This paper aims to resolve two important issues for reliable large-scale numerical simulation of three-dimensional time-dependent magnetohydrodynamic（MHD） equations:the divergence-free constraint for the magnetic in...
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This paper aims to resolve two important issues for reliable large-scale numerical simulation of three-dimensional time-dependent magnetohydrodynamic（MHD） equations:the divergence-free constraint for the magnetic induction and,fast and robust solvers for the discrete *** first issue is treated by adopting a mixed formulation based on magnetic vector potential $\BA$,in which RDSHAROLLAR（Bu,p）$ pair is discretized by $\BP-P$ Taylor-Hood element and $\BA$ is approximated by second order N\'{e}d\'{e}lec's edge *** roved the energy law,uniqueness and existence of the discrete *** solve the second issue,we developed a parallel implicit iterative solver for the discrete problem to allow for long time stable simulation and relative big time-step *** make the iterative solvers easy to implement and the nonlinear iteration fast to converge,in every nonlinear iteration we solve two special sub-problems:one is convection-diffusion problem for the magnetic vector potential and the other one is the disturbed incompressible Navier-Stokes equation which accounts for coupled effect between the fluid and magnetic *** efficiently solve the resulted linear algebraic systems,preconditioned GMRES methods using modified auxiliary space preconditioning for edge element and modified least-squares commutator（LSC） preconditioner for Navier-Stokes part are *** numerical examples are conducted to show validity of mixed finite element methods,the quasi-optimality with respect to the degree of freedoms and the scalability of the proposed parallel implicit Picard-Krylov solver.
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