Structure-Preserving Hybrid Finite Element Methods

有限元方法是现代科学计算的基石之一，它将高性能与精度和稳定性的强大理论保证相结合。它们的应用包括联邦战略关注的几个领域：材料和制造、生物医学工程和生物技术、结构工程和土木基础设施、环境工程等等。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。然而，为了使这些方法产生的模拟在物理上有意义和值得信赖，它们需要捕获原始系统中存在的某些基本物理定律。某些系统满足“守恒定律”，即某些量（如质量、能量或电荷）可以在空间中移动，但不会自发地出现或消失，而其他系统满足“耗散定律”，要求这些量以一定的速率增长或衰减（例如，由于摩擦引起的能量衰减）。尽管经典的有限元方法很难表达这些物理定律，但PI已经表明，“混合”有限元方法提供了一种方法。在这个项目中，PI将进一步研究混合有限元方法，开发和分析保护这些重要物理结构的计算技术。该项目的成功将导致新的计算方法，并提高对各种高价值科学应用的现有方法的理解，对计算物理和工程具有重要影响。许多偏微分方程，特别是在物理应用中遇到的偏微分方程，包含局部对称性、守恒定律和相关结构。乍一看，这些局部结构中的一些似乎难以用有限元方法捕获，因此，在这个方向上的许多研究都局限于矩形网格的有限差分方法。然而，近年来，PI开发了一个成功的研究项目，表明杂交提供了一条绕过局部结构保留有限元素的明显障碍的途径。这个项目将围绕两个主题，在这个方向上研究几个新问题。(1)二次密度的局部守恒律在哈密顿偏微分方程中很常见，因为它们是由线性点对称产生的。然而，传统的有限元离散破坏了局部对称性，只得到全局守恒定律。PI旨在通过使用混合方法来规避这一障碍，并将其扩展到非保守系统中，其中二次量是耗散的而不是守恒的。(2) PI和他的合作者最近成功地混合了有限元外演算（FEEC），它使用微分形式统一了拉普拉斯型算子的几种方法。PI将研究杂化FEEC方法在哈密顿偏微分方程中的应用，特别是关注多辛结构的保存，并研究具有强（而不仅仅是弱）保守通量的杂化FEEC方法，包括不符合和杂化不连续伽辽金（HDG）方法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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 cornerstones of modern scientific computing, combining high performance with strong theoretical guarantees of accuracy and stability. 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. However, in order for the simulations produced by these methods to be physically meaningful and trustworthy, it is desirable that they capture certain fundamental physical laws present in the original systems. Certain systems satisfy "conservation laws," which state that some quantity (like mass, energy, or charge) can move through space but not appear or disappear spontaneously, while other systems satisfy "dissipation laws" that require these quantities to grow or decay at a certain rate (e.g., decay of energy due to friction). Although classical finite element methods make it difficult to express these physical laws, the PI has shown that "hybrid" finite element methods provide a way to do so. In this project, the PI will conduct further investigations into hybrid finite element methods, developing and analyzing computational techniques for preserving these important physical structures. The success of this project would lead to new computational methods and improved understanding of current methods for a wide variety of high-value scientific applications, with important ramifications for computational physics and engineering.Many partial differential equations (PDEs), particularly those encountered in physical applications, contain local symmetries, conservation laws, and related structures. At first glance, some of these local structures appear difficult for a finite element method to capture, and consequently, much of the research in this direction has been restricted to finite difference methods on rectangular grids. However, in recent years, the PI has developed a successful research program showing that hybridization provides a route around the apparent obstacles to local-structure-preserving finite elements. This project will investigate several new questions in this direction, organized around two main themes. (1) Local conservation laws with quadratic densities are common in Hamiltonian PDEs, because they arise from linear point symmetries. However, conventional finite element discretization breaks local symmetry, resulting only in global conservation laws. The PI aims to circumvent this obstacle by using hybrid methods, and to extend this to non-conservative systems where a quadratic quantity is dissipated rather than conserved. (2) The PI and his collaborators have recently succeeded in hybridizing finite element exterior calculus (FEEC), which uses differential forms to unify several families of methods for Laplace-type operators. The PI will investigate the application of hybridized FEEC methods to Hamiltonian PDEs, specifically focusing on multisymplectic structure preservation, and to investigate hybrid FEEC methods with strongly (rather than merely weakly) conservative fluxes, including nonconforming and hybridizable discontinuous Galerkin (HDG) methods.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.