The Finite Element Method on Polygonal Domains

Hengguang Li
Pennsylvania State University

Abstract: Since most geometric objects we are interested in consist of corners (2-D) or corners and edges (3-D), it is necessary to develop a good numerical method to evaluate the solutions of the PDEs with singularities on such domains. To fix ideas, the Poisson's equation will be shown as the example. The existence, uniqueness and regularity property of the solution will be discussed during the talk in terms of weighted Sobolev spaces. Based on regularity results, the development of the corresponding FEM represents the natural extension of the theoretical analysis. Numerical results will verify our theoretical prediction. This method can be also applied to some degenerate elliptic PDEs in which case new numerical difficulties may arise from the triangulation.