具有curl、div算子的MHD方程在非光滑区域上的解具有非H1空间奇异性而造成许多数值方法不能获得正确、最优收敛的逼近解，也给理论分析带来极大的挑战。本项目提出研究一类新的间断有限元方法数值求解MHD方程：有限元L2投影重构间断有限元稳定化方法，其中通过有限元L2投影局部重构curl、div算子来正确、最优逼近非H1空间解，也通过有限元L2投影设计稳定化方法来稳定逼近具有边界层现象的解。研究目标是发展二阶、四阶curl型MHD方程有源问题（稳态与依赖时间问题）和相关的二阶、四阶curl特征值问题及不定方程的这类新的间断有限元方法及其理论（稳定性、收敛性、误差估计），研究重点是非H1空间解的逼近、离散紧性理论。还做基准问题的数值实验，检验方法的有效性和理论的正确性。由于许多重要的数学物理问题都涉及curl、div算子，从而本项目的研究具有重要科学意义与广泛应用背景。

Magnetohydrodynamic equations(MHD equations) couples Maxwell equations and Navier-Stokes equations. In addition to the Laplace operator, the main partial differential operators are curl and div(divergence) operators. Under complex domains(e.g., nonconvex domains with reentrant corners and edges) , the magnetic field solution has strong singularity and does not belong to H1 space(the gradient is not L2 integrable), while high Reynolds number causes boundary layer effects on the solution, i.e., the solution oscillates drastically in some very narrow regions. The consequence is that many finite element methods cannot correctly solve the non H1 space magnetic field solution and lead to highly oscillatory approximations to the velocity and the pressure solutions. Also, the theory and analysis would be extremely difficult. This project is to propose a class of discontinuous finite element method, using the finite element L2 projection to reconstruct the curl and div operators in the discrete distributional sense so that the magnetic field is only required to belong to L2 space, and using the finite element L2 projection to construct the stabilization method for efficiently approximate the velocity and the pressure. Consequently, the non H1 space magnetic field solution can be correctly and optimally approximated while the finite element solutions of the velocity and the pressure are more stable and have higher accuracy even if the mesh is coarser. We shall study the second-order and the fourth-order curl MHD equations and the related second-order and fourth-order curl eigenvalue problems and the corresponding indefinite equations by this class of discontinuous finite element method, and at the same time, we shall rigorously analyze the stability and the convergence, and establish the error bounds. In particular, we shall provide a theory for the so-called discrete compactness, which is the key for the spectral-correct and spurious-free approximations in the discrete eigenvalue problems. It is also the key for the well-posedness and the convergence of the finite element discretizations of the indefinite MHD equations and the time-dependent MHD equations. Since MHD equations has extensive applications in various circumstances of electromagnetic fields and relates to many important mathematical and physical problems, this project has important scientific significance and practical applications.