A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

A new mixed variational formulation of the equations of stationary incompressible magneto–hydrodynamics is introduced and analyzed. The formulation is based on curl-conforming Sobolev spaces for the magnetic variables and is shown to be well-posed in (possibly non-convex) Lipschitz polyhedra. A finite element approximation is proposed where the hydrodynamic unknowns are discretized by standard inf-sup stable velocity-pressure space pairs and the magnetic ones by a mixed approach using Nedelec’s elements of the first kind. An error analysis is carried out that shows that the proposed finite element approximation leads to quasi-optimal error bounds in the mesh-size.

Mixed finite element methods for stationary incompressible magneto–hydrodynamics

We propose and analyze a stabilized finite element method for the incompressible magnetohydrodynamic equations. The numerical results that we present show a good behavior of our approximation in experiments which are relevant from an industrial viewpoint. We explain in particular in the proof of our convergence theorem why it may be interesting to stabilize the magnetic equation as soon as the hydrodynamic diffusion is small and even if the magnetic diffusion is large. This observation is confirmed by our numerical tests.

A stabilized finite element method for the incompressible magnetohydrodynamic equations

https://www.cs.ubc.ca/%7Egreif/Publications/glsw2010.pdf

A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics

Discretization of the Primitive Variable Formulation: A Primitive Variable Formulation. The Finite Element Problem and the Div-St abi lity Condition. Finite Element Spaces. Alternate Weak Forms, Boundary Conditions and Numerical Integration. Penalty Methods. Solution of the Discrete Equations: Newton's Method and Other Iterative Methods. Solving the Linear Systems. Solution Methods for Large Reynolds Numbers. Time Dependent Problems: A Weak Formulation and Spatial Discretizations. Time Discretizations. The Streamfunction-Vorticity Formulation: Algorithms for the Streamfunction-Vorticity Equations. Solution Techniques for Multiply Connected Domains. The Streamfunction Formulation: Algorithms for Determining Streamfunction Approximations. Eigenvalue Problems Connected with Stability Studies for Viscous Flows: Energy Stability Analysis of Viscous Flows. Linearized Stability Analysis of Stationary Viscous Flows. Exterior Problems: Truncated Domain-Artificial Boundary Condition Methods. Nonlinear Constitutive Relations: A Ladyzhenskaya Model and Algebraic Turbulence Models. Bingham Fluids. Electromagnetically or Thermally Coupled Flows: Flows of Liquid Metals. The Boussinesq Equations. Remarks on Some Topics That Have Not Been Considered: Problems, Formulations, Algorithms, and Other Issues That Have Not Been Considered. Bibliography. Glossary of Symbols. Index.

Finite element methods for viscous incompressible flows: A guide to theory, practice, and algorithms

Nous etudions divers problemes mathematiques et numeriques poses par la modelisation de la fabrication de l'aluminium par electrolyse (procede Hall-Heroult) De nombreux phenomenes physiques entrent en jeu : ecoulements magnetohydrodynamiques (MHD) et multiphasiques, phenomenes thermiques, reactions electrochimiques, aimantation. La premiere partie consiste en une presentation de ces phenomenes; nous nous consacrons ensuite essentiellement a la MHD. La deuxieme partie est dediee a l'etude theorique de systemes d'equations aux derivees partielles modelisant certains aspects de l'electrolyse de l'aluminium. Nous prouvons l'existence de solutions faibles pour le systeme de la MHD incompressible avec densite, viscosite et conductivite electrique variables. Nous etudions ensuite un systeme MHD simplifie consistant en un couplage entre les equations de Navier-Stokes transitoires et les equations de Maxwell stationnaires. Nous nous interessons enfin au comportement en temps long d'un systeme de deux fluides incompressibles non miscibles separes par une interface libre. L'objet principal de la troisieme partie est d'etudier des methodes numeriques pour les equations de la MHD. La question que nous abordons dans un premier temps est de nature plus generale : nous traitons d'un probleme de vitesses parasites pouvant survenir dans la simulation d'un ecoulement incompressible soumis a des forces exterieures. Nous illustrons ce phenomene par plusieurs exemples et proposons une maniere de l'attenuer. Nous nous interessons ensuite a deux methodes pour la resolution numerique d'un probleme de magnetostatique. L'une d'elle repose sur la notion de matrice d'influence. Puis, nous presentons quelques manieres de traiter le couplage entre les equations de Maxwell et les equations de Navier-Stokes dans le systeme de la MHD. Nous etendons enfin a la MHD les techniques d'elements finis stabilises de type "streamline diffusion. [Resume de l'auteur]

https://pastel.archives-ouvertes.fr/tel-00005593/document

Problèmes mathématiques et numériques posés par la modélisation de l'electrolyse de l'aluminium

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Problèmes mathématiques et numériques posés par la modélisation de l'electrolyse de l'aluminium