On an unconditionally convergent stabilized finite element approximation of resistive magnetohydrodynamics

Badia, Santiago; Codina, Ramon; Planas, Ramón

doi:10.1016/j.jcp.2012.09.031

Cited by 60 publications

(79 citation statements)

References 47 publications

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“…Even if tailored approximations for Maxwell's problem may be afforded at a reasonable computational cost when it is an isolated problem, it is obvious that a classical Lagrangian type approximation greatly simplifies its implementation in situations where this problem is coupled to others, as in MHD (see [5]). On the other hand, our approach may be viewed as an alternative to the use of the so called compatible discretization, satisfying the appropriate inf-sup conditions.…”

Section: Discussionmentioning

confidence: 99%

“…We refer to [4] for detailed numerical experiments about the effect of having a suitable macro-element structure in the convergence of the method. In [5] we have extended this work to three-dimensions in the frame of MHD applications; we have considered both the 3d Powell-Sabin element and a 3d extension of the crossbox; both choices exhibit excellent convergence properties. …”

Section: Allows Us To Obtainmentioning

confidence: 99%

“…It can be implemented in a stabilized FE solver for the Navier-Stokes equations with minor modifications. Thus, the method is an excellent candidate for being used in MHD; we have developed a nodal-based FE formulation of the visco-resistive MHD problem where the magnetic sub-problem is approximated following the ideas in this work in [5], reporting excellent results.…”

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confidence: 99%

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A Nodal-based Finite Element Approximation of the Maxwell Problem Suitable for Singular Solutions

Badia¹,

Codina²

2012

SIAM J. Numer. Anal.

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Abstract.A new mixed finite element approximation of Maxwell's problem is proposed, its main features being that it is based on a novel augmented formulation of the continuous problem and the introduction of a mesh dependent stabilizing term, which yields a very weak control on the divergence of the unknown. The method is shown to be stable and convergent in the natural H(curl; Ω) norm for this unknown. In particular, convergence also applies to singular solutions, for which classical nodal based interpolations are known to suffer from spurious convergence upon mesh refinement.1. Introduction. The simulation of electromagnetic phenomena with increasing complexity demands accurate and efficient numerical methods suitable for large-scale computing. Finite element (FE) methods are commonly used in this context because they can easily handle complicated geometries by using unstructured grids, provide a rigorous mathematical framework and allow adaptation.In many applications of current interest, the electromagnetic problem is coupled to other physical processes. Salient examples of multiphysics phenomena that include electromagnetics are magnetohydrodynamics (MHD) and plasma physics. These two problems have experienced increasing attention due to the need to develop numerical laboratories in fusion technology design. The simulation of these problems (and many others) would benefit of an all-purpose FE method that would be suitable for the different sub-problems at hand, simplifying the implementation issues and the enforcement of the coupling conditions. In particular, an all-purpose continuous nodal-based formulation would be a favored candidate. E.g. the Navier-Stokes equations are commonly solved with stabilized FE approximations that can deal with the singularly perturbed nature of the system for high Reynolds numbers and circumvent the restrictions related to the corresponding inf-sup condition (see e.g. [13,14]). In plasma physics, fields computed by discontinuous FE Maxwell solvers create a considerable numerical noise when embedded in a plasma code, e.g. using the particle-in-cell method (see [2]). Furthermore, nodal approximations are particularly well-suited for time-dependent electromagnetic problems because the mass matrix can be consistently lumped without loss of accuracy, leading to inexpensive transient solvers.The Maxwell operator has a saddle-point structure, with the particularity that the Lagrange multiplier introduced to enforce the divergence-free constraint is identically zero. Existing FE methods that satisfy the discrete counterpart of the inherent inf-sup condition for this problem are based on Nedelec's or edge elements (see e.g. [27,33]); edge elements lead to fields with discontinuous normal component on element edges or faces. We also refer to alternative formulations based on discontinuous Galerkin approximations [28,24,23,34]. With the aim to solve the Maxwell problem with Lagrangian finite elements (FEs), the differential operator of the problem can be transformed into an elliptic on...

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Section: Discussionmentioning

confidence: 99%

Section: Allows Us To Obtainmentioning

confidence: 99%

mentioning

confidence: 99%

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A Nodal-based Finite Element Approximation of the Maxwell Problem Suitable for Singular Solutions

Badia¹,

Codina²

2012

SIAM J. Numer. Anal.

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“…For non-convex polygonal (polyhedral) domains, re-entrant corners may lead to singularities in the magnetic field that are missed by continuous H 1 -conforming finite elements approximation. To solve this difficulty, please refer [17,[42][43][44].…”

Section: Finite Element Spatial Discretization and Stabilitymentioning

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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“…This idea has been applied to our target problem, (thermally coupled) inductionless MHD problem, (where the unknowns are the velocity, pressure, current density and electric potential) but can also be applied to other problems like resistive MHD [4] or liquid crystal problems [5]. We consider different preconditioners based on approximations of the resulting Schur complement matrices.…”

Section: Discussionmentioning

confidence: 99%

Block recursive LU preconditioners for the thermally coupled incompressible inductionless MHD problem

Badia

Martı́n

Planas

2014

Journal of Computational Physics

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The thermally coupled incompressible inductionless magnetohydrodynamics (MHD) problem models the flow of an electrically charged fluid under the influence of an external electromagnetic field with thermal coupling. This system of partial differential equations is strongly coupled and highly nonlinear for real cases of interest. Therefore, fully implicit time integration schemes are very desirable in order to capture the different physical scales of the problem at hand. However, solving the multiphysics linear systems of equations resulting from such algorithms is a very challenging task which requires efficient and scalable preconditioners. In this work, a new family of recursive block LU preconditioners is designed and tested for solving the thermally coupled inductionless MHD equations. These preconditioners are obtained after splitting the fully coupled matrix into one-physics problems for every variable (velocity, pressure, current density, electric potential and temperature) that can be optimally solved, e.g., using preconditioned domain decomposition algorithms. The main idea is to arrange the original matrix into an (arbitrary) 2 × 2 block matrix, and consider a LU preconditioner obtained by approximating the corresponding Schur complement. For every one of the diagonal blocks in the LU preconditioner, if it involves more than one type of unknown, we proceed the same way in a recursive fashion. This approach is stated in an abstract way, and can be straightforwardly applied to other multiphysics problems. Further, we precisely explain a flexible and general software design for the code implementation of this type of preconditioners.

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On an unconditionally convergent stabilized finite element approximation of resistive magnetohydrodynamics

Cited by 60 publications

References 47 publications

A Nodal-based Finite Element Approximation of the Maxwell Problem Suitable for Singular Solutions

A Nodal-based Finite Element Approximation of the Maxwell Problem Suitable for Singular Solutions

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

Block recursive LU preconditioners for the thermally coupled incompressible inductionless MHD problem

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