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We proposed a divergence-free and $H(div)$-conforming embedded-hybridizeddiscontinuous Galerkin (E-HDG) method for solving stationary incompressibleviso-resistive magnetohydrodynamic (MHD) equations. In particular, the E-HDGmethod is computationally far more advantageous over the hybridizeddiscontinuous Galerkin (HDG) counterpart in general. The benefit is evensignificant in the three-dimensional/high-order/fine mesh scenario. On asimplicial mesh, our method with a specific choice of the approximation spacesis proved to be well-posed for the linear case. Additionally, the velocity andmagnetic fields are divergence-free and $H(div)$-conforming for both linear andnonlinear cases. Moreover, the results of well-posedness analysis,divergence-free property, and $H(div)$-conformity can be directly applied tothe HDG version of the proposed approach. The HDG or E-HDG method for thelinearized MHD equations can be incorporated into the fixed point Picarditeration to solve the nonlinear MHD equations in an iterative manner. Weexamine the accuracy and convergence of our E-HDG method for both linear andnonlinear cases through various numerical experiments including two- andthree-dimensional problems with smooth and singular solutions. For smoothproblems, the results indicate that convergence rates in the $L^2$ norm for thevelocity and magnetic fields are optimal in the regime of low Reynolds numberand magnetic Reynolds number. Furthermore, the divergence error is machine zerofor both smooth and singular problems. Finally, we numerically demonstratedthat our proposed method is pressure-robust.

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