On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics

Gunzburger, Max; Meir, A. J.; Peterson, Janet

doi:10.1090/s0025-5718-1991-1066834-0

Cited by 268 publications

(167 citation statements)

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“…In this work, we consider an exact penalty formulation of the MHD equations [14]. This formulation, although limited to convex domains, leads to a discrete threevariable block structure which we will show to be amenable to block preconditioning strategies of a type used successfully for the Navier-Stokes equations.…”

Section: Phillips Elman Cyr Shadid and Pawlowskimentioning

confidence: 99%

“…Many strategies exist for incorporating the solenoidal condition (2.1d) into the other three equations to ensure the solvability of the system. These methods include exact penalty [11,14], Lagrange multiplier [2,22], vector potential [17,23], and divergence cleaning [5] formulations. For this work, we will use an exact penalty formulation in which the solenoidal condition is weakly enforced in the weak form.…”

Section: Phillips Elman Cyr Shadid and Pawlowskimentioning

confidence: 99%

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A Block Preconditioner for an Exact Penalty Formulation for Stationary MHD

Phillips¹,

Elman²,

Cyr³

et al. 2014

SIAM J. Sci. Comput.

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Abstract. The magnetohydrodynamics (MHD) equations are used to model the flow of electrically conducting fluids in such applications as liquid metals and plasmas. This system of non-self adjoint, nonlinear PDEs couples the Navier-Stokes equations for fluids and Maxwell's equations for electromagnetics. There has been recent interest in fully coupled solvers for the MHD system because they allow for fast steady-state solutions that do not require pseudo-time stepping. When the fully coupled system is discretized, the strong coupling can make the resulting algebraic systems difficult to solve, requiring effective preconditioning of iterative methods for efficiency. In this work, we consider a finite element discretization of an exact penalty formulation for the stationary MHD equations. This formulation has the benefit of implicitly enforcing the divergence free condition on the magnetic field without requiring a Lagrange multiplier. We consider extending block preconditioning techniques developed for the Navier-Stokes equations to the full MHD system. We analyze operators arising in block decompositions from a continuous perspective and apply arguments based on the existence of approximate commutators to develop new preconditioners that account for the physical coupling. This results in a family of parameterized block preconditioners for both Picard and Newton linearizations. We develop an automated method for choosing the relevant parameters and demonstrate the robustness of these preconditioners for a range of the physical non-dimensional parameters and with respect to mesh refinement. Key words. magnetohydrodynamics, iterative methods, preconditionersAMS subject classifications. 76W05, 65F08, 65M221. Introduction. The magnetohydrodynamics (MHD) model describes the flow of electrically conducting fluids in the presence of magnetic fields. A principal application of MHD is the modeling of plasma physics, ranging from plasma confinement for thermonuclear fusion to astrophysical plasma dynamics [13]. MHD is also used to model the flow of liquid metals, for instance in magnetic pumps, liquid metal blankets in fusion reactor concepts, and aluminum electrolysis [19]. The model consists of a non-self-adjoint, nonlinear system of partial differential equations (PDEs) that couple the Navier-Stokes equations for fluid flow to a reduced set of Maxwell's equations for electromagnetics. Because multiple physical processes are represented in the model, the PDEs can span over a range of length-and time-scales, making the equations difficult to solve and requiring a robust, accurate means of approximating the solution. Decoupled solution methods which solve the fluid and magnetic systems separately and possibly couple the systems by an outer iteration have been commonly employed as solvers for the transient and steady MHD systems. In the context of transient systems these methods are commonly used in operator splitting techniques, for steady state solves a fixed point iteration serves to couple the system (see e.g. [2], and th...

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Section: Phillips Elman Cyr Shadid and Pawlowskimentioning

confidence: 99%

Section: Phillips Elman Cyr Shadid and Pawlowskimentioning

confidence: 99%

A Block Preconditioner for an Exact Penalty Formulation for Stationary MHD

Phillips¹,

Elman²,

Cyr³

et al. 2014

SIAM J. Sci. Comput.

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“…However, using a condition on a boundary vector k in (iii) vector E can be (10) so that the first equation in (2) holds a.e. in Ω (see details in [15]).…”

Section: Lemma 21 Under Condition (I) There Exist Constantsmentioning

confidence: 99%

Study of control problems for the stationary MHD equations

Brizitskii

2017

J. Phys.: Conf. Ser.

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Abstract.The optimal control problems for the stationary magnetohydrodynamic equations under inhomogeneous mixed boundary conditions for a magnetic field are considered. The role of control in control xs under study is played by normal component of the magnetic field on the part of the boundary. In the capacity of cost fucntionals quadratic tracking-type functionals for a velocity, magnetic field or pressure are taken. Introduction. Statement of the boundary value problemGreat attention has been paid recently to the optimal control problems for the models of magnetohydrodynamics (MHD) for a viscous conducting incompressible fluid (see [1][2][3][4][5][6]).In this paper we study control problems for the new MHD model in which stationary equations of magnetohydrodynamic are considered under mixed boundary conditions for a magnetic field and under Dirichlet boundary condition for a velocity with tracking-type functionals. The role of controls in control problems under study is played by a normal component of the magnetic field on the part of the boundary.One of the features of this work consist in proving of the solvability of optimal control problem under minimum requirements on a normal component of the magnetic field given on the part of boundary and playing the role of control. This result, in contrast to [3], allows to use a simple L 2 -norm for the normal component of the magnetic field instead of the H 1/2 -norm on the part of the boundary by Tikhonov regularization of considered incorrect control problem. This regularization is needed to prove stability and uniqueness of the control problem's solution.Let Ω be a bounded domain of space R 3 with boundary ∂Ω consisting of two parts Σ ν and Σ τ . In this paper we study control problems for the stationary magnetohydrodynamic equations of viscous incompressible fluid

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“…Nevertheless, there is also the possibility of relying on the mathematical structure of the equations and to expect that the original problem will already yield a magnetic field close enough to solenoidal. This is the idea followed in [22], which probably contains the first analysis of a finite element approximation to the MHD problem, and it is also used in [15], among other papers.…”

Section: Introductionmentioning

confidence: 99%