Mixed finite element methods for stationary incompressible magneto-hydrodynamics

Summary.: A new mixed variational formulation of the equations of stationary incompressible magneto-hydrodynamics is introduced and analyzed. The formulation is based on curl-conforming Sobolev spaces for the magnetic variables and is shown to be well-posed in (possibly non-convex) Lipschitz polyhedra. A finite element approximation is proposed where the hydrodynamic unknowns are discretized by standard inf-sup stable velocity-pressure space pairs and the magnetic ones by a mixed approach using Nédélec's elements of the first kind. An error analysis is carried out that shows that the proposed finite element approximation leads to quasi-optimal error bounds in the mesh-siz

Stabilized finite element approximation of the stationary magneto-hydrodynamics equations

In this work we present a stabilized finite element method for the stationary magneto-hydrodynamic equations based on a simple algebraic version of the subgrid scale variational concept. The linearization that yields a well posed linear problem is first identified, and for this linear problem the stabilization method is designed. The key point is the correct behavior of the stabilization parameters on which the formulation depends. It is shown that their expression can be obtained only on the basis of having a correct error estimate. For the stabilization parameters chosen, a stability estimate is proved in detail, as well as the convergence of the numerical solution to the continuous one. The method is then extended to nonlinear problems and its performance checked through numerical experiments

A Conservative Finite Element Solver for MHD Kinematics equations: Vector Potential method and Constraint Preconditioning

A new conservative finite element solver for the three-dimensional steady magnetohydrodynamic (MHD) kinematics equations is presented.The solver utilizes magnetic vector potential and current density as solution variables, which are discretized by H(curl)-conforming edge-element and H(div)-conforming face element respectively. As a result, the divergence-free constraints of discrete current density and magnetic induction are both satisfied. Moreover the solutions also preserve the total magnetic helicity. The generated linear algebraic equation is a typical dual saddle-point problem that is ill-conditioned and indefinite. To efficiently solve it, we develop a block preconditioner based on constraint preconditioning framework and devise a preconditioned FGMRES solver. Numerical experiments verify the conservative properties, the convergence rate of the discrete solutions and the robustness of the preconditioner.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1712.0892

hp-Finite element solution of coupled stationary magnetohydrodynamics problems including magnetostrictive effects

We extend our existing hp-finite element framework for non-conducting magnetic fluids (Jin et al., 2014) to the treatment of conducting magnetic fluids including magnetostriction effects in both two- and three-dimensions. In particular, we present, to the best of our knowledge, the first computational treatment of magnetostrictive effects in conducting fluids. We propose a consistent linearisation of the coupled system of non-linear equations and solve the resulting discretised equations by means of the Newton–Raphson algorithm. Our treatment allows the simulation of complex flow problems, with non-homogeneous permeability and conductivity, and, apart from benchmarking against established analytical solutions for problems with homogeneous material parameters, we present a series of simulations of multiphase flows in two- and three-dimensions to show the predicative capability of the approach as well as the importance of including these effects

Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamics system

The incompressible MHD equations couple Navier-Stokes equations with Maxwell's equations to describe the flow of a viscous, incompressible, and electrically conducting fluid in a Lipschitz domain $\Omega \subset \mathbb{R}^3$. We verify convergence of iterates of different coupling and decoupling fully discrete schemes towards weak solutions for vanishing discretization parameters. Optimal first order of convergence is shown in the presence of strong solutions for a splitting scheme which decouples the computation of velocity field, pressure, and magnetic fields at every iteration step