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

University Of Georgia Research Foundation Inc, Athens GA

Investigators

Abstract

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.