A unified and general framework for enriching finite element approximations

Abstract

In this chapter our goal is to develop a unified and general framework for enriching finite element approximations via the use of additional enrichment functions. A crucial point in such an approach is to determine conditions on enrichment functions which guarantee that they generate a well-defined finite element. We start by giving under some conditions an abstract general theorem characterizing the existence of any enriched finite element approximation. After proving four key lemmas, we then establish under a unisolvence condition a more practical characterization result. We show that this proposed method easily allows us to establish a new class of enriched non-conforming finite elements in any dimension. This new family is inspired by the Han rectangular element and the nonconforming rotated element of Rannacher and Turek. They are all obtained as applications of a new family of multivariate trapezoidal, midpoint, and Simpson type cubature formulas,which employ integrals over facets. In addition, we provide analogously a general class of perturbed trapezoidal and midpoint cubature formulas, and use them to build a new enriched nonconforming finite element of Wilson-type.