Collaborative Research: Adaptive Methods and Finite Element Exterior Calculus for Nonlinear Geometric PDE

该项目的主要技术目标是发展一般的逼近理论和可靠的，收敛的自适应方法的内在离散的一般类的非线性几何椭圆和发展偏微分方程的黎曼2-和3-流形。 研究人员将利用他们为有限元外微积分（FEEC）开发的变分犯罪框架，将FEEC扩展到非线性椭圆问题，超曲面问题以及非线性抛物和双曲问题。 这个框架将有助于设计，开发和收敛分析的AFEM算法与FEEC使用。 这种方法将允许一个更自然和一般的治疗几何错误，由于变分犯罪的后验分析，他们最近的方法进行先验分析。在获得后验分析的坚实理论框架，产生后验误差估计和局部指标后，他们将在扩展FEEC框架内开发和分析自适应有限元方法（AFEM）。收敛性分析方法将基于他们最近发表的关于线性椭圆问题混合公式的AFEM收敛性分析的工作。总体目标是在FEEC中开发一个完整的AFEM收敛理论，补充最近开发的泊松型问题和半线性推广的非混合配方的收缩框架。将使用开源FETK ToolKit生成原型和产品实现，并将在与物理科学家和工程师的持续合作中使用所生成的软件。研究人员将研究和开发数学物理和几何分析交叉点上产生的稳态和演化偏微分方程（PDE）系统的近似解方法。这样的方程组，被称为几何偏微分方程，出现在广泛的物理和数学问题中;例子包括麦克斯韦方程（或更一般的杨-米尔斯方程），爱因斯坦场方程和其他哈密顿系统。这种系统的柯西（或初值）公式产生一个包含非动力学方程的约束演化系统。这些非动力学几何偏微分方程本身就有很大的意义，例如Yamabe问题、爱因斯坦方程中的哈密顿量和动量约束、Monge-Ampere方程等。如果我们的目标得以实现，这个项目的结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的影响。本文所提出的方法将有助于复杂三维约束非线性动力学数值模拟方法的发展。我们生产的模拟技术将为天体物理学和相对论以及一些纯数学领域（如几何分析）的模型探索提供强大的工具。参与该项目的研究生将由两名研究人员共同培训;这将涉及两个合作机构的团队成员之间的定期互动。 PI以前曾在这样一个共享的培训结构上进行过合作，在过去的项目中取得了巨大的成功;这两个研究小组之间的知识和技能的共享培训和转让将成为两个小组的宝贵研究资源。

The primary technical aim of this project is to develop general approximation theory and reliable, convergent adaptive methods for the intrinsic discretization of a general class of nonlinear geometric elliptic and evolution PDE on Riemannian 2- and 3-manifolds. The investigators will exploit the variational crimes framework they have developed for the finite element exterior calculus (FEEC), extending the FEEC to nonlinear elliptic problems, to problems on hypersurfaces, and to nonlinear parabolic and hyperbolic problems. This framework will aid in the design, development, and convergence analysis of AFEM algorithms for use with FEEC. This approach will allow for a more natural and general treatment of geometric error due to variational crimes in a posteriori analysis, following their recent approach for a priori analysis. After obtaining a solid theoretical framework for a posteriori analysis, yielding a posteriori error estimates and local indicators, they will develop and analyze adaptive finite element methods (AFEM) within the extended FEEC framework. The convergence analysis approach will be based on their recent published work on AFEM convergence analysis for mixed formulations of linear elliptic problems. The overall goal is to develop a complete AFEM convergence theory in FEEC, complementing the recently developed contraction frameworks for non-mixed formulations of Poisson-type problems and semilinear generalizations. Both prototype and production implementations will be produced, using the opensource FETK ToolKit, and the resulting software will be used in ongoing collaborations with physical scientists and engineers.The investigators will study and develop methods for 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; examples include Maxwell's equations (or more generally the Yang-Mills equations), Einstein's field equations, and other Hamiltonian systems. The Cauchy (or initial-value) formulation for such systems yields a constrained evolution system containing non-dynamical equations. These non-dynamical geometric PDE are of great interest in their own right; examples include the Yamabe problem, the Hamiltonian and momentum constraints in the Einstein equations, and the Monge-Ampere equations, among others. If our goals are achieved, the results of this project will have a broad impact on areas of mathematics such as geometric analysis, as well as in astrophysics and general relativity. The methods developed here will contribute to the advancement of numerical methods for complex three-dimensional constrained nonlinear dynamical simulations. The simulation technology we 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. Graduate students involved in the project will be co-trained by both investigators; this will involve regular interaction between the members of the teams at both partner institutions. The PI has previously collaborated on such a shared training structure with great success on past projects; this shared training and transfer of knowledge and skills between the two research groups will be an invaluable research resource to both groups.