Direct Finite Elements on Convex Polygons and Polyhedra

数学建模和计算仿真是理解和测试自然和工程系统的一种安全且经济有效的方法。人们常常需要确定某一利率量的变化率。在这种情况下，模型通常包括偏微分方程（PDEs），可以使用有限元方法进行计算求解。感兴趣的数量是空间的函数，它近似于称为元素的小简单形状，它们一起形成计算网格。网格通常由简单的三角形和矩形（或3D中的简单体和砖块）组成，但这些网格有各种限制。多边形（或3D中的多面体）的网格更灵活，但对于这些网格来说，没有多少有限元可以准确地近似该函数。这个项目将在多边形和简单多面体上开发实用的有限元，这些多边形和简单多面体连续合并在一起，并且可以证明是准确的。新的有限单元预计将对科学和工程的广泛领域产生影响，使使用多边形网格更容易。它们还可能在计算机图形和数据表示中的插值和函数可视化方面产生更广泛的影响。具体应用到地球科学（地下建模能源和水资源管理）的计划。该项目将支持将在跨学科环境中工作的学生的研究。这样训练的学生在工业和政府实验室以及学术界都有很高的需求。许多计算科学家对在非标准的多面体元素（多边形和多面体）上定义有限元很感兴趣。人们通常需要一个具有明确的有限元基础的一致性近似。后者在处理非线性偏微分方程和耦合方程组时特别有用。可以在参考元素上定义有限元素，并将其映射到物理元素。然而，由于映射是非仿射的，因此存在精度损失，并且映射的混合元素通常不能在点向意义上保持无散度特性。提出的研究包括在多边形和简单多面体上明确和实际地构建最小自由度有限元，以及它们的近似性质和数值应用的数学分析。该方法是用直接作用于物理元素上的多项式来定义符合h1的形状函数，并用少量显式定义的非多项式函数来补充空间。这些补充函数将被定义为允许我们定义节点基的有理函数。然后，可以用de Rham理论来定义混合有限元。目标是：1)在二维凸多边形上开发直接偶然性和混合有限元；2)在三维立方体六面体上发展直接偶然性和向量值H（旋度）和H（div）有限元；3)在一般三维凸多面体上开发直接的偶然性和向量值有限元（但不期望在项目的三年内完全解决这个目标）；4)使用新的有限元来解决地下流体应用中的问题，包括一些可以利用和探索多边形单元的hp-细化特性的问题。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。

Mathematical modeling and computational simulation is a safe and cost-effective way to understand and test natural and engineered systems. Often one needs to determine the rate of change of some quantity of interest. In that case, the models will generally include partial differential equations (PDEs), which can be solved computationally using finite element methods. The quantity of interest is a function of, say, space that is approximated over small simple shapes called elements which together form a computational mesh. Meshes are often composed of simple triangles and rectangles (or simplices and bricks in 3D), but these meshes have various limitations. Meshes of polygons (or polyhedra in 3D) are more flexible, but there are not many finite elements available for these meshes that accurately approximate the function. This project will develop practical finite elements on polygons and simple polytopes that merge together continuously and are provably accurate. The new finite elements are expected to have an impact on broad areas of science and engineering by making the use of polytopal meshes more accessible. They may also have a broader impact in terms of interpolation and visualization of functions in computer graphics and in the representation of data. Specific applications to the geosciences (subsurface modeling for energy and water resource management) are planned. The project will support the research of students who will work in an interdisciplinary environment. Students thus trained are in high demand in industrial and governmental labs, as well as in academia.Many computational scientists are interested in defining finite elements on nonstandard, polytopal elements (polygons and polyhedra). Often one desires a conforming approximation with an explicit finite element basis. The latter is particularly helpful when dealing with nonlinear PDEs and coupled systems of equations. One could define a finite element on a reference element and map it to the physical element. However, there is a loss in accuracy due to the fact that the map is non-affine, and also mapped mixed elements generally do not preserve the divergence free property in a pointwise sense. The proposed research involves explicit and practical construction of minimal degree of freedom finite elements on polygons and simple polytopes, as well as the mathematical analysis of their approximation properties and numerical applications. The approach is to define H1-conforming shape functions in terms of polynomials posed directly on the physical element and supplement the space with a small number of explicitly defined nonpolynomial functions. These supplemental functions will be defined as rational functions that allow us to define a nodal basis. The de Rham theory can then be used to define mixed finite elements. The objectives are to: 1) Develop direct serendipity and mixed finite elements on 2D convex polygons; 2) Develop direct serendipity and vector-valued H(curl) and H(div) finite elements on 3D cuboidal hexahedra; 3) Develop direct serendipity and vector-valued finite elements on general 3D convex polyhedra (however, it is not expected that this objective will be fully resolved within the three year duration of the project); and 4) Use the new finite elements to solve problems in subsurface flow applications, including some that can use and explore hp-refinement properties of the polygonal elements.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.