Analysis of coupling iterations based on the finite element method for stationary magnetohydrodynamics on a general domain

Zhang, Guodong; He, Ya‐Ling; Yang, Di

doi:10.1016/j.camwa.2014.07.025

Cited by 46 publications

(32 citation statements)

References 27 publications

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“…Remark 3.2. The above results indicate that our method obtains same convergence rate as existing conforming methods [39,21]. Namely, if the exact solution is sufficient smooth, we have optimal convergence for u, b, p in the energy norms.…”

Section: )supporting

confidence: 68%

“…[38] is the first paper which tried to obtain optimal approximation under the minimal regularity assumption on the exact solution. However, according to [39,Section 1], the author of [38] failed to prove [38,Proposition 3.2] and [38,Corollary 3.1], which are indispensable to give an error estimate. In [39], a correct error analysis is given to show that with the first type of boundary (or constraint) conditions (1.2), optimal convergence is achieved by the conforming method in [38] under the minimal regularity assumption.…”

Section: B)mentioning

confidence: 99%

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A mixed DG method and an HDG method for incompressible magnetohydrodynamics

Qiu

Shi

2019

IMA Journal of Numerical Analysis

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In this paper we propose and analyze a mixed DG method and an HDG method for the stationary Magnetohydrodynamics (MHD) equations with two types of boundary (or constraint) conditions. The mixed DG method is based on a recent work proposed by Houston et. al. in [25] for the linearized MHD. With two novel discrete Sobolev embedding type estimates for the discontinuous polynomials, we provide a priori error estimates for the method on the nonlinear MHD equations. In the smooth case, we have optimal convergence rate for the velocity, magnetic field and pressure in the energy norm, the Lagrange multiplier only has suboptimal convergence order. With the minimal regularity assumption on the exact solution, the approximation is optimal for all unknowns. To the best of our knowledge, this is the first a priori error estimates of DG methods for the nonlinear MHD equations. In addition, we also propose and analyze the first divergence-free HDG method for the problem with several unique features comparing with the mixed DG method.2000 Mathematics Subject Classification. 65N30, 65L12.

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Section: )supporting

confidence: 68%

Section: B)mentioning

confidence: 99%

A mixed DG method and an HDG method for incompressible magnetohydrodynamics

Qiu

Shi

2019

IMA Journal of Numerical Analysis

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“…From the above result we can see that it suffices to bound the terms on the right hand side of the energy identity to get the error estimates in the energy norm. The first four terms can be handled with standard tools for Navier-Stokes equations, see [14,32] for instance. In particular, we need the following continuity result for the advection term, see [32]:…”

Section: Convergence Of Finite Element Methodsmentioning

confidence: 99%

Convergence of a B-E based finite element method for MHD models on Lipschitz domains

Qiu

Shi

2020

Journal of Computational and Applied Mathematics

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We discuss a class of magnetic-electric fields based finite element schemes for stationary magnetohydrodynamics (MHD) systems with two types of boundary conditions. We establish a key L 3 estimate for divergence-free finite element functions for a new type of boundary conditions. With this estimate and a similar one in [21], we rigorously prove the convergence of Picard iterations and the finite element schemes with weak regularity assumptions. These results demonstrate the convergence of the finite element methods for singular solutions.2000 Mathematics Subject Classification. Primary: 65N30, 76W05.Key words and phrases. magnetohydrodynamics, finite element method, structure-preserving, de Rham complex. The title of this article in its original version (see arXiv:1711.11330) is "Magnetic-Electric Formulations for Stationary Magnetohydrodynamics Models".

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“…The application of the scheme (32) with (29), (31) and (33) to Eqs. (11) and (12) obtains, respectively,…”

Section: Ehoc Scheme For 2d Mhd Flowmentioning

confidence: 97%

“…This equation is consistent with Eqs. (11) and (12) which are originated from the coupled system of equations for the MHD duct flow. The…”

Section: Nonuniform Ehoc Scheme Of Convection-diffusion Modelmentioning

confidence: 99%

Exponential high-order compact scheme on nonuniform grids for the steady MHD duct flow problems with high Hartmann numbers

Zhou

Tian

2015

Computer Physics Communications

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Analysis of coupling iterations based on the finite element method for stationary magnetohydrodynamics on a general domain

Cited by 46 publications

References 27 publications

A mixed DG method and an HDG method for incompressible magnetohydrodynamics

A mixed DG method and an HDG method for incompressible magnetohydrodynamics

Convergence of a B-E based finite element method for MHD models on Lipschitz domains

Exponential high-order compact scheme on nonuniform grids for the steady MHD duct flow problems with high Hartmann numbers

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