Numerical Methods for Geometric PDE on Manifolds with Arbitrary Topology

这个项目是关于在数学物理和几何分析的交叉点上产生的定常和发展偏微分方程组的近似解。这种被称为几何偏微分方程组的方程组出现在广泛的物理和数学问题中；这个项目的主要动机之一是爱因斯坦，这对引力波科学至关重要。这类问题最具挑战性的特征之一，无论是数学分析还是计算模拟，都是潜在的空间域，它具有潜在复杂的流形结构，而不是三维空间中的简单形状。此外，这种流形的几何和拓扑都可能随着时间的推移而演变，这取决于特定的模型。研究成果将对几何分析等数学领域以及天体物理学和广义相对论产生广泛影响。所开发的方法将有助于复杂三维约束非线性动力学模拟的数值方法的进步。PI产生的模拟技术将为探索天体物理学和相对论的模型以及几何分析等纯数学领域的模型提供强大的工具。这个项目的主要技术目标是开发一个通用的逼近理论框架，以及可靠和可证明收敛的自适应方法，用于黎曼2-和3-流形上一类一般的非线性几何椭圆型和发展偏微分方程组的内在离散化。虽然这类偏微分方程解的理论在过去的三十年里得到了深入的研究，但在发展具有相应逼近理论的稳健数值方法方面取得了较新的进展。到目前为止，大多数方法，如二维问题的曲面有限元方法，都是基于将曲面嵌入到三维空间中。对于广义相对论这样的应用，需要一种更一般的方法，它不依赖于这种嵌入的存在。在这个项目中，PI将研究真正的内在离散化的发展，这种离散化不使用外部信息来产生离散化，以允许在具有任意拓扑的黎曼2-和3-流形上发展数值方法。PIS的方法是使用项目团队开发的多立方体框架和局部单纯形近似技术等技术，开发基于地图集的离散化。为了开发相应的误差分析框架，PI将利用曲面方法的变分犯罪框架，例如基于有限元外部微积分的方法。

This project is concerned with the approximate solution of systems of stationary and evolution partial differential equations (PDE) arising at the intersection of mathematical physics and geometric analysis. Such systems of equations, known as Geometric PDE, appear in a wide range of physical and mathematical problems; one of the primary motivations for this project is the Einstein, which are of central importance to gravitational wave science. One of the most challenging features of this class of problems, for both mathematical analysis and computational simulation, is the underlying spatial domain which has the structure of a potentially complicated manifold rather than a simple shape in 3-space. Moreover, both the geometry and the topology of this manifold may evolve over time, depending on the particular model. The research results will have a broad impact on areas of mathematics such as geometric analysis, as well as in astrophysics and general relativity. The methods developed will contribute to the advancement of numerical methods for complex three-dimensional constrained nonlinear dynamical simulations. The simulation technology the PIs produce will provide powerful tools for the exploration of models in astrophysics and relativity as well as in some areas of pure mathematics such as geometric analysis. The two graduate students involved in the project will be co-trained by both investigators; this will involve regular interaction between all four members of the team.The primary technical aim of this project is to develop a general approximation theory framework, together with reliable and provably convergent adaptive methods, for the intrinsic discretization of a general class of nonlinear geometric elliptic and evolution PDE on Riemannian 2- and 3-manifolds. While the solution theory for this class of PDE has been intensively studied over the last thirty years, progress on the development of robust numerical methods with a corresponding approximation theory has been a more recent development. Most of the approaches to date, such as surface finite element methods for two-dimensional problems, are based on exploiting the embedding of the surface into 3-dimension. For applications such as general relativity, a more general approach is needed that does not rely on the existence of such an embedding. In this project, the PIs will study the development of truly intrinsic discretizations that use no extrinsic information to produce a discretization, to allow for the development of numerical methods on Riemannian 2- and 3-manifolds with arbitrary topology. The PIs' approach is to develop an atlas-based discretization using techniques such as the multi-cube framework and the local simplex approximation techniques developed by the project team. To develop a corresponding error analysis framework, the PIs will exploit the variational crimes framework for methods in surfaces, such as methods based on finite element exterior calculus.