An MHD Stokes eigenvalue problem and its approximation by a spectral collocation method

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

“…Consider the following problem (see Türk 4, (6) ): $$$$ \left\{\begin{array}{cc}-\Delta \mathbf{u}+\nabla p-{(Ha)}^2\left(-\nabla V+\mathbf{u}\times \mathbf{H}\right)\times \mathbf{H}=\lambda \mathbf{u},\kern0.30em & in\kern0.20em \Omega, \\ {}\nabla \cdotp \mathbf{u}=0,\kern0.30em & in\kern0.20em \Omega, \\ {}\mathbf{u}=0,\kern0.30em & on\kern0.20em \mathrm{\partial \Omega },\end{array}\right. $$$$ where $$$ \Omega \subset {R}^2 $$$ stands for the cross‐section of channel that the flow takes place, $$$ \mathbf{u} $$$ is the velocity field, $$$ \mathbf{H} $$$ is the magnetic field and $$$ \mathbf{H}=\left(0,{H}_0,0\right) $$$ when the magnetic field is applied vertically and $$$ \left({H}_0,0,0\right) $$$ when the magnetic field is applied horizontally, $$$ p $$$ is the pressure, $$$ V $$$ is the electric potential, and …”

“…As described in the literature, 4 making use of Ohm's law and the conservation of the electric current results in $$$ \Delta V $$$ = 0 (see Gürbüz & Tezer‐Sezgin 3 and Davidson 22 ), and homogenous boundary conditions for $$$ V $$$ implies that the electric potential is identically zero over the problem domain. Thus, the term $$$ \nabla V $$$ in the first equation of () drops out.…”

“…Merve Gürbüz and Tezer‐Sezgin 3 solved the MHD Stokes flow equation in a lid‐driven cavity and backward‐facing step channel in the presence of a uniform magnetic field with different orientations. Türk 4 proposed the MHD Stokes eigenvalue problem and studied its numerical approximation by a Chebyshev spectral collocation method. The flow considered in Türk 4 is assumed to take place in a channel with a square cross‐section, and the author pointed out that an extension to more general geometries can be made by coordinate transformations or domain decomposition approaches.…”

“…Türk 4 proposed the MHD Stokes eigenvalue problem and studied its numerical approximation by a Chebyshev spectral collocation method. The flow considered in Türk 4 is assumed to take place in a channel with a square cross‐section, and the author pointed out that an extension to more general geometries can be made by coordinate transformations or domain decomposition approaches. Based on the above work, in this paper, we adopt the discontinuous Galerkin finite element method (abbreviated as DGFEM) and the nonconforming enriched Crouzeix‐Raviart (ECR) finite element method to solve the MHD Stokes eigenvalue problem on domains of more general shape, which stand for the cross‐section of the channel that the flow takes place.…”

Least-Squares Finite Element Method for Solving Stokes Flow under Point Source Magnetic Field

“…They implemented DRBEM in solving this problem. In [17], the solution of the MHD Stokes eigenvalue problem was approximated by using the Chebyshev spectral collocation method (CSCM).…”