Construction of Finite Elements Using Generalized Barycentric Coordinates and Its Application for Numerical Solution of Partial Differential Equations

我们将研究在一个区域上分段定义的连续光滑函数，该区域是一个多边形或多面体的并。这些函数称为多边形样条线。传统上，定义在由三角形或四面体组成的区域上的函数被研究并用于许多应用，如散乱数据的插补或拟合以及偏微分方程组的数值求解。这些被称为有限元或多元样条的传统函数在物理、化学、工程和生物学研究中的许多应用问题中发挥了重要而成功的作用。定义在多边形而不是三角形上的多边形样条曲线可以扩展我们更有效地处理复杂域上的应用问题的能力。例如，不规则立方体并集上的多边形样条线将比三变量样条线或有限元更有效，因为每个立方体必须分割成至少5个四面体，以便在立方体上定义有限元或三元样条函数。为了在应用程序中很好地使用它们，人们必须了解几个基本属性。例如，多边形样条曲线在一个多边形集上有多少个基函数？另一个问题是多边形样条曲线能多好地逼近任意函数。有限元方法是偏微分方程组数值求解的基本工具，在所有应用科学中都有广泛的应用。PI将开发一种新的方法，使用多边形或多面体上的广义重心坐标来构造随机有限元。随机单元的优点之一是减少了偏微分方程组数值求解所用的单元数，从而提高了计算效率。在这个项目中开发的多边形样条线可以在任何不规则立方体上构造。PI将推广一些现有的在3D凸多面体上构造意外有限元的想法，例如以六个2D四边形作为其边界面的不规则立方体以及3D环境中的其他更一般的凸多面体。PI将为3D环境中的一些线性和非线性偏微分方程组实现这些意外的有限元。此外，PI将在二维和三维凸多边形上构造可微的高阶随机元素，这对于高阶偏微分方程组是有用的。

Continuous and smooth functions piecewisely defined over a domain which is a union of polygons or polyhedrons will be studied. These functions are called polygonal splines. Traditionally, functions defined on a domain consisting of triangles or tetrahedra were studied and used for many applications such as scattered data interpolation or fitting and numerical solution of partial differential equations. These traditional functions called finite elements or multivariate splines have been playing a significant and successful role in many applied problems in physics, chemistry, engineering, and biological studies. Polygonal splines defined on polygons instead of triangles can extend our ability to handle applied problems over complicated domains more effectively. For example, polygonal splines over a union of irregular cubes will be more efficient than trivariate splines or finite elements since each cube has to be split into at least 5 tetrahedra in order to define finite element or trivariate spline functions over the cube. There are several fundamental properties one must understand in order to use them well for applications. For example, how many basis functions of polygonal splines over a collection of polygons are there? Another problem is how well polygonal splines can approximate arbitrary functions. The finite element method is the fundamental tool for numerical solution of partial differential equations (PDE) which finds extensive applications in all applied sciences. The PI will develop a new approach to construct serendipity finite elements using generalized barycentric coordinates over polygons or polyhedrons. One of the advantages of serendipity elements is to reduce the number of elements used for the numerical solution of PDE so that the computation will be more efficient. Polygonal splines developed in this project can be constructed over any irregular cubes. The PI will generalize some of the existing ideas to construct serendipity finite elements over a 3D convex polyhedron, e.g. irregular cube with six 2D quadrilaterals as its boundary faces as well as other more general convex polyhedra in the 3D setting. The PI will implement these serendipity finite elements for some linear and nonlinear PDEs in the 3D setting. Furthermore, the PI will construct differentiable high order serendipity elements over convex polygons in 2D and 3D which will be useful for higher order PDEs.