebraic multilevel methods. These preconditioners include a new augmentation-based approximation for the magnetic induction saddle point system as well as efficient approximations of the Schur complements that arise from the complex coupling between the Navier--Stokes equations and the Maxwell equations.},
doi = {10.1137/16M1074084},
journal = {SIAM Journal on Scientific Computing},
number = 6,
volume = 38,
place = {United States},
year = {2016},
month = {11}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Citation Metrics:
Cited by: 12 works
Citation information provided by
Web of Science

Save / Share:

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

An Improved Algebraic Multigrid Method for Solving Maxwell's Equations
journal, January 2003

  • Bochev, Pavel B.; Garasi, Christopher J.; Hu, Jonathan J.
  • SIAM Journal on Scientific Computing, Vol. 25, Issue 2
  • DOI: 10.1137/S1064827502407706

Scalable parallel implicit solvers for 3D magnetohydrodynamics
journal, July 2008


An Implicit, Nonlinear Reduced Resistive MHD Solver
journal, May 2002

  • Chacón, L.; Knoll, D. A.; Finn, J. M.
  • Journal of Computational Physics, Vol. 178, Issue 1
  • DOI: 10.1006/jcph.2002.7015

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

A Supernodal Approach to Sparse Partial Pivoting
journal, January 1999

  • Demmel, James W.; Eisenstat, Stanley C.; Gilbert, John R.
  • SIAM Journal on Matrix Analysis and Applications, Vol. 20, Issue 3
  • DOI: 10.1137/S0895479895291765

A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier–Stokes equations
journal, January 2008

  • Elman, Howard; Howle, V. E.; Shadid, John
  • Journal of Computational Physics, Vol. 227, Issue 3
  • DOI: 10.1016/j.jcp.2007.09.026

Preconditioners for the discretized time-harmonic Maxwell equations in mixed form
journal, January 2007

  • Greif, Chen; Schötzau, Dominik
  • Numerical Linear Algebra with Applications, Vol. 14, Issue 4
  • DOI: 10.1002/nla.515

An overview of the Trilinos project
journal, September 2005

  • Heroux, Michael A.; Phipps, Eric T.; Salinger, Andrew G.
  • ACM Transactions on Mathematical Software, Vol. 31, Issue 3
  • DOI: 10.1145/1089014.1089021

Nodal Auxiliary Space Preconditioning in H(curl) and H(div) Spaces
journal, January 2007

  • Hiptmair, Ralf; Xu, Jinchao
  • SIAM Journal on Numerical Analysis, Vol. 45, Issue 6
  • DOI: 10.1137/060660588

Toward an h-Independent Algebraic Multigrid Method for Maxwell's Equations
journal, January 2006

  • Hu, Jonathan J.; Tuminaro, Raymond S.; Bochev, Pavel B.
  • SIAM Journal on Scientific Computing, Vol. 27, Issue 5
  • DOI: 10.1137/040608118

A parallel fully coupled algebraic multilevel preconditioner applied to multiphysics PDE applications: Drift-diffusion, flow/transport/reaction, resistive MHD
journal, September 2010

  • Lin, Paul T.; Shadid, John N.; Tuminaro, Raymond S.
  • International Journal for Numerical Methods in Fluids, Vol. 64, Issue 10-12
  • DOI: 10.1002/fld.2402

A Note on Preconditioning for Indefinite Linear Systems
journal, January 2000

  • Murphy, Malcolm F.; Golub, Gene H.; Wathen, Andrew J.
  • SIAM Journal on Scientific Computing, Vol. 21, Issue 6
  • DOI: 10.1137/S1064827599355153

Mixed finite elements in ?3
journal, September 1980


A stochastic approach to uncertainty in the equations of MHD kinematics
journal, March 2015


A Block Preconditioner for an Exact Penalty Formulation for Stationary MHD
journal, January 2014

  • Phillips, Edward G.; Elman, Howard C.; Cyr, Eric C.
  • SIAM Journal on Scientific Computing, Vol. 36, Issue 6
  • DOI: 10.1137/140955082

GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
journal, July 1986

  • Saad, Youcef; Schultz, Martin H.
  • SIAM Journal on Scientific and Statistical Computing, Vol. 7, Issue 3
  • DOI: 10.1137/0907058

Mixed finite elements for incompressible magneto-hydrodynamics
journal, July 2003


Mixed finite element methods for stationary incompressible magneto?hydrodynamics
journal, February 2004


Towards a scalable fully-implicit fully-coupled resistive MHD formulation with stabilized FE methods
journal, October 2010

  • Shadid, J. N.; Pawlowski, R. P.; Banks, J. W.
  • Journal of Computational Physics, Vol. 229, Issue 20
  • DOI: 10.1016/j.jcp.2010.06.018

Scalable implicit incompressible resistive MHD with stabilized FE and fully-coupled Newton–Krylov-AMG
journal, June 2016

  • Shadid, J. N.; Pawlowski, R. P.; Cyr, E. C.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 304
  • DOI: 10.1016/j.cma.2016.01.019

Three-dimensional linear stability analysis of lid-driven magnetohydrodynamic cavity flow
journal, August 2003

  • Shatrov, V.; Mutschke, G.; Gerbeth, G.
  • Physics of Fluids, Vol. 15, Issue 8
  • DOI: 10.1063/1.1582184

Modified block preconditioners for the discretized time-harmonic Maxwell equations in mixed form
journal, January 2013

  • Wu, Shi-Liang; Huang, Ting-Zhu; Li, Cui-Xia
  • Journal of Computational and Applied Mathematics, Vol. 237, Issue 1
  • DOI: 10.1016/j.cam.2012.06.011

Block triangular preconditioner for static Maxwell equations
journal, January 2011


    Works referencing / citing this record:

    A face‐based monolithic approach for the incompressible magnetohydrodynamics equations
    journal, May 2020

    • Ata, Kayhan; Sahin, Mehmet
    • International Journal for Numerical Methods in Fluids, Vol. 92, Issue 5
    • DOI: 10.1002/fld.4786

    Second order unconditionally convergent and energy stable linearized scheme for MHD equations
    journal, August 2017

    • Zhang, Guo-Dong; Yang, Jinjin; Bi, Chunjia
    • Advances in Computational Mathematics, Vol. 44, Issue 2
    • DOI: 10.1007/s10444-017-9552-x