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

The scalable iterative solution of strongly coupled three-dimensional incompressible resistive magnetohydrodynamics (MHD) equations  is very 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 presents 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 eq...

[1]  John N. Shadid,et al.  Scalable implicit incompressible resistive MHD with stabilized FE and fully-coupled Newton–Krylov-AMG , 2016 .

[2]  Scott P. MacLachlan,et al.  Monolithic Multigrid Methods for Two-Dimensional Resistive Magnetohydrodynamics , 2016, SIAM J. Sci. Comput..

[3]  Howard C. Elman,et al.  A stochastic approach to uncertainty in the equations of MHD kinematics , 2015, J. Comput. Phys..

[4]  Ting-Zhu Huang,et al.  Modified block preconditioners for the discretized time-harmonic Maxwell equations in mixed form , 2013, J. Comput. Appl. Math..

[5]  John N. Shadid,et al.  A Block Preconditioner for an Exact Penalty Formulation for Stationary MHD , 2014, SIAM J. Sci. Comput..

[6]  Dana A. Knoll,et al.  An Implicit, Nonlinear Reduced Resistive MHD Solver , 2002 .

[7]  Tingzhu Huang,et al.  Block triangular preconditioner for static Maxwell equations , 2011 .

[8]  John N. Shadid,et al.  A New Approximate Block Factorization Preconditioner for Two-Dimensional Incompressible (Reduced) Resistive MHD , 2013, SIAM J. Sci. Comput..

[9]  James Demmel,et al.  A Supernodal Approach to Sparse Partial Pivoting , 1999, SIAM J. Matrix Anal. Appl..

[10]  John N. Shadid,et al.  Teko: A Block Preconditioning Capability with Concrete Example Applications in Navier-Stokes and MHD , 2016, SIAM J. Sci. Comput..

[11]  John N. Shadid,et al.  A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier-Stokes equations , 2008, J. Comput. Phys..

[12]  Luis Chacon,et al.  An optimal, parallel, fully implicit Newton–Krylov solver for three-dimensional viscoresistive magnetohydrodynamicsa) , 2008 .

[13]  Gunter Gerbeth,et al.  Three-dimensional linear stability analysis of lid-driven magnetohydrodynamic cavity flow , 2003 .

[14]  Stefaan Poedts,et al.  Principles of Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas , 2004 .

[15]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[16]  T. Lelièvre,et al.  Mathematical Methods for the Magnetohydrodynamics of Liquid Metals , 2006 .

[17]  Gene H. Golub,et al.  A Note on Preconditioning for Indefinite Linear Systems , 1999, SIAM J. Sci. Comput..

[18]  Tamara G. Kolda,et al.  An overview of the Trilinos project , 2005, TOMS.

[19]  Paul Lin,et al.  A parallel fully coupled algebraic multilevel preconditioner applied to multiphysics PDE applications: Drift‐diffusion, flow/transport/reaction, resistive MHD , 2010 .

[20]  John N. Shadid,et al.  Aztec user`s guide. Version 1 , 1995 .

[21]  Dominik Schötzau,et al.  Mixed finite elements for incompressible magneto-hydrodynamics , 2003 .

[22]  Jonathan J. Hu,et al.  ML 5.0 Smoothed Aggregation Users's Guide , 2006 .

[23]  J. Nédélec.  Mixed finite elements in ℝ3 , 1980 .

[24]  Allen C. Robinson,et al.  An Improved Algebraic Multigrid Method for Solving Maxwell's Equations , 2003, SIAM J. Sci. Comput..

[25]  Rony Keppens,et al.  Implicit and semi-implicit schemes: Algorithms , 1999 .

[26]  L Chacón,et al.  Scalable parallel implicit solvers for 3D magnetohydrodynamics , 2008 .

[27]  Howard C. Elman,et al.  BOUNDARY CONDITIONS IN APPROXIMATE COMMUTATOR PRECONDITIONERS FOR THE NAVIER-STOKES EQUATIONS ∗ , 2009 .

[28]  Barry Lee,et al.  Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics , 2006, Math. Comput..

[29]  Allen C. Robinson,et al.  Toward an h-Independent Algebraic Multigrid Method for Maxwell's Equations , 2006, SIAM J. Sci. Comput..

[30]  Dominik Schötzau,et al.  Mixed finite element methods for stationary incompressible magneto–hydrodynamics , 2004, Numerische Mathematik.

[31]  Paul T. Lin,et al.  Towards a scalable fully-implicit fully-coupled resistive MHD formulation with stabilized FE methods , 2009, J. Comput. Phys..

[32]  Jinchao Xu,et al.  Nodal Auxiliary Space Preconditioning in H(curl) and H(div) Spaces , 2007, SIAM J. Numer. Anal..

[33]  R. Keppens,et al.  Implicit and semi-implicit schemes in the Versatile Advection Code : numerical tests , 1998 .

[34]  Michael A. Heroux.  AztecOO user guide. , 2004 .

[35]  Chen Greif,et al.  Preconditioners for the discretized time-harmonic Maxwell equations in mixed form , 2007, Numer. Linear Algebra Appl..