Recent Advances in Macro-Element Methods

 

Abstract

We show how tools from spline theory can be used to analyze various classical finite elements which consist of piecewise polynomials on triangulations. We then apply the tools to derive new macro elements of arbitrary smoothness and least possible degree and/or degrees of freedom. The resulting elements are of interest for solving boundary-value problems for PDE's, and in scattered data fitting. The constructions also lead to stable local bases for various spline spaces, which in turn provide optimal order approximation results.