Computational and Applied Mathematics seminar

Abstract: The title of this talk refers to a question frequently asked by Ivo Babuska, from the late 1980s and on, following various talks on finite element methods. More precisely, what can we say about the properties of finite element methods as we raise the polynomial degree? Even today the so called $p$-method is less understood,and the corresponding analysis is less canonical, than the traditional approach of mesh refinement, i.e., the $h$-method. In recent years Rick Falk and I have developed a theory which represents a new tool to analyze finite element methods of  high polynomial degree, which we refer to as  the bubble transform. The key idea is to construct a decomposition into local bubbles which simultaneously covers all possible polynomial degrees. The purpose of this talk is to give a review of this theory, and to discuss potential applications.