基于高精度技术的自适应有限元方法及应用

自适应有限元方法已成为数值求解科学工程计算的最有效方法之一，特别对存在奇性或者多尺度性质的问题有很强的适应性。理论成熟的自适应有限元方法因误差估计子的非渐近准确性、低效的网格自适应加密策略等因素，使得自适应迭代步数较多，从而计算效率较低。本项目以提高自适应有限元方法的计算效率为目标，针对自适应有限元方法开展系统研究。获得更加精细的有限元误差估计，给出可计算的参数度量自适应网格上超收敛现象；提出新的有限元重构方法，构造渐近准确的后验误差估计；设计高效的网格自适应策略；将超收敛和重构技术、高质量网格生成与优化方法等有限元高精度技术与自适应等有机结合，发展高效且具有数学理论支撑的高效自适应有限元方法，进一步发展薛定谔方程和相场问题的高效自适应方法，并研究高效自适应算法的程序实现。

Adaptive finite element methods is one of the efficient numerical methods for scientific and engineering computation, especially for the problems with singularities and multiscale properties. Due to the use of non-asymptotically exact a posteriori error estimate and ineffective mesh adaptive strategy, the normal adaptive finite element methods are usually require many adaptive iteration steps, and thus reduce the computational efficiency of the adaptive algorithms. This project conducts a systematic investigation on adaptive finite element methods, with the aim of improving the computational efficiency of adaptive methods. The research topics are including deriving more detailed finite element error estimations, proposed computable parameters to measure the superconvergence on adaptive meshes, develop new finite element recovery techniques and construct asymptotically exact a posteriori error estimates, design an efficient mesh adaptive strategy. We plan to integrate the finite element high accuracy techniques including the superconvergence and recovery techniques, high quality mesh generation and optimization in mesh adaptive, to develop an efficient adaptive finite element methods and establish their mathematical theories. Then apply the newly developed adaptive methods for solving Schrödinger equation and phase field problems. This project also includes implementation of the adaptive methods as computer codes.