A mixed DG method and an HDG method for incompressible magnetohydrodynamics

Abstract

In this paper we propose and analyze a mixed DG method and an HDG method for the stationary Magnetohydrodynamics (MHD) equations with two types of boundary (or constraint) conditions. The mixed DG method is based on a recent work proposed by Houston et. al. in [25] for the linearized MHD. With two novel discrete Sobolev embedding type estimates for the discontinuous polynomials, we provide a priori error estimates for the method on the nonlinear MHD equations. In the smooth case, we have optimal convergence rate for the velocity, magnetic field and pressure in the energy norm, the Lagrange multiplier only has suboptimal convergence order. With the minimal regularity assumption on the exact solution, the approximation is optimal for all unknowns. To the best of our knowledge, this is the first a priori error estimates of DG methods for the nonlinear MHD equations. In addition, we also propose and analyze the first divergence-free HDG method for the problem with several unique features comparing with the mixed DG method.2000 Mathematics Subject Classification. 65N30, 65L12.

A Divergence-Conforming Hybridized Discontinuous Galerkin Method for the Incompressible Magnetohydrodynamics Equations

“…The magnetic pressure may be interpreted as a Lagrange multiplier associated with the ∇ • B = 0 condition and, for a suitable choice of boundary conditions, it is identically zero [30]. Inclusion of a magnetic pressure to enforce the ∇ • B = 0 condition is common in both continuous Galerkin [6,9,30,31] and discontinuous Galerkin [21,29,32] finite element methods.…”

“…This requires special treatment of the terms that couple the velocity and magnetic fields to arrive at a method that is energy stable. It should be noted that a previously presented HDG method for the incompressible MHD equations does yield pointwise divergence-free velocity fields [29], but it yields only discretely divergence-free magnetic fields.…”

A new error analysis of a mixed finite element method for the quad-curl problem

“…for real number j ∈ [1, k +3] and p ∈ H j (Ω). Next, we introduce an interpolation ( [6,29]): for any v ∈ H s (curl; Ω) with s > 1 2 , we define…”

“…There are vast literatures on numerical methods solving the MHD model without the quad-curl term ∇×(∇×(∇×(∇×u))), see [3,10,11,15,19,20,23,29] and references therein for detailed information. However, when the quad-curl term ∇ × (∇ × (∇ × (∇ × u))) is present, the design and analysis of numerical methods for the MHD model becomes more difficult and challenging.…”

Regularity results for a model in magnetohydrodynamics with imposed pressure

“…Also in [1,3], the authors have studied the stationary magnetohydrodynamic equations of electrically and heat conducting fluid. For the discretization approaches of (M H D), a few related contributions include mixed finite elements [13,14,16], discontinuous galerkin finite elements [15] or iterative penalty finite element methods [12] and so on. The boundary condition under the form P = P 0 + α i on Γ i , i = 1, .…”