Analysis of geometric discretization methods

拟议项目的主题是几何偏微分方程相容离散技术的理论理解。更具体地说，我的兴趣是在离散化方法，继承基本的几何和拓扑性质的基本微分方程。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，我们想研究离散外部演算（DEC）的收敛性质。DEC是一种构造外部微分对象离散版本的框架，广泛应用于计算机图形学和科学计算。然而，严格的收敛性分析DEC一直缺乏。最近，申请人和他的学生证明了任意维泊松问题的DEC解收敛。我们计划研究一般k-形式的收敛性。这是一个短期目标，可以在2-3年内完成的博士生。杨-米尔斯方程的DEC的类似物是所谓的格点规范理论，其量子版本包括格点量子色动力学。因此，一个好的理解DEC肯定会给深入了解格点规范理论。利用这种联系，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的类似物是所谓的Regge演算。最近，爱因斯坦场方程的Regge离散被发现是数值不稳定的。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不稳定。最后两段所述的项目是无限期的长期项目。第二个子集是有限元外演算（FEEC）。这是一个理论框架来处理抽象希尔伯特复形的混合有限元类型离散化，以及越来越多的具体实现理论。我想在这个方向上做的第一个项目是开发FEEC的LP理论。这是一个定义明确的问题，有已知的方法，我希望在1-2年内有具体的结果。一个重要的未决问题是适应性。在这个方向上的最大障碍是，所谓的准正交性（这是一个推广的伽辽金正交性）是很难建立在混合有限元设置。然而，最近，这个问题已经在斯托克斯问题的背景下得到解决。我们希望将他们的技术应用到FEEC环境中。这个项目在技术上是具有挑战性的，但不是那么开放，因为关于其解决方案的一般想法刚刚开始出现。

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.