项目名称: 若干超导数学模型的自适应有限元方法
项目编号: No.11201307
项目类型: 青年科学基金项目
立项/批准年度: 2013
项目学科: 数理科学和化学
项目作者: 徐一峰
作者单位: 上海师范大学
项目金额: 22万元
中文摘要: 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。
中文关键词: 后验误差估计;自适应有限元方法;收敛性;变分不等式;
英文摘要: 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.
英文关键词: a posteriori error estimation;adaptive finite element method;convergence;variational inequality;