Convergence analysis of three finite element iterative methods for the 2D/3D stationary incompressible magnetohydrodynamics

Abstract Three finite element iterative methods are designed and analyzed for solving 2D/3D stationary incompressible magnetohydrodynamics (MHD). By a new technique, strong uniqueness conditions for both Stokes type iterative method (Iterative method I) and Newton iterative method (Iterative method II) are obtained, which are weaker than the ones reported in open literature. Stability and optimal convergence rates for the above two methods are derived, where the Iterative method II has an exponential convergent part with respect to iterative step m. Moreover, Oseen type iterative method (Iterative method III) is unconditionally stable and convergent under a uniqueness condition. Finally, performance of the three proposed methods is investigated by numerical experiments.

[1]  D. Schötzau,et al.  A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics , 2010 .

[2]  Yinnian He,et al.  Some iterative finite element methods for steady Navier-Stokes equations with different viscosities , 2013, J. Comput. Phys..

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

[4]  Jean-Frédéric Gerbeau,et al.  A stabilized finite element method for the incompressible magnetohydrodynamic equations , 2000, Numerische Mathematik.

[5]  Yinnian He.  Unconditional convergence of the Euler semi-implicit scheme for the three-dimensional incompressible MHD equations , 2015 .

[6]  Yinnian He,et al.  Convergence of three iterative methods based on the finite element discretization for the stationary Navier–Stokes equations☆ , 2009 .

[7]  Rolf Rannacher,et al.  Finite element approximation of the nonstationary Navier-Stokes problem, part III. Smoothing property and higher order error estimates for spatial discretization , 1988 .

[8]  R. Rannacher,et al.  Finite element approximation of the nonstationary Navier-Stokes problem. I : Regularity of solutions and second-order error estimates for spatial discretization , 1982 .

[9]  S. Ravindran.  Linear feedback control and approximation for a system governed by unsteady MHD equations , 2008 .

[10]  Yinnian He,et al.  Two-Level Newton’s Method for Nonlinear Elliptic PDEs , 2013, J. Sci. Comput..

[11]  Yin-nianHe,et al.  A TWO-LEVEL FINITE ELEMENT GALERKIN METHOD FOR THE NONSTATIONARY NAVIER-STOKES EQUATIONS I： SPATIAL DISCRETIZATION , 2004 .

[12]  Janet S. Peterson,et al.  A two-level newton, finite element algorithm for approximating electrically conducting incompressible fluid flows , 1994 .

[13]  Santiago Badia,et al.  Approximation of the inductionless MHD problem using a stabilized finite element method , 2011, J. Comput. Phys..

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

[15]  Yinnian He,et al.  Two-Level Method Based on Finite Element and Crank-Nicolson Extrapolation for the Time-Dependent Navier-Stokes Equations , 2003, SIAM J. Numer. Anal..

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

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

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

[19]  Yinnian He,et al.  Unconditional convergence of the Euler semi-implicit scheme for the 3D incompressible MHD equations , 2015 .

[20]  William Layton,et al.  A TWO-LEVEL DISCRETIZATION METHOD FOR THE STATIONARY MHD EQUATIONS , 1997 .

[21]  Yinnian He.  Euler implicit/explicit iterative scheme for the stationary Navier–Stokes equations , 2013, Numerische Mathematik.

[22]  Jinchao Xu,et al.  A Novel Two-Grid Method for Semilinear Elliptic Equations , 1994, SIAM J. Sci. Comput..

[23]  Yinnian He,et al.  Two‐level Newton iterative method for the 2D/3D steady Navier‐Stokes equations , 2012 .