基于磁势变量磁流体方程组的有限元方法研究

With the need of engineering technology, the development of high efficiency numerical algorithm for magnetohydrodynamics (MHD) has become one of the research hotspots in the field of Computational Mathematics. As far as the traditional magnetic field formulation incompressible magnetohydrodynamic equations are concerned, its variational formulation is less stable than the corresponding Navier-Stokes equation which is caused by the nonlinear Lorenz force item. Thus it is more difficult to solve numerically. Moreover, the realization of divergence-free discrete magnetic field (satisfy Gauss's law) is also not easy. Instead of the traditional magnetic field formulation, we will study a magnetic potential formulation incompressible MHD equations. Its varitional fromulation has the following advantages: more stable than the traditional one, reflecting the inhibition of Lorenz force to the fluid filed, easy to design stable and efficient solvers, achieving natural divergence free numerical solutions. This project will focus on in-depth study based on the magnetic potential formulation incompressible magnetohydrodynamic equations. We prove the existence and uniqueness of the weak solution of the equations. Some stable and conservative finite element discrete schemes and their corresponding theoretical analysis will be emphatically studied. Meanwhile, we will design efficient space-time adaptive finite element method for the magnetic potential formulation incompressible magnetohydrodynamic equations. The research of this variational model and its finite element method can effectively improve the defects of traditional magnetic formulation MHD computation, and will play an important role in the future computational magnetohydrodynamics.

近年来，发展磁流体力学（MHD）的高效数值算法成为计算数学领域的研究热点之一。对于传统的磁场变量形式不可压缩磁流体力学方程组来说，由于非线性项洛伦兹力的存在，其变分形式上比相应的Navier-Stokes方程表现更不稳定，更加难以数值求解，而且不太容易实现无散度的离散磁场数值解（满足磁场高斯定律）。本项目研究一种采用磁势变量替代传统磁场变量的不可压缩磁流体力学方程组，该模型的变分形式比传统形式多一个稳定项，正好体现了洛伦兹力对流体的抑制作用，更容易设计高效稳定的求解器，而且可以实现自然的无散度数值解。本课题将围绕基于磁势变量的磁流体方程组进行深入研究，证明该方程组弱解的存在唯一性，着重研究该模型的稳定、守恒的有限元离散数值格式及其相应理论分析，并发展高效的时空自适应有限元算法。该变分模型及其有限元方法的研究能够有效改善传统基于磁场变量MHD计算的缺陷，在计算磁流体力学中能够发挥重要作用。

磁流体力学在等离子体物理,天体物理，冶金化工，航空航天以及受控热核聚变等技术科学都有着重要的应用基础，复杂几何、多物理场耦合的不可压缩磁流体力学系统的高效数值计算是一个挑战性的研究课题。另一方面，离散磁场变量保持无散度性质（满足磁场高斯定律）是一个十分重要的物理定律。但是传统的基于磁感应强度变量形式的不可压缩磁流体力学模型难以实现无散度的离散磁场数值解（满足磁场高斯定律）。本课题主要研究了一种采用磁势变量替代传统磁场变量的不可压缩磁流体力学模型及其数值方法。我们在一定假设条件下给出了该模型系统解的适定性证明。然后对模型采用全离散混合有限元方法求解，该离散系统产生一个完全无散度的磁感应离散解。通过复杂理论分析严格地建立了全离散格式的误差分析理论。开发了大量三维并行程序验证了模型的有效性和数值方法的可靠性，并揭示了方腔流体中随着流体导电系数增大会大幅增加涡度数量的物理规律。基于相场方法提出了扩散界面的两相磁流体模型，针对模型设计了无条件能量稳定有限元数值格式并建立了收敛性理论。针对实际热磁流体物理参数依赖于热场的问题，我们研究了各个系数依赖于温度的三维不可压缩变系数热磁流体模型的数值方法和理论。该项目的研究有效地丰富了磁流体力学的计算方法和理论。