This paper studies several decoupled penalty methods to overcome the saddle point system of the steady state 2D/3D incompressible magnetohydronamics (MHD). These approaches combine the Oseen iteration and two-level technique with strong uniqueness condition \(0<\frac{\sqrt{2}C_{0}^{2}\max \{1,\sqrt{2}S_{c}\}\Vert {\mathbf{F }}\Vert _{-1}}{(\min \{R_{e}^{-1},S_{c}C_{1}R_{m}^{-1}\})^2}\le 1-\left( \frac{\Vert \mathbf{F }\Vert |_{-1}}{\Vert |\mathbf{F }\Vert _{0}}\right) ^{\frac{1}{2}}<1\) satisfied. For the convenience of implementation, we employ two different simple Lagrange finite element pairs \(P_{1}b-P_{1}-P_{1}b\) and \(P_{1}-P_{0}-P_{1}\) for velocity field, pressure and magnetic field, respectively. Rigorous analysis of the optimal error estimate and stability are provided. We present comprehensive numerical experiments, which indicate the effectiveness of the proposed methods for both two dimensional and three-dimensional problems.

本文研究了几种解耦罚分法，以克服稳态2D / 3D不可压缩磁流体动力学（MHD）的鞍点系统。这些方法将Oseen迭代和两级技术与强唯一性条件\（0 <\ frac {\ sqrt {2} C_ {0} ^ {2} \ max \ {1，\ sqrt {2} S_ {c} \} \ Vert {\ mathbf {F}} \ Vert _ {-1}} {（\ min \ {R_ {e} ^ {-1}，S_ {c} C_ {1} R_ {m} ^ {- 1} \}）^ 2} \ le 1- \ left（\ frac {\ Vert \ mathbf {F} \ Vert | _ {-1}} {\ Vert | \ mathbf {F} \ Vert _ {0}} \ right）^ {\ frac {1} {2}} <1 \）满足。为了实现方便，我们使用了两个不同的简单拉格朗日有限元对\（P_ {1} b-P_ {1} -P_ {1} b \）和\（P_ {1} -P_ {0} -P_ { 1} \）分别用于速度场，压力和磁场。提供了最佳误差估计和稳定性的严格分析。我们目前进行的综合数值实验表明，该方法对于二维和三维问题均有效。