[Paper Review] A Mixed DG method and an HDG method for incompressible magnetohydrodynamics

Weifeng Qiu, Ke Shi|arXiv (Cornell University)|Feb 6, 2017

Advanced Numerical Methods in Computational Mathematics20 references3 citations

TL;DR

This paper proposes a mixed discontinuous Galerkin (DG) method and a hybridized DG (HDG) method for stationary incompressible magnetohydrodynamics (MHD) with two types of boundary conditions. Using novel discrete Sobolev embedding estimates for discontinuous polynomials, the authors establish optimal a priori error estimates for all variables—velocity, magnetic field, pressure, and Lagrange multiplier—under minimal regularity assumptions, marking the first such analysis for DG methods applied to nonlinear MHD systems.

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 a recent work proposed by Houston et. al. 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 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.

Motivation & Objective

• To develop a mixed DG method for the stationary incompressible MHD system with two distinct types of boundary or constraint conditions.
• To establish a priori error estimates for the nonlinear MHD equations using discontinuous Galerkin methods, addressing the challenge of nonlinear coupling between fluid and magnetic fields.
• To achieve optimal convergence rates for all unknowns (velocity, pressure, magnetic field, and Lagrange multiplier) under minimal regularity assumptions on the exact solution.
• To propose and analyze the first divergence-free HDG method for incompressible MHD, ensuring local conservation and enhanced stability.
• To extend existing linearized MHD DG analysis to the full nonlinear system via new discrete Sobolev-type estimates for the $L^3$-norm of the discrete magnetic field.

Proposed method

• Adapts an interior penalty DG (IP-DG) scheme from a prior work on linearized MHD to the nonlinear stationary MHD system.
• Introduces two novel discrete Sobolev embedding-type estimates for discontinuous polynomial approximations to control the $L^3$-norm of the magnetic field, essential for handling nonlinear terms.
• Employs a mixed finite element formulation with mixed variables: velocity $\mathbf{u}$, pressure $p$, magnetic field $\mathbf{b}$, and Lagrange multiplier $r$ for the divergence-free constraint.
• For the HDG method, applies hybridization to eliminate element-wise degrees of freedom, leading to a globally coupled system only on the skeleton of the mesh, improving efficiency.
• Uses edge, face, and volume degrees of freedom for the magnetic field approximation, with reconstruction of a globally continuous-like field $\tilde{\mathbf{b}}_h$ to facilitate error analysis.
• Applies shape-regularity assumptions and Cauchy-Schwarz inequalities to bound the difference between the discrete magnetic field and its reconstructed version, enabling stability and convergence analysis.

Experimental results

Research questions

• RQ1 Can optimal convergence rates be achieved for all unknowns in a DG method for the nonlinear incompressible MHD system under minimal regularity assumptions?
• RQ2 How can discrete Sobolev embedding estimates be constructed for discontinuous polynomial approximations to control the $L^3$-norm of the magnetic field in nonlinear MHD?
• RQ3 What are the convergence properties of a mixed DG method for nonlinear MHD with two distinct types of boundary conditions for the magnetic field and Lagrange multiplier?
• RQ4 Can a divergence-free HDG method be formulated for incompressible MHD, and how does it compare to the mixed DG method in terms of stability and convergence?
• RQ5 What are the implications of local conservation of velocity and magnetic field in DG and HDG formulations for the accuracy and physical consistency of MHD simulations?

Key findings

• The mixed DG method achieves optimal convergence rates in the energy norm for velocity, pressure, and magnetic field under minimal regularity assumptions on the exact solution.
• The Lagrange multiplier associated with the divergence constraint on the magnetic field exhibits suboptimal convergence order, though the method remains stable and accurate.
• The proposed discrete Sobolev embedding estimates for discontinuous polynomials enable control of the $L^3$-norm of the magnetic field, which is critical for analyzing nonlinear coupling terms.
• The first divergence-free HDG method for incompressible MHD is proposed, preserving local conservation and offering a competitive alternative to the mixed DG method.
• The analysis confirms optimal convergence for all unknowns in the HDG method, with the same convergence rates as in the mixed DG method under the same regularity assumptions.
• This work provides the first a priori error estimates for DG methods applied to the nonlinear stationary incompressible MHD equations, filling a significant gap in the literature.