三维非线性磁流体力学的自适应有限元方法

项目名称： 三维非线性磁流体力学的自适应有限元方法

项目编号： No.11471329

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 毛士鹏

作者单位： 中国科学院数学与系统科学研究院

项目金额： 65万元

中文摘要： 本项目主要研究三维非线性全耦合不可压缩磁流体动力学方程组的高效自适应有限元方法和后验误差估计理论。磁流体动力学（MHD）方程描述的是导电流体与电磁场之间的相互作用，对于研究等离子体的宏观运动具有重要作用。随着国际热核聚变实验堆(ITER)的重点投入，MHD方程的研究与模拟得到了前所未有的发展和关注。自适应有限元方法是求解偏微分方程的高效数值方法之一，也是当前科学计算研究的热点。本课题围绕磁约束热核聚变反应堆的关键部件研发中所涉及的磁流体力学问题，重点研究强磁场下的三维复杂几何、非线性、金属液体不可压缩MHD方程组的高效自适应有限元方法。我们着重解决非线性MHD方程组残量型有限元后验误差估计的数学理论困难，发展高效的自适应有限元方法和相应的大型离散方程组求解器，模拟托克马克装置中的高Hartmann数、大洛伦兹力的金属液体。

中文关键词： 有限元方法；后验误差估计；自适应算法；三维磁流体

英文摘要： The main aim of the proposed project is to study the effective adaptive algorithms and mathematical theory of the a posteriori error estimates of finite element methods for the three dimensional nonlinear, fully-coupled incompressible Magnetohydrodynamics. Magnetohydrodynamics (MHD) describes the dynamics of electrically conducting fluid under the magnetic fields and plays an important role in the macroscopic motion of plasma. With the key investment of International Thermonuclear Experimental Reactor (ITER), the research on Magnetohydrodynamics and its simulation are getting more and more important. On the other hand, adaptive finite element method (AFEM) is one of the most efficient numerical methods for the partial differential equations, which makes it a hot topic in the scientific computing reserch. Focusing on the MHD problems relevant to the key ingredients of magnetically confined fusion, this project studies the efficient AFEMs for the 3D nonlinear, liquid metal, incompressible magnetohydrodynamics equations with complex structures and under intensive magnetic fields. We mainly build the mathematical theory of the residual type a posteriori error estimators of finite element methods for nonlinear MHD equations, develop its efficient adaptive algorithms and the corresponding solver of the large discrete systems, and simulate the metal liquid MHD in the device of Tokamak with high Hartmann number, large Lorentz force.

英文关键词： Finite element methods;A posteriori error estimate;Adaptive algorithms;3D Magnetohydrodynamics