不可压缩磁流体动力学方程的非协调自适应有限元高效算法研究

Incompressible magnetohydrodynamic (MHD) equations is the mathematical and physical formula which used to describe the interaction between a incompressible, electrically conducting fluid and the magnetic field. In this project, we focus on devising the adaptive nonconforming finite element algorithm for the 2D/3D incompressible MHD equations for its nonlinear and strong coupling properties. Firstly, the low order Crouzeix-Raviart nonconforming finite element pair is used to the velocity and pressure spatial discretization. The R-T finite element is used for the magnetic field. For nonlinearity and strong coupling, we combine Stokes, Newton, Oseen iterative scheme and penalty finite element method, two level finite element method to design the decoupling iterative method which is suitable for different physical parameters. And in order to improve stability and simplify complexity of the decoupling iterative method, we develop the residual and recovery type nonconforming a posteriori error estimators and to design efficiently novel adaptive nonconforming finite element method. Secondly, the stability, convergence and error estimates of the decoupling iterative method are analyzed. And on this bais to analyze the efficiency and reliability for the adaptive nonconforming finite element method. Finally, we develop application software package with good portability and highly adaptability for the numerical implentation of adaptive finite element method on high performance computer systems.

不可压缩磁流体动力学（MHD）方程是描述不可压缩导电流体与磁场之间相互作用的非线性耦合数学物理方程。本项目主要针对2D/3D不可压缩MHD方程的非线性和强耦合性，研究其数值逼近中的非协调自适应有限元算法和理论分析。首先，速度和压力空间离散采用低阶Crouzeix-Raviart非协调有限元配对，磁场离散采用R-T元。针对其非线性和强耦合性，结合Stokes、Newton、Oseen迭代和加罚有限元、两网格思想，设计适合不同物理参数的解耦迭代算法。为提高解耦迭代算法的稳定性和降低计算的复杂性，构造残量型非协调后验误差估计子和重构型非协调后验误差估计子，设计新型的高效非协调自适应有限元算法。其次，给出解耦迭代算法的稳定性、收敛性和最优误差估计，并分析非调自适应有限元算法的可靠性和有效性。最后，研制具备良好可移植性、适应性强的自适应计算程序软件包，从而可在高性能计算机上实现数值模拟。

不可压磁流体动力学（MHD）方程是描述不可压缩导电流体与磁场之间相互作用的非线性耦合数学物理方程。本项目主要针对2D/3D不可压MHD方程的非线性和强耦合性，研究其数值逼近中的非协调自适应有限元算法和理论分析。首先，基于协调元和非协调元，针对控制方程组的强耦合性和非线性，提出适合不同粘性系数的加罚迭代算法，以及为了进一步提高计算效率，提出以上算法所对应的两水平算法。分析了每种算法的稳定性、收敛性和最优误差估计，进行若干数值实验，验证了所提出算法的有效性。其次，为了进一步地降低求解规模，提出基于Gauge-Uzawa算法及旋转速度校正算法的全解耦算法，证明了数值解的能量稳定性和收敛性，并给出数值算例，验证了理论分析的有效性。最后，基于梯度恢复型后验误差估计子，提出较为高效的自适应算法，分析了算法的有效性和可靠性，数值算了验证了理论分析。该项目的研究工作将有助于非线性科学研究的深入发展和磁流体动力学在工程技术中的广泛应用。