外微积分理论及高阶椭圆型方程高效数值算法的研究

Higher-order partial differential equations are widely used in the fields of elasticity problems, image processing, micro-electromechanical systems, phase field crystal models, etc. The finite element exterior calculus theories have been becoming the key tools in designing stable numerical methods for partial differential equations. This project is intended to study the applications of the exterior calculus theories in devising efficient numerical methods for higher-order elliptic equations, which is both theoretically and practically important. Firstly, using a discrete commutative diagram, the finite element methods for the biharmonic equation in two dimensions are decomposed into two finite element methods for the Poisson equation and one mixed finite element method for the linear elasticity, based on which a fast auxiliary space preconditioner is developed for the finite element methods for the biharmonic equation in two dimensions. Secondly, we decompose the biharmonic equation in three dimensions into two Poisson equations and one three-term Stokes-type equation by using a commutative diagram. Then a second variational formulation for the biharmonic equation in three dimensions is derived by transforming the three-term Stokes-type equation into a three-term elasticity-type equation. Stable finite element methods and their fast auxiliary space preconditioners are advanced for these two new variational formulations for the biharmonic equation in three dimensions. Finally, the triharmonic equation in two and three dimensions is decomposed into two biharmonic equations and one Stokes-type equation with the help of the commutative diagrams. Based on this decomposition, we construct stable finite element methods and their fast auxiliary space preconditioners for the triharmonic equation in two and three dimensions. It is worth mentioning that all of the fast solvers developed in this project work for the shape-regular unstructured grids.

高阶偏微分方程广泛应用于弹性力学、微机电系统、图像处理、晶体相场模型等领域。有限元外微积分理论已成为设计偏微分方程稳定数值离散方法的重要工具。本项目拟研究外微积分理论在高阶椭圆型方程高效数值方法中的应用，这具有重要的理论意义和实用价值。首先，利用离散交换图将二维重调和方程有限元方法分解成两个Poisson方程有限元方法和一个线弹性问题混合元方法，由此设计快速辅助空间预条件子。然后，利用交换图将三维重调和方程分解成两个Poisson方程和一个三变量Stokes型方程，并改写三变量Stokes型方程为三变量线弹性型方程得到新的变分形式，设计两种变分形式的稳定有限元方法及快速辅助空间预条件子。最后，利用交换图将二维和三维三重调和方程分解成两个重调和方程和一个Stokes型方程，由此设计三重调和方程的稳定有限元方法及快速辅助空间预条件子。本项目设计的所有快速求解算法都适用于形状正则无结构网格。

高阶偏微分方程广泛应用于弹性力学、微机电系统、图像处理、晶体相场模型等领域。有限元外微积分理论已成为设计偏微分方程稳定数值离散方法的重要工具。本项目主要研究外微积分理论在高阶椭圆型方程高效数值方法中的应用，这具有重要的理论意义和实用价值。首先，利用微分复形和交换图给出了构造广义Helmholtz分解的统一方法，并由此系统地提出了将高阶椭圆型微分方程解耦成多个Poisson型方程和Stokes型方程的一般框架。接着，借助Stokes复形提出两维重调和方程基于伪应力的高效数值方法。借助Hellan-Herrmann-Johnson (HHJ)复形设计了重调和方程HHJ混合有限元方法的V循环多重网格算法，该算法的收缩数关于网格尺寸是一致小于1的。借助光滑的de Rham复形设计了三维重调和方程基于三变量解耦变分形式的数值离散方法和快速求解算法。通过修改右端项提出四阶椭圆奇异扰动问题关于小参数一致收敛的Morley-Wang-Xu (MWX)元方法，两维情形可以解耦成两个Poisson方程的Morley元方法和一个Brinkman方程的非协调P1-P0元方法，并应用Nitsche技巧得到具有最优收敛阶的修正MWX方法。借助光滑的弹性复形设计了两维三重调和方程基于解耦变分形式的数值离散方法和快速求解算法。借助光滑的Hessian复形设计了三维三重调和方程基于三变量解耦变分形式的数值离散方法和快速求解算法。提出了任意维欧式空间中任意形状多面体上任意高阶可微的协调和完全非协调的任意次虚拟元的统一构造方法。借助Stokes复形构造出四阶curl方程基于解耦变分形式的有限元方法，并设计自适应算法。此外，构造出线弹性问题Hu-Zhang混合元方法关于网格尺寸和Lame系数一致稳定的近似块分解预处理广义极小残差法。借助两维有限元弹性复形构造线弹性问题Hu-Zhang混合元方法可靠有效的残差型后验误差估计子，并设计自适应算法。设计了重调和方程自适应杂交C0间断有限元方法，并分析了最优收敛性和复杂度。本项目研究成果创新性强，给高阶偏微分方程数值方法的研究带来系统的新进展。