四阶偏微分方程的杂交间断有限元方法及自适应算法

The fourth-order partial differential equations are widely used in the fields of elastic multi-structures, phase separation models, image denoising, electrostatic micro-electromechanical systems, fluid mechanics, etc. Moreover, fourth-order partial differential equations also play a vital role in designing efficient numerical methods for general multiscale problems. Therefore, it is both theoretically and practically important to investigate efficient numerical methods for fourth-order partial differential equations. This project is intended to develop hybridizable discontinuous Galerkin finite element methods and associated adaptive algorithms for fourth-order partial differential equations. Firstly, optimal discontinuous Galerkin finite element methods based on Hermann-Miyoshi formulation will be proposed for fourth-order partial differential equations, and a new superconvergent numerical solution will be obtained by postprocessing. Then, we present a unified framework of hybridizable finite element methods for fourth-order partial differential equations and characterize the hybridized degrees of freedom, based on which some internal relations of various hybridizable finite element methods will be established. We also provide a unified a priori error analysis for these hybridizable methods and postprocess the numerical solution to obtain a new superconvergent numerical solution. Finally, some reliable and efficient a posteriori error estimators are constructed for superconvergent discontinuous Galerkin finite element methods and hybridizable finite element methods for fourth-order partial differential equations, based on which we design adaptive algorithms and study the quasi-optimal convergence and optimal complexity. It is expected that after implementation of the project, systematic and high-level progresses will be made in the field of numerical methods for the fourth-order partial differential equations.

四阶偏微分方程不仅在组合弹性结构、相分离模型、图像去噪、静电驱动微机电系统、流体力学等领域有着重要应用，而且在一般多尺度问题高效求解算法中起关键作用。因此研究和构造四阶偏微分方程的高效数值求解方法具有重要的理论意义和直接的应用价值。本项目主要研究四阶偏微分方程的杂交间断有限元方法及自适应算法，包括：基于Hermann-Miyoshi形式，构造四阶偏微分方程具有最优收敛阶的间断有限元方法，对数值解进行后处理获得新的超收敛数值解；提出四阶偏微分方程杂交有限元方法的统一框架，刻画杂交化后的整体自由度并由此建立各种可杂交化有限元方法的内在联系，给出统一的先验误差分析，后处理杂交元法数值解获得新的超收敛数值解；构造四阶偏微分方程超收敛间断有限元方法和杂交有限元方法的后验误差估计子，设计自适应算法并开展拟最优收敛性和最优复杂度的理论分析研究。可望为四阶偏微分方程数值解的研究带来国际前沿的系统的新进展。

四阶偏微分方程不仅在组合弹性结构、相分离模型、图像去噪、静电驱动微机电系统、流体力学等领域有着重要应用，而且在一般多尺度问题高效求解算法中起关键作用。因此研究和构造四阶偏微分方程的高效数值求解方法具有重要的理论意义和直接的应用价值。本项目研究了四阶偏微分方程的杂交间断有限元方法及自适应算法。首先，提出了Kirchhoff板弯问题带最少惩罚项的约化局部C0间断(RLCDG)有限元方法。基于RLCDG方法可以看成是Hellan-Herrmann-Johnson（HHJ）方法的局部化的事实，分析了RLCDG方法的适定性和先验误差估计。借助Zienkiewicz-Guzman-Neilan元空间，证明了RLCDG方法残差型后验误差估计子的可靠性和有效性。其次，通过消去局部C0间断(LCDG)有限元方法中的全局提升算子惩罚项以及增强局部提升算子惩罚项，设计了Kirchhoff板弯问题的紧凑型C0间断(CCDG)有限元方法。第三、构造了Kirchhoff板弯问题LCDG方法可靠和有效的残差型后验误差估计子，并由此设计了LCDG方法的自适应算法。第四、设计了Kirchhoff板弯问题的超收敛C0间断(SCDG)有限元方法。当稳定化参数满足一定的等式时，我们出给了SCDG方法的杂交化（HCDG）。杂交化后的SCDG方法具有以下两个优点：（1）未知量只有位移和Lagrangian乘子，自由度比原来的SCDH方法要少的多；（2）杂交化后的SCDG方法离散得到的线性系统是对称正定的，可用PCG来求解。第五、研究了Kirchhoff板弯问题自适应杂交C0间断有限元（AHCDG）方法的最优收敛性和复杂度。首先构造了HCDG方法可靠和有效的后验误差估计子，然后利用后处理弯矩和离散Helmholtz分解建立了拟正交性和误差估计子的离散可靠性。由此彻底地分析了Kirchhoff板弯问题自适应杂交C0间断有限元方法的拟最优收敛性和最优复杂度。第六、借助于法向弯矩、扭矩和有效剪切力等物理量，提出了Kirchhoff板弯问题的具有最少整体自由度的超收敛杂交间断有限元（HDG）方法。借助网格依赖范数下的inf-sup条件和一些局部下界后验误差估计，在真解最小正则性假设下证明了HDG方法具有最优收敛阶和超收敛性。