Analysis of geometric discretization methods

几何离散化方法分析

• 批准号: RGPIN-2020-04389
• 负责人: Tsogtgerel, Gantumur
• 金额: $ 1.75万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=739935
• 关键词: Analysis; geometric; discretization; methods

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.

拟议项目的主题是对几何部分微分方程的兼容离散技术的理论理解。更具体地说，我的利益是基于基本微分方程的基本几何和拓扑特性的离散方法。我的长期目标是帮助发明具有良好几何特性的爱因斯坦场方程的离散方法，并为晶格量子染色体动力学的设计和分析做出贡献。在该项目的框架内，我们想研究离散外部微积分（DEC）的收敛性。 DEC是用于构建离散版本的外部微分积分对象的框架，并广泛用于计算机图形和科学计算中。但是，对DEC的严格合并分析始终缺乏。最近，申请人和他的学生证明了在任意维度中对泊松问题的解决方案汇聚在一起。我们计划研究一般K形式的收敛。这是一个短期目标，可以在2 - 3年内由博士生实现。 Yang-Mills方程的DEC的类似物是所谓的晶格仪理论，其量子版本包括晶格量子染色体动力学。因此，对DEC的良好理解肯定会深入了解晶格仪的理论。利用这种联系，我计划研究Yang-Mills方程的（经典）晶格计理论的收敛性。爱因斯坦方程的DEC的类似物是所谓的regge演算。最近，已经发现爱因斯坦场方程的恢复离散化在数值上是不稳定的。我们想将regge演算扩展到包括四边形，棱镜等在内的一般缝线，而不仅仅是三角形。我们的希望是，这将减轻上述不稳定。最后两段中描述的项目是开放式的长期项目。第二个子集在有限元外观（FEEC）上。这是一个理论框架，可以处理抽象希尔伯特综合体的混合有限元类型离散，以及越来越多的关于具体实现的理论。我想朝这个方向进行的第一个项目是开发FEEC的LP理论。这是一个明确定义的问题，具有已知的方法论，我希望在1 - 2年内取得具体的结果。这里一个重要的待处理问题是适应性。在这个方向上的最大障碍是，在混合有限元设置中很难确定所谓的准正交性属性（这是Galerkin正交性的概括）。但是，最近，在Stokes问题的背景下解决了这个问题。我们希望将他们的技术调整到FEEC环境中。该项目在技术上具有挑战性，但没有太多的结局，因为关于其解决方案的一般思想才刚刚开始出现。