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主力。对于每个特定的应用，需要不同类型的有限元。在1980年，Nederlands在不同的有限元之间建立了一个里程碑式的联系，并找到了更多的应用。这个项目将把这些连接推广到不同的光滑类型的有限元。这些联系将使我们能够解决新的应用。本研究将在有限元外演算（FEEC）的框架下，在任意维的单纯网格上建立和分析有限元空间，这些网格适合于有限元复合体。这个方案的显著特点是，我们将建造比传统空间更平滑的空间（例如惠特尼/内德维德形式）。平滑的空间对于某些应用来说更自然：板问题，流体流动问题。我们将通过使用单纯形的分裂来实现这一点，这种分裂比区域的任意单纯分解提供更多的结构。复形的最左边的空间将与样条社区中已经研究过的函数空间相一致。最右边的空间是与流体流动问题的inf-sup稳定有限元空间相关联的空间。因此，我们计划以自然的方式将这些函数空间连接成有限元复形。此外，我们将探索与样条和应用代数几何社区的联系。该奖项反映了NSF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.