Direct Finite Elements on Convex Polygons and Polyhedra

数学建模和计算模拟是理解和测试自然和工程系统的安全和经济有效的方法。人们常常需要确定某个利息量的变化率。在这种情况下，模型通常包括偏微分方程（PDE），可以使用有限元方法计算求解。 感兴趣的量是一个函数，比如说，空间是近似的小的简单形状称为元素，共同形成一个计算网格。网格通常由简单的三角形和矩形（或3D中的单纯形和砖块）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，但没有太多的有限元可用于这些网格，精确地近似函数。这个项目将在多边形和简单的多面体上开发实用的有限元，这些多边形和多面体连续地合并在一起，并且可以证明是准确的。 新的有限元预计将产生广泛的科学和工程领域的影响，使使用多面体网格更容易。它们还可能在计算机图形中的插值和可视化功能以及数据表示方面产生更广泛的影响。 地球科学的具体应用（能源和水资源管理的地下建模）正在计划中。 该项目将支持学生谁将在跨学科环境中工作的研究。许多计算科学家对在非标准多面体元素（多边形和多面体）上定义有限元很感兴趣。人们常常希望用显式的有限元基得到一致逼近. 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，并将其映射到物理元素。然而，由于映射是非仿射的，并且映射的混合元素通常不保持逐点意义上的发散自由属性，因此存在精度损失。所提出的研究涉及多边形和简单多面体上的最小自由度有限元的明确和实际的构造，以及它们的逼近性质和数值应用的数学分析。 该方法是定义H1-一致的形状函数的多项式直接对物理元素和补充的空间与少量明确定义的非多项式函数。这些补充函数将被定义为有理函数，允许我们定义一个节点基。然后可以使用de Rham理论来定义混合有限元。其目标是：1）在二维凸多边形上建立了直接偶然性和混合有限元; 2）在三维立方体六面体上建立了直接偶然性和向量值H（curl）和H（div）有限元; 3）在一般三维凸多面体上建立了直接偶然性和向量值有限元（然而，预计这一目标不会在项目的三年期限内得到充分解决）;以及4）使用新的有限元来解决地下流动应用中的问题，包括一些可以使用和探索hp的，该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的学术价值和更广泛的影响审查标准。

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.