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Mathematical Methods for the Magnetohydrodynamics of Liquid Metals$
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Jean-Frédéric Gerbeau, Claude Le Bris, and Tony Lelièvre

Print publication date: 2006

Print ISBN-13: 9780198566656

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198566656.001.0001

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NUMERICAL APPROXIMATION OF ONE-FLUID PROBLEMS

NUMERICAL APPROXIMATION OF ONE-FLUID PROBLEMS

Chapter:
(p.83) 3 NUMERICAL APPROXIMATION OF ONE-FLUID PROBLEMS
Source:
Mathematical Methods for the Magnetohydrodynamics of Liquid Metals
Author(s):

Jean-Frédéric Gerbeau

Claude Le Bris

Tony Lelièvre

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198566656.003.0003

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.

Keywords:   coupled problem, mixed formulations, stabilized finite elements, saddle point problems

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