Structure-preserving finite element methods for stationary MHD models

Abstract

In this paper, we develop a class of mixed finite element scheme for stationary magnetohydrodynamics (MHD) models, using magnetic field B and current density j as the discretization variables. We show that the Gauss's law for the magnetic field, namely ∇·B = 0, and the energy law for the entire system are exactly preserved in the finite element schemes. Based on some new basic estimates for H h (div), we show that the new finite element scheme is well-posed. Furthermore, we show the existence of solutions to the nonlinear problems and the convergence of Picard iterations and finite element methods under some conditions.2010 Mathematics Subject Classification. Primary 65N30, 65N12.

Convergence of a B-E based finite element method for MHD models on Lipschitz domains

“…We also show another strategy to impose the strong divergence-free condition, instead of using Lagrange multipliers as in the previous work [21] by one of the authors and collaborator. We introduce an augmented term (∇ · B, ∇ · C) in the variational formulation.…”

“…To preserve magnetic Gauss's law precisely on the discrete level with electric and magnetic fields as variables, a class of finite element schemes was developed in [20,21] for the time dependent and the stationary MHD systems respectively. The magnetic field B is discretized by the H(div) conforming Raviart-Tomas [27] or BDM [9] elements.…”

“…The magnetic field B is discretized by the H(div) conforming Raviart-Tomas [27] or BDM [9] elements. An electric variable, either the electric field E in [20] or the current density j in [21], is retained in the formulation and discretized by the H(curl) conforming elements in the same discrete de Rham complex.…”

“…Several variants of this type of schemes exist, and we choose to consider a B-E based formulation in the discussions below. This formulation is the stationary case of [20] and differs from the B-j formulation in [21] by a projection of nonlinear terms (see Section 4 below for details). Therefore we do not claim the discretization studied in this paper as a brand new method, although the precise formulation has not appeared in the literature to the best of our knowledge.…”

“…We prove the convergence of the finite element scheme by carefully choosing interpolation functions (see (.) below). Comparing with the convergence analysis in [21] for the B-j based finite element methods, we adopt a new strategy and, as a result, only assume weak regularity of the solutions in this paper ((.) below). This demonstrates the convergence of the Picard iterations and the finite element schemes for singular solutions.…”