Hybrid Finite Element Methods for Geometric Partial Differential Equations

有限元方法是科学计算中最重要、最强大的工具之一。它们的应用包括联邦战略利益的几个领域：材料和制造、生物医学工程和生物技术、结构工程和民用基础设施、环境工程等。它们被学术界、工业界和国家实验室的科学家和工程师广泛用于模拟大型、复杂的物理系统。这些物理系统通常遵循“守恒定律”，即某些量--如质量、能量或电荷--可以在空间中移动，但不能自发地出现或消失。希望这些系统的模拟也遵守这些守恒定律，因为它们是如此基本；否则，模拟的结果可能没有物理意义或可信。然而，目前的方法并不总是这样。在这个项目中，PI将开发和分析遵守这些守恒定律并保留相关物理性质的有限元方法。该项目的成功将导致新的计算方法，并提高对各种科学应用的现有方法的理解。由于该研究的具体应用具有很高的科学价值，这可能会对计算物理和工程产生重要的影响。PI建议研究包含局部对称性、不变量和守恒律的偏微分方程组的保结构杂交有限元方法。在应用中，这些局部不变的几何结构通常具有重要的物理意义(例如，电磁学中的电荷守恒)，因此需要设计一种保守的数值方法来精确地而不是近似地保持这些结构。混合方法为此提供了一个自然的框架，因为人们可以逐个元素地检查局部不变量，根据它们边界上的数值痕迹和通量。拟议的研究由两个主要部分组成。(1)在物理应用中普遍存在的哈密顿偏微分方程组满足多辛守恒定律，这与物理上重要的互易现象、行波、色散关系和分叉密切相关。PI将把他最近关于空间离散的多辛杂交间断Galerkin(HDG)方法的联合工作扩展到时间演化问题，使用空间HDG半离散和时空HDG方法。(2)虽然麦克斯韦方程的时间演化自动保留了与电荷守恒相关的发散约束，但对于有限元离散通常不是这样。初步结果表明，这可以用一类混合方法来解决，其中电荷守恒在数值电通量的意义上成立。PI建议通过分析Maxwell特征值问题的非协调混合方法以及使用有限元外部计算的Yang-Mills理论的混合方法来扩展这一点。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Finite element methods are among the most important, powerful tools in scientific computing. Their applications include several areas of Federal strategic interest: materials and manufacturing, biomedical engineering and biotechnology, structural engineering and civil infrastructure, environmental engineering, and more. They are widely used by scientists and engineers in academia, industry, and national laboratories to simulate large, complex physical systems. These physical systems often obey "conservation laws," which state that some quantity -- like mass, or energy, or electric charge -- can move around in space, but cannot appear or disappear spontaneously. It is desirable that simulations of these systems also obey these conservation laws, because they are so fundamental; otherwise, the simulated results may not be physically meaningful or trustworthy. However, this is not always the case with current methods. In this project, the PI will develop and analyze finite element methods that obey these conservation laws and preserve related physical properties. The success of this project would lead to new computational methods and improved understanding of current methods for a wide variety of scientific applications. Because the specific applications addressed by the proposed research are of high scientific value, this could have important ramifications for computational physics and engineering.The PI proposes to investigate structure-preserving hybrid finite element methods for partial differential equations (PDEs) containing local symmetries, invariants, and conservation laws. In applications, these locally-invariant geometric structures often have important physical meaning (e.g., conservation of charge in electromagnetics), so it is desirable to devise conservative numerical methods that preserve these structures exactly rather than approximately. Hybrid methods provide a natural framework for this, since one may examine local invariants, element-by-element, in terms of numerical traces and fluxes on their boundaries. The proposed research consists of two main components. (1) Hamiltonian PDEs, which are ubiquitous in physical applications, satisfy the multisymplectic conservation law, which is closely tied to physically-important reciprocity phenomena, traveling waves, dispersion relations, and bifurcations. The PI will extend his recent joint work on multisymplectic hybridizable discontinuous Galerkin (HDG) methods for spatial discretization to time-evolution problems, using spatial HDG semidiscretization and spacetime HDG methods. (2) While the time evolution of Maxwell's equations automatically preserves a divergence constraint associated with charge conservation, this is generally not true for finite element discretizations. Preliminary results show that this can be resolved using a class of hybrid methods, where charge conservation holds in the sense of the numerical electric flux. The PI proposes to extend this by analyzing nonconforming hybrid methods for the Maxwell eigenvalue problem, as well as hybrid methods for Yang-Mills theory using finite element exterior calculus.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.