Finite Element Exterior Calculus with Smoother Piecewise Polynomials

有限元方法是模拟工程、物理、化学和生物学中许多问题的计算工具。对于每个特定的应用，需要不同类型的有限元。1980年，Nedelec在不同的有限元之间建立了巨大的联系，并找到了更多的应用。本项目将把这些联系推广到不同的光滑型有限元中。这些连接将使我们能够处理新的应用程序。本研究将遵循有限元外演算（FEEC）的框架，在适合于有限元复合体的任意维度的简单网格上建立和分析有限元空间。这个方案的独特之处在于，我们将建造比传统空间更平滑的空间（例如Whitney/Nedelec形式）。平滑的空间对于某些应用来说更自然：板材问题，流体流动问题。我们将通过使用简单分解来实现这一点，它提供了比域的任意简单分解更多的结构。复合体最左侧的空间将与样条群落中研究过的功能空间相吻合。最右边的空间是与流体流动问题的中稳定有限元空间有关的空间。因此，我们计划以一种自然的方式将这些功能空间连接成一个有限元复合体。此外，我们将探索与样条和应用代数几何群体的联系。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 the computational workhorse in simulating many problems in engineering, physics, chemistry and biology. For each particular application, different type of finite elements are needed. In 1980, Nedelec made a monumental connection between different finite elements that has found even more applications. This project will generalize these connections to different finite elements of smoother type. These connections will allow us to tackle new applications.The research will build and analyze finite element spaces on simplicial meshes in arbitrary dimension that fit in a finite element complex following the framework of the finite element exterior calculus (FEEC). The distinctive feature of this proposal is that we will build spaces that are smoother than traditional spaces (e.g. Whitney/Nedelec forms). Smoother spaces are more natural for some applications: plate problems, fluid flow problems. We will accomplish this by using splits of simplices that provide more structure than an arbitrary simplicial decomposition of a domain. The left most spaces of the complex will coincide with functions spaces that have been studied in the spline community. The far right spaces are ones that are associated with inf-sup stable finite element spaces for fluid flow problems. Thus, we plan to connect these function spaces in a natural way into a finite element complex. In addition, we will explore connections with the spline and applied algebraic geometry communities.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.