*Jean-Frédéric Gerbeau, Claude Le Bris, and Tony Lelièvre*

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198566656
- eISBN:
- 9780191718014
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566656.003.0003
- Subject:
- Mathematics, Mathematical Physics

This chapter addresses the discretization of the one-fluid magnetohydrodynamics equations. At the numerical level, the difficulties due to nonlinearities (coming both from the hydrodynamics equations ...
More

This chapter addresses the discretization of the one-fluid magnetohydrodynamics equations. At the numerical level, the difficulties due to nonlinearities (coming both from the hydrodynamics equations and the coupling between the Navier-Stokes equations and the Maxwell equations) translate into several numerical challenges. Various variational formulations of the magnetohydrodynamics equations are presented, along with their finite element discretization and algorithms to solve the resulting nonlinear problems. Dedicated finite elements methods, with appropriate mixed formulations or stabilization techniques have to be introduced. For clarity of exposition and for pedagogic purposes, the Stokes problem is first considered, and a concise presentation of the general theory in an abstract framework is provided.Less

This chapter addresses the discretization of the one-fluid magnetohydrodynamics equations. At the numerical level, the difficulties due to nonlinearities (coming both from the hydrodynamics equations and the coupling between the Navier-Stokes equations and the Maxwell equations) translate into several numerical challenges. Various variational formulations of the magnetohydrodynamics equations are presented, along with their finite element discretization and algorithms to solve the resulting nonlinear problems. Dedicated finite elements methods, with appropriate mixed formulations or stabilization techniques have to be introduced. For clarity of exposition and for pedagogic purposes, the Stokes problem is first considered, and a concise presentation of the general theory in an abstract framework is provided.

*Jean-Frédéric Gerbeau, Claude Le Bris, and Tony Lelièvre*

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198566656
- eISBN:
- 9780191718014
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566656.003.0002
- Subject:
- Mathematics, Mathematical Physics

This chapter focuses on the modelling of one-fluid magnetohydrodynamics problems. The crucial point under consideration is the coupling between hydrodynamics phenomena and electromagnetic phenomena. ...
More

This chapter focuses on the modelling of one-fluid magnetohydrodynamics problems. The crucial point under consideration is the coupling between hydrodynamics phenomena and electromagnetic phenomena. From a mathematical viewpoint, the coupling induces a nonlinearity, additional to the nonlinearities already present in the hydrodynamics. A series of difficult, thus interesting, problems follow. With a reasonable amount of theoretical efforts, these problems can be dealt with. For instance, it can be shown that a system coupling the time-dependent incompressible Navier-Stokes equations with a simplified form of the Maxwell equations (the so-called low-frequency approximation) is well-posed when the electromagnetic equation is taken time-dependent, in parabolic form. In contrast, the same model is likely to be ill-posed when the electromagnetic equation is taken time-independent, in elliptic form.Less

This chapter focuses on the modelling of one-fluid magnetohydrodynamics problems. The crucial point under consideration is the coupling between hydrodynamics phenomena and electromagnetic phenomena. From a mathematical viewpoint, the coupling induces a nonlinearity, additional to the nonlinearities already present in the hydrodynamics. A series of difficult, thus interesting, problems follow. With a reasonable amount of theoretical efforts, these problems can be dealt with. For instance, it can be shown that a system coupling the time-dependent incompressible Navier-Stokes equations with a simplified form of the Maxwell equations (the so-called low-frequency approximation) is well-posed when the electromagnetic equation is taken time-dependent, in parabolic form. In contrast, the same model is likely to be ill-posed when the electromagnetic equation is taken time-independent, in elliptic form.