Stable finite element methods preserving ∇·B=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla \cdot \varvec{B}=

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

[2]  M. Tezer-Sezgin,et al.  DRBEM Solution of Incompressible MHD Flow withMagnetic Potential , 2013 .

[3]  M. Fortin,et al.  Mixed Finite Element Methods and Applications , 2013 .

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

[5]  W. Cai,et al.  Divergence-Free $\boldsymbol{\mathcal{H}}(\mathbf{div})$-Conforming Hierarchical Bases for Magnetohydrodynamics (MHD) , 2013 .

[6]  Santiago Badia,et al.  On an unconditionally convergent stabilized finite element approximation of resistive magnetohydrodynamics , 2013, J. Comput. Phys..

[7]  W. Cai,et al.  Divergence-Free H(div)-Conforming Hierarchical Bases for Magnetohydrodynamics (MHD) , 2012, 1210.5575.

[8]  Blanca Ayuso de Dios,et al.  A Simple Preconditioner for a Discontinuous Galerkin Method for the Stokes Problem , 2012, Journal of Scientific Computing.

[9]  Liwei Xu,et al.  Arbitrary order exactly divergence-free central discontinuous Galerkin methods for ideal MHD equations , 2012, J. Comput. Phys..

[10]  Anders Logg,et al.  Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book , 2012 .

[11]  Jinchao Xu,et al.  Global existence, uniqueness and optimal solvers of discretized viscoelastic flow models , 2011 .

[12]  Ramon Codina,et al.  Approximation of the thermally coupled MHD problem using a stabilized finite element method , 2011, J. Comput. Phys..

[13]  Bertram Taetz,et al.  An unstaggered constrained transport method for the 3D ideal magnetohydrodynamic equations , 2010, J. Comput. Phys..

[14]  Stephen C. Jardin,et al.  Computational Methods in Plasma Physics , 2010 .

[15]  Andreas Prohl,et al.  Convergent finite element discretization of the multi-fluid nonstationary incompressible magnetohydrodynamics equations , 2010, Math. Comput..

[16]  Paul Houston,et al.  A Mixed DG Method for Linearized Incompressible Magnetohydrodynamics , 2009, J. Sci. Comput..

[17]  D. Arnold,et al.  Finite element exterior calculus: From hodge theory to numerical stability , 2009, 0906.4325.

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

[19]  Chun Liu,et al.  An Introduction of Elastic Complex Fluids: An Energetic Variational Approach , 2009 .

[20]  A. I. Nesliturk,et al.  Two‐level finite element method with a stabilizing subgrid for the incompressible MHD equations , 2009 .

[21]  A. Prohl Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamics system , 2008 .

[22]  James A. Rossmanith,et al.  An Unstaggered, High-Resolution Constrained Transport Method for Magnetohydrodynamic Flows , 2006, SIAM J. Sci. Comput..

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

[24]  Paul Lin,et al.  Performance of fully coupled algebraic multilevel domain decomposition preconditioners for incompressible flow and transport , 2006 .

[25]  D. Arnold,et al.  Finite element exterior calculus, homological techniques, and applications , 2006, Acta Numerica.

[26]  John N. Shadid,et al.  Block Preconditioners Based on Approximate Commutators , 2005, SIAM J. Sci. Comput..

[27]  T. Rylander,et al.  Computational Electromagnetics , 2005 .

[28]  Chi-Wang Shu,et al.  Locally Divergence-Free Discontinuous Galerkin Methods for MHD Equations , 2005, J. Sci. Comput..

[29]  Jian-Guo Liu Energy and helicity preserving schemes for hydro- and magnetohydro-dynamics flows with symmetry , 2004 .

[30]  Dominik Schötzau,et al.  Mixed finite element approximation of incompressible MHD problems based on weighted regularization , 2004 .

[31]  Chi-Wang Shu,et al.  Locally divergence-free discontinuous Galerkin methods for the Maxwell equations , 2004, Journal of Computational Physics.

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

[33]  J. Guermond,et al.  Mixed finite element approximation of an MHD problem involving conducting and insulating regions: The 3D case , 2003 .

[34]  D. Balsara,et al.  A Comparison between Divergence-Cleaning and Staggered-Mesh Formulations for Numerical Magnetohydrodynamics , 2003, astro-ph/0310728.

[35]  L. Zanna,et al.  On the divergence-free condition in Godunov-type schemes for ideal magnetohydrodynamics: the upwind constrained transport method , 2003, astro-ph/0310183.

[36]  Dinshaw Balsara,et al.  Second-Order-accurate Schemes for Magnetohydrodynamics with Divergence-free Reconstruction , 2003, astro-ph/0308249.

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

[38]  John N. Shadid,et al.  On a multilevel preconditioning module for unstructured mesh Krylov solvers: two-level Schwarz , 2002 .

[39]  C. Munz,et al.  Hyperbolic divergence cleaning for the MHD equations , 2002 .

[40]  R. Hiptmair Finite elements in computational electromagnetism , 2002, Acta Numerica.

[41]  Jian-Guo Liu,et al.  An energy-preserving MAC-Yee scheme for the incompressible MHD equation , 2001 .

[42]  W. Habashi,et al.  A finite element method for magnetohydrodynamics , 2001 .

[43]  G. Tóth The ∇·B=0 Constraint in Shock-Capturing Magnetohydrodynamics Codes , 2000 .

[44]  Michel Fortin,et al.  A conservative stabilized finite element method for the magneto-hydrodynamic equations , 1999 .

[45]  D. Balsara,et al.  A Staggered Mesh Algorithm Using High Order Godunov Fluxes to Ensure Solenoidal Magnetic Fields in Magnetohydrodynamic Simulations , 1999 .

[46]  L. Demkowicz,et al.  hp-adaptive finite elements in electromagnetics , 1999 .

[47]  A. Frank,et al.  A Divergence-free Upwind Code for Multidimensional Magnetohydrodynamic Flows , 1998, astro-ph/9807228.

[48]  Paul R. Woodward,et al.  A Simple Finite Difference Scheme for Multidimensional Magnetohydrodynamical Equations , 1998 .

[49]  Paul R. Woodward,et al.  On the Divergence-free Condition and Conservation Laws in Numerical Simulations for Supersonic Magnetohydrodynamical Flows , 1998 .

[50]  L. Demkowicz,et al.  Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements , 1998 .

[51]  Xiu Ye,et al.  A discrete divergence-free basis for finite element methods , 1997, Numerical Algorithms.

[52]  J. C. Simo,et al.  Long-term dissipativity of time-stepping algorithms for an abstract evolution equation with applications to the incompressible MHD and Navier-Stokes equations☆ , 1996 .

[53]  Bo-nan Jiang,et al.  The Origin of Spurious Solutions in Computational Electromagnetics , 1996 .

[54]  H. Conraths EDDY CURRENT AND TEMPERATURE SIMULATION IN THIN MOVING METAL STRIPS , 1996 .

[55]  M. Gunzburger,et al.  On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics , 1991 .

[56]  C. Richard DeVore,et al.  Flux-corrected transport techniques for multidimensional compressible magnetohydrodynamics , 1989 .

[57]  J. Hawley,et al.  Simulation of magnetohydrodynamic flows: A Constrained transport method , 1988 .

[58]  Michael L. Norman,et al.  Numerical Simulations of a Magnetically Confined Jet , 1986 .

[59]  J. Nédélec A new family of mixed finite elements in ℝ3 , 1986 .

[60]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[61]  Y. R. Fautrelle,et al.  Analytical and numerical aspects of the electromagnetic stirring induced by alternating magnetic fields , 1981, Journal of Fluid Mechanics.

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

[63]  J. Brackbill,et al.  The Effect of Nonzero ∇ · B on the numerical solution of the magnetohydrodynamic equations☆ , 1980 .

[64]  M. Avila,et al.  Magnetohydrodynamics , 2017 .

[65]  Sergey Yakovlev,et al.  Locally divergence-free central discontinuous Galerkin methods for ideal MHD equations , 2013, J. Comput. Sci..

[66]  Paul Lin,et al.  INITIAL PERFORMANCE OF FULLY-COUPLED AMG AND APPROXIMATE BLOCK FACTORIZATION PRECONDITIONERS FOR SOLUTION OF IMPLICIT FE RESISTIVE MHD , 2010 .

[67]  Shangyou Zhang Bases for C 0-P 1 divergence-free elements and for C 1P 2 finite elements on union jack grids , 2009 .

[68]  A. Bossavit Discretization of Electromagnetic Problems: The “Generalized Finite Differences” Approach , 2005 .

[69]  Ronald H. W. Hoppe,et al.  Finite element methods for Maxwell's equations , 2005, Math. Comput..

[70]  L. Driel-Gesztelyi An Introduction to Magnetohydrodynamics , 2004 .

[71]  Manuel Torrilhon,et al.  A Constrained Transport Upwind Scheme for Divergence-free Advection , 2003 .

[72]  Jean-Frédéric Gerbeau,et al.  Simulations of MHD flows with moving interfaces , 2003 .

[73]  Ralf Hiptmair,et al.  Symmetric Coupling for Eddy Current Problems , 2002, SIAM J. Numer. Anal..

[74]  Matthias Wiedmer,et al.  Finite element approximation for equations of magnetohydrodynamics , 2000, Math. Comput..

[75]  K. Powell An Approximate Riemann Solver for Magnetohydrodynamics , 1997 .

[76]  N. Ida,et al.  Electromagnetics and calculation of fields , 1992 .

[77]  J. Brackbill Fluid modeling of magnetized plasmas , 1985 .

[78]  P. Raviart,et al.  A mixed finite element method for 2-nd order elliptic problems , 1977 .

[79]  F. Brezzi On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[80]  S.,et al.  Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media , 1966 .