The discontinuous Galerkin and the nonconforming ECR element approximations for an MHD Stokes eigenvalue problem

In this paper, we adopt the discontinuous Galerkin finite element method and the enriched Crouzeix-Raviart finite element method to study the magnetohydrodynamic (MHD) Stokes eigenvalue problem describing the flow of a viscous and electrically conducting fluid in a duct under the influence of a uniform magnetic field. We give the convergence and error analysis for the approximations, and the theoretical analysis and numerical experiments show that the methods are effective and can be applied to general domains. We also explore the influence of the Hartmann number on the eigenpairs and the consequential variation of the eigenstructure with the magnetic field by numerical experiments.