Unconditional convergence of the Euler semi-implicit scheme for the three-dimensional incompressible MHD equations

He, Yunze

doi:10.1093/imanum/dru015

Cited by 182 publications

(104 citation statements)

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“…A. Prohl [22] studied a coupled semi-implicit scheme and a decoupled scheme, in which he just proved the convergence under a strong condition t ≤ Ch 3 . Very recently, He [23,24] studied the first order Euler semi-implicit scheme for MHD equations. He proved the Euler semi-implicit scheme for MHD flows is unconditionally stable and convergent.…”

Section: Introductionmentioning

confidence: 99%

Decoupled schemes for unsteady MHD equations II: Finite element spatial discretization and numerical implementation

Zhang

2015

Computers & Mathematics with Applications

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a b s t r a c tIn this article, a decoupled fully discrete scheme for solving 2D magnetohydrodynamics (MHD) equations is proposed. The decoupled scheme is used for time discretization, and the finite element method is used for spatial discretization. Firstly, the almost unconditional stability ( t ≤ C ) of this scheme is established. Then optimal L 2 and H 1 error estimates of numerical solution are provided. Finally, a numerical example is presented to confirm our theoretical results and show the high efficiency of the decoupled scheme.

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Section: Introductionmentioning

confidence: 99%

Decoupled schemes for unsteady MHD equations II: Finite element spatial discretization and numerical implementation

Zhang

2015

Computers & Mathematics with Applications

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“…In [33] and [32], the H −1 , i.e., μ = 1, negative norm error estimates of the finite element methods were obtained for the nonstationary Navier-Stokes equations and the incompressible MHD equations, respectively. In this paper, we generalize their negative norm error estimates to the fourth order problem and the case μ = 2.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

“…We generalize the error estimate in the H −1 norm for the second order elliptic equations; see, e.g. [33] and [32], to both H −1 and H −2 norm for the fourth order equations.…”

Section: Introductionmentioning

confidence: 99%

Postprocessing Mixed Finite Element Methods For Solving Cahn–Hilliard Equation: Methods and Error Analysis

2015

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A postprocessing technique for mixed finite element methods for the Cahn-Hilliard equation is developed and analyzed. Once the mixed finite element approximations have been computed at a fixed time on the coarser mesh, the approximations are postprocessed by solving two decoupled Poisson equations in an enriched finite element space (either on a finer grid or a higher-order space) for which many fast Poisson solvers can be applied. The nonlinear iteration is only applied to a much smaller size problem and the computational cost using Newton and direct solvers is negligible compared with the cost of the linear problem. The analysis presented here shows that this technique remains the optimal rate of convergence for both the concentration and the chemical potential approximations. The corresponding error estimate obtained in our paper, especially the negative norm error estimates, are non-trivial and different with the existing results in the literatures.

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“…Similarly, define the discrete operator A 2μ B μ = R 0μ (∇ μ ×∇ ×B μ + ∇ μ ∇ ·B μ ) ∈ W μ as follows (see [16,36])…”

Section: The Stability and Convergence Of Galerkin Finite Element Methodsmentioning

confidence: 99%

“…Recently, the classical iterative methods (Stokes, Newton and Oseen type iterative methods) have been analyzed for the steady Navier-Stokes equations in [17]. Furthermore, these iterative schemes have been designed and considered for the steady MHD problem in [5].…”

Section: The Stability and Convergence Of Three Iterative Methodsmentioning

confidence: 99%

Stability and convergence of two-level iterative methods for the stationary incompressible magnetohydrodynamics with different Reynolds numbers

Tao

Zhang

2015

Journal of Mathematical Analysis and Applications

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In the paper we develop some two-level finite element iterative methods and use these methods to solve the stationary incompressible magnetohydrodynamics (MHD) with different Reynolds numbers under some different uniqueness conditions. Firstly, we use the Stokes type iterative method, Newton type iterative method and Oseen type iterative method of m times on a coarse mesh with mesh size H and then solve a linear problem with the Stokes type, Newton type and Oseen type correction of one time on a fine grid with mesh sizes h H. Furthermore, we analyze the uniform stability and convergence of these methods with respect to Reynolds numbers, mesh sizes h and H and iterative times m. Finally, the stationary incompressible MHD equations with large Reynolds number are researched by the one-level finite element method based on the Oseen type iteration on a fine mesh under a weak uniqueness condition.

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Unconditional convergence of the Euler semi-implicit scheme for the three-dimensional incompressible MHD equations

Cited by 182 publications

References 0 publications

Decoupled schemes for unsteady MHD equations II: Finite element spatial discretization and numerical implementation

Decoupled schemes for unsteady MHD equations II: Finite element spatial discretization and numerical implementation

Postprocessing Mixed Finite Element Methods For Solving Cahn–Hilliard Equation: Methods and Error Analysis

Stability and convergence of two-level iterative methods for the stationary incompressible magnetohydrodynamics with different Reynolds numbers

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