若干超导数学模型的自适应有限元方法

Superconductivity has a wide range of applications in science and industry. Therefore, it is of great value to investigate relevant mathematical theory and computational methods. Crtical-state models, including the Bean model, the modified Bean model and the power-law model, provide macroscopic description of an electromagnetization field in a superconductor and are extensively used in physics and engineering community with large-scale applications of superconducting materials involved. Thanks to low regularity of solutions to critica-state models, it is a top priority to develop efficient adaptive finite element methods for real computations. The goal of this program is twofold: residual-based a posteriori error estimation for finite element approximations of some variational problems associated with these models; design of adaptive finite element methods as well as convergence and complexity analysis of some paricular ones. It is expected that several fast and efficient algorithms will be available for numerical simulation in applied superconductivity and some advances will be made in the theory of adaptive finite element methods for nonlinear problems.

超导材料在科学研究和工业生产中有广泛的应用前景，相关数学问题的理论分析和计算具有重要的应用价值。临界态模型是一类描述超导材料电磁性质的宏观数学模型，在工程应用和科学研究中，此类模型经常被工程师和物理学家用来研究大尺度超导材料的物理特性。由于临界态模型解的光滑性较低，亟需发展高效的自适应有限元方法来数值求解。本项目拟针对三个具体模型（Bean 模型、修正Bean模型和幂律模型）对应的变分问题，借鉴已有的相关研究成果，获得有限元方法的残差型后验误差估计理论，进而提出相应的自适应有限元方法，并对某些具体算法进行收敛性和计算复杂度分析。本课题既有望为超导问题的数值模拟提供若干快速、高效的计算方法，也有望在非线性问题的自适应有限元方法理论方面取得新的进展。

本项目针对Robin系数重构问题，热流分布重构问题和四阶弹性薄板弯曲问题研究了相关的自适应有限元方法，给出了具体的残量型后验误差估计，在此基础上设计了自适应方法并分析了算法的收敛性。数值例子说明了理论分析是可靠的。本课题的研究在非线性问题的自适应有限元方法理论基础方面取得了一定的进展。其中的一个重要创新点是利用自适应算法的收敛性分析导出后验误差估计子。这一研究思路有别于传统的方法，不但对本项目所涉及的Robin系数重构问题的自适应方法有效，还可以运用到其他非线性问题自适应有限元方法的研究。