Analysis of geometric discretization methods

拟议项目的主题是从理论上理解几何偏微分方程组的相容离散化技术。更具体地说，我感兴趣的是继承基本微分方程的基本几何和拓扑性质的离散化方法。我的长期目标是帮助发明具有良好几何性质的爱因斯坦场方程的离散化方法，并为晶格量子色动力学的设计和分析做出贡献。在这个项目的框架内，我们想要研究离散外微积分(DEC)的收敛性质。DEC是一个构造外微分对象的离散版本的框架，广泛应用于计算机图形学和科学计算中。然而，一直缺乏对DEC进行严格的收敛分析。最近，申请人和他的学生证明了任意维泊松问题的DEC解是收敛的。我们计划研究一般k-型的收敛。这是一个短期目标，博士生可以在2-3年内完成。杨-米尔斯方程的DEC类似于所谓的晶格规范理论，其量子版本包括晶格量子色动力学。因此，对DEC的深入理解将有助于我们对格点规范理论的深入研究。利用这个联系，我计划研究杨-米尔斯方程的(经典)格点规范理论的收敛性质。爱因斯坦方程的DEC类似于所谓的雷格微积分。最近，人们发现爱因斯坦场方程的雷格离散化在数值上是不稳定的。我们希望将Regge演算扩展到一般的镶嵌，包括四边形、棱柱等，而不仅仅是三角形。我们希望这将缓解上述的不稳定。最后两段所述的项目是不限成员名额的长期项目。第二个子集是有限元外积分(FEEC)。这是一个处理抽象希尔伯特复形的混合有限元离散的理论框架，以及关于具体实现的不断增长的理论体系。我想在这个方向上工作的第一个项目是开发FEEC的LP-理论。这是一个用已知的方法明确定义的问题，我预计在1-2年内会有具体的结果。这里一个重要的悬而未决的问题是适应性。这一方向的最大障碍是很难在混合有限元环境中建立所谓的准正交性(它是Galerkin正交性的推广)。然而，最近，这个问题在斯托克斯问题的背景下得到了解决。我们希望将他们的技术适应FEEC的环境。这个项目在技术上具有挑战性，但并不是无止境的，因为关于其解决方案的一般想法才刚刚开始出现。

The theme of the proposed project is the theoretical understanding of compatible discretization techniques for geometric partial differential equations. More specifically, my interests are in discretization methods that inherit fundamental geometric and topological properties of the underlying differential equation. My long term goals are to help invent discretization methods for the Einstein field equations with nice geometric properties, and to make contributions to the design and analysis of lattice quantum chromodynamics. Within the framework of this project, we would like to investigate convergence properties of discrete exterior calculus (DEC). DEC is a framework for constructing discrete versions of exterior differential calculus objects, and is widely used in computer graphics and scientific computing. However, a rigorous convergence analysis of DEC has always been lacking. Recently, the applicant and his student proved that DEC solutions to the Poisson problem in arbitrary dimensions converge. We plan to study convergence for general k-forms. This is a short term goal that can be accomplished in 2-3 years by a PhD student. The analogue of DEC for the Yang-Mills equations is the so called lattice gauge theories, whose quantum version include lattice quantum chromodynamics. Thus, a good understanding of DEC will certainly give insights into lattice gauge theories. Making use of this connection, I plan to study convergence properties of (classical) lattice gauge theory for the Yang-Mills equations. The analogue of DEC for the Einstein equations is the so called Regge calculus. Very recently, the Regge discretization of the Einstein field equations has been found to be numerically unstable. We would like to extend the Regge calculus to general tessellations including quadrilaterals, prisms, etc, instead of only triangles. Our hope is that this would alleviate the aforementioned instability. The projects described in the last 2 paragraphs are open-ended long term projects. The second subset is on finite element exterior calculus (FEEC). This is a theoretical framework to handle mixed finite element type discretizations of abstract Hilbert complexes, together with a growing body of theory on concrete realizations. The first project I want to work on in this direction is to develop an Lp-theory of FEEC. This is a well-defined question with known methodologies and I expect to have concrete results in 1-2 years. An important pending issue here is adaptivity. The biggest obstacle in this direction has been that the so called quasi-orthogonality property (which is a generalization of Galerkin orthogonality) is hard to establish in the mixed finite element setting. However, very recently, this problem has been solved in the context of the Stokes problem. We hope to adapt their techniques into the FEEC setting. This project is technically challenging, but not so much open ended, as general ideas on its resolution are just beginning to emerge.