We proposed a divergence-free and $H(div)$-conforming embedded-hybridized discontinuous Galerkin (E-HDG) method for solving stationary incompressible viso-resistive magnetohydrodynamic (MHD) equations. In particular, the E-HDG method is computationally far more advantageous over the hybridized discontinuous Galerkin (HDG) counterpart in general. The benefit is even significant in the three-dimensional/high-order/fine mesh scenario. On a simplicial mesh, our method with a specific choice of the approximation spaces is proved to be well-posed for the linear case. Additionally, the velocity and magnetic fields are divergence-free and $H(div)$-conforming for both linear and nonlinear cases. Moreover, the results of well-posedness analysis, divergence-free property, and $H(div)$-conformity can be directly applied to the HDG version of the proposed approach. The HDG or E-HDG method for the linearized MHD equations can be incorporated into the fixed point Picard iteration to solve the nonlinear MHD equations in an iterative manner. We examine the accuracy and convergence of our E-HDG method for both linear and nonlinear cases through various numerical experiments including two- and three-dimensional problems with smooth and singular solutions. For smooth problems, the results indicate that convergence rates in the $L^2$ norm for the velocity and magnetic fields are optimal in the regime of low Reynolds number and magnetic Reynolds number. Furthermore, the divergence error is machine zero for both smooth and singular problems. Finally, we numerically demonstrated that our proposed method is pressure-robust.

翻译：针对定常不可压缩粘性电阻磁流体动力学（MHD）方程，本文提出了一种无散度且$H(div)$-协调的嵌入杂交间断伽辽金（E-HDG）方法。特别地，E-HDG方法在计算效率上普遍优于杂交间断伽辽金（HDG）方法，在三维/高阶/精细网格场景下优势尤为显著。在单纯形网格上，采用特定近似空间组合时，该方法在线性情形下被证明具有适定性。此外，速度和磁场在线性与非线性情形下均保持无散度特性与$H(div)$-协调性。适定性分析、无散度性质及$H(div)$-协调性的结论可直接推广至本方法的HDG版本。线化MHD方程的HDG或E-HDG方法可结合不动点Picard迭代，以迭代方式求解非线性MHD方程。通过包含二维/三维光滑解与奇异解问题的多组数值实验，我们检验了E-HDG方法在线性与非线性情形下的精度与收敛性。对于光滑问题，结果表明在低雷诺数与低磁雷诺数区域，速度和磁场的$L^2$范数收敛阶达到最优。此外，无论光滑解还是奇异解问题，散度误差均达到机器零量级。最后，数值实验证明该方法具有压力鲁棒性。