A Divergence-Free and $H(div)$-Conforming Embedded-Hybridized DG Method for the Incompressible Resistive MHD equations

TL;DR

This paper introduces a divergence-free, $H(div)$-conforming hybridized DG method and an embedded variant for stationary incompressible resistive MHD equations, offering computational efficiency and preserving key physical properties.

Contribution

The paper develops a novel embedded-HDG method with reduced degrees of freedom that maintains divergence-free and $H(div)$-conforming properties for nonlinear MHD equations.

Findings

Method is pressure robust and divergence-free for both smooth and singular solutions.

Numerical experiments confirm accuracy and convergence for 2D and 3D problems.

Embedded-HDG significantly reduces computational cost compared to traditional HDG.

Abstract

We present a divergence-free and $H(div)$-conforming hybridized discontinuous Galerkin (HDG) method and a computationally efficient variant called embedded-HDG (E-HDG) for solving stationary incompressible viso-resistive magnetohydrodynamic (MHD) equations. The proposed E-HDG approach uses continuous facet unknowns for the vector-valued solutions (velocity and magnetic fields) while it uses discontinuous facet unknowns for the scalar variable (pressure and magnetic pressure). This choice of function spaces makes E-HDG computationally far more advantageous, due to the much smaller number of degrees of freedom, compared to the HDG counterpart. The benefit is even more significant for three-dimensional/high-order/fine mesh scenarios. On simplicial meshes, the proposed methods with a specific choice of approximation spaces are well-posed for linear(ized) MHD equations. For nonlinear MHD problems, we present a simple approach exploiting the proposed linear discretizations by using a Picard iteration. The beauty of this approach is that the divergence-free and $H(div)$-conforming properties of the velocity and magnetic fields are automatically carried over for nonlinear MHD equations. We study the accuracy and convergence of our E-HDG method for both linear and nonlinear MHD cases through various numerical experiments, including two- and three-dimensional problems with smooth and singular solutions. The numerical examples show that the proposed methods are pressure robust, and the divergence of the resulting velocity and magnetic fields is machine zero for both smooth and singular problems.