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.

翻译：我们提出了一种无散且满足$H(div)$相容性的混合间断伽辽金（HDG）方法，以及一种计算高效的变体，称为嵌入式HDG（E-HDG），用于求解稳态不可压缩黏性电阻磁流体动力学（MHD）方程。所提出的E-HDG方法对矢量值解（速度场和磁场）使用连续的面未知量，而对标量变量（压力和磁压）使用间断的面未知量。由于自由度数量显著减少，与HDG方法相比，这种函数空间的选择使得E-HDG在计算上更具优势。对于三维/高阶/精细网格情形，这一优势更为显著。在单纯形网格上，采用特定近似空间选择的所提方法对于线性（化）MHD方程是适定的。对于非线性MHD问题，我们提出了一种利用皮卡迭代结合所提线性离散化的简单方法。该方法的优点在于，速度场和磁场的无散性及$H(div)$相容性对于非线性MHD方程自动得以保持。我们通过多种数值实验（包括具有光滑解和奇异解的二维及三维问题），研究了所提E-HDG方法在线性和非线性MHD情形下的精度与收敛性。数值算例表明，所提方法具有压力鲁棒性，并且对于光滑和奇异问题，所得速度场和磁场的散度均为机器精度零。