Convergence of a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e480" altimg="si2.svg"><mml:mi mathvariant="bold-italic">B</mml:mi></mml:math>-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e485" altimg="si3.svg"><mml:mi mathvariant="bold-italic">E</mml:mi></mml:math> based finite element method for MHD models on Lipschitz domains

Hu, Kaibo; Qiu, Weifeng; Shi, Ke

doi:10.1016/j.cam.2019.112477

Cited by 16 publications

(17 citation statements)

References 34 publications

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“…The above estimates is due to Sobolev inequality Lemma 3.2 and Theorem 1 in [27], stability of the solution Theorem 2.1 and the estimates for u, B. This completes the proof for (2.7) with a simple triangle inequality and projection error estimates for E in Lemma 3.3.…”

Section: Error Estimatessupporting

confidence: 60%

Analysis of a semi-implicit structure-preserving finite element method for the nonstationary incompressible Magnetohydrodynamics equations

Qiu

Shi

2020

Preprint

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We revise the structure-preserving finite element method in [K. Hu, Y. MA and J. Xu. (2017) Stable finite element methods preserving ∇ • B = 0 exactly for MHD models. Numer. Math., 135,. The revised method is semi-implicit in time-discretization. We prove the linearized scheme preserves the divergence free property for the magnetic field exactly at each time step. Further, we showed the linearized scheme is unconditionally stable and we obtain optimal convergence in the energy norm of the revised method even for solutions with low regularity.

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Section: Error Estimatessupporting

confidence: 60%

Analysis of a semi-implicit structure-preserving finite element method for the nonstationary incompressible Magnetohydrodynamics equations

Qiu

Shi

2020

Preprint

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“…Proof. As in [16], the stationary Faraday's law ∇×E h = 0 follows from testing (.c) with C h = ∇×E h , and the magnetic Gauss' law ∇ • B h = 0 then follows from testing (.c) with C h = B h . The proof of the energy law follows from testing (.d) with j h .…”

Section: Nonlinear Schemementioning

confidence: 89%

“…These formulations either eliminate E or j with help of (.f) or (.b). Here, the augmented Lagrangian formulation in [16] seems a natural approach, as it only includes u, p, B and E as unknowns and enforces ∇ • B = 0 without the need for a Lagrange multiplier. Our proposed formulation with both j and E as unknowns tries to use the fewest number of unknown variables for the Hall system while still enforcing the magnetic Gauss's law precisely.…”

Section: Introductionmentioning

confidence: 99%

Structure-preserving and helicity-conserving finite element approximations and preconditioning for the Hall MHD equations

Fabian¹,

Farrell²,

Hu³

2022

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We develop structure-preserving finite element methods for the incompressible, resistive Hall magnetohydrodynamics (MHD) equations. These equations incorporate the Hall current term in Ohm's law and provide a more appropriate description of fully ionized plasmas than the standard MHD equations on length scales close to or smaller than the ion skin depth. We introduce a stationary discrete variational formulation of Hall MHD that enforces the magnetic Gauss's law exactly (up to solver tolerances) and prove the well-posedness and convergence of a Picard linearization. For the transient problem, we present time discretizations that preserve the energy and magnetic and hybrid helicity precisely in the ideal limit for two types of boundary conditions. Additionally, we present an augmented Lagrangian preconditioning technique for both the stationary and transient cases. We confirm our findings with several numerical experiments.

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“…More recently, a stabilised FEM for stationary MHD is designed in [40] which preserves the divergence free constraints of both the velocity and magnetic fields at the discrete level. We also make note of the recent publications [41,50,38]. A non-conforming approximation of the linearised model is proposed in [39] using a mixed discontinuous Galerkin (DG) approach.…”

Section: Introductionmentioning

confidence: 99%

A hybrid high-order scheme for the stationary, incompressible magnetohydrodynamics equations

Droniou¹,

Liam²

2022

Preprint

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We propose and analyse a hybrid high-order (HHO) scheme for stationary incompressible magnetohydrodynamics equations. The scheme has an arbitrary order of accuracy and is applicable on generic polyhedral meshes. For sources that are small enough, we prove error estimates in energy norm for the velocity and magnetic field, and L 2 -norm for the pressure; these estimates are fully robust with respect to small faces, and of optimal order with respect to the mesh size. Using compactness techniques, we also prove that the scheme converges to a solution of the continuous problem, irrespective of the source being small or large. Finally, we illustrate our theoretical results through 3D numerical tests on tetrahedral and Voronoi mesh families.

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

Cited by 16 publications

References 34 publications

Analysis of a semi-implicit structure-preserving finite element method for the nonstationary incompressible Magnetohydrodynamics equations

Analysis of a semi-implicit structure-preserving finite element method for the nonstationary incompressible Magnetohydrodynamics equations

Structure-preserving and helicity-conserving finite element approximations and preconditioning for the Hall MHD equations

A hybrid high-order scheme for the stationary, incompressible magnetohydrodynamics equations

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