有限元重构方法及在Cahn-Hilliard方程高效计算方法中的应用

The project is concerned with the development of efficient and reliable numerical methods for the Cahn-Hilliard equation. The finite element recovery techniques and mesh adaption will be employed to overcome these difficulties due to the small parameter and the high order derivative appear in the Cahn-Hilliard equation. The research topics will include study the finite element recovery techniques (such as the value recovery, direction derivative recovery and high order derivative recovery), design new recovery-based (discontinuous Galerkin) finite element methods and the recovery type a posteriori error estimates, develop the efficient and robust adaptive methods for the high order partial differential equations, establish the convergence of adaptive finite element methods based on the recovery-type estimators and the mesh optimization, expand the applications of the finite element recovery techniques. The main goal of this project is integrating the finite element recovery techniques, discontinuous Galerkin methods, a posteriori error estimation and adaptive methods to develop some robust and high accuracy numerical methods for the Cahn-Hilliard equation, and establish the related mathematical theory of the newly developed numerical methods. The project also includes implementation of the newly recovery method and adaptive methods as computer codes.

本项目主要针对Cahn-Hilliard方程的高效数值方法开展研究。拟结合有限元重构技术和网格自适应技术来处理Cahn-Hilliard方程的数值计算中由小参数和高阶引起的困难。系统研究有限元重构技术（包括函数值重构、方向导数重构及高阶导数重构），设计基于重构的新型（间断）有限元方法和后验误差估计，发展高阶偏微分方程的高效自适应算法，研究基于重构型后验误差估计和网格优化的自适应算法的收敛性，拓展有限元重构方法的应用范围。目的是希望将有限元重构算法、间断有限元方法、后验误差估计、自适应算法等有机结合，发展Cahn-Hilliard方程的高效数值方法以及相关数学理论，并研究新的高效算法的程序实现。

本项目主要研究有限元方法的重构技术及其在后验误差估计、自适应计算和相场模型的高效数值求解中的应用。针对各类有限元（包括Lagrange元、Crouzeix-Raviart元、棱有限元、间断有限元等），提出了几种新的超收敛重构方法；发展了基于梯度重构技术的线性有限元求解四阶问题；提出了可计算的网格几何参数来度量网格的对称性，进而刻画网格上的超收敛性；将梯度重构技术和网格优化技术与自适应技术结合，设计了新的自适应有限元方法，改进经典自适应有限元方法迭代步数多、误差估计不准确的缺点；针对相场模型中的Allen-Cahn方程、Cahn-Hilliard方程和Cahn-Hilliard-Navier-Stokes方程，建立了保持能量耗散的全离散格式及其后验误差估计理论，发展了相应的时空自适应方法；针对薛定谔方程和非线性变分波动方程等，构造了保持方程守恒/耗散特征的高阶间断有限元方法，适用于长时间数值模拟。

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