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

本文将研究定义在多边形或多面体并域上的连续光滑函数。这些函数称为多边形样条。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，例如离散数据插值或拟合以及偏微分方程的数值解。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可以更有效地扩展我们处理复杂域上应用问题的能力。 例如，不规则立方体的并集上的多边形样条将比三元样条或有限元更有效，因为每个立方体必须被分成至少5个四面体，以便在立方体上定义有限元或三元样条函数。有几个基本属性必须了解，以便在应用程序中很好地使用它们。例如，多边形集合上的多边形样条有多少个基函数？另一个问题是多边形样条曲线如何逼近任意函数。有限元法是求解偏微分方程的基本工具，在各个应用科学领域都有着广泛的应用。PI将开发一种新的方法来构建偶然有限元使用广义重心坐标多边形或多面体。偶然性单元的优点之一是减少了用于偏微分方程数值解的单元数目，从而提高了计算效率。在这个项目中开发的多边形样条可以在任何不规则的立方体上构造。PI将推广一些现有的想法来构建三维凸多面体上的偶然有限元，例如，以六个二维四边形作为边界面的不规则立方体以及其他更一般的三维凸多面体。PI将在3D设置中为一些线性和非线性偏微分方程实现这些偶然发现的有限元。此外，PI将在2D和3D凸多边形上构造可微的高阶偶然元素，这对高阶偏微分方程很有用。

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.