Mathematics

This talk is a very naive and gentle introduction on how to enhance standard Galerkin methods by particular non-polynomial functions. We first review the history of some classical approaches (Extended and Generalized Finite Element Methods). These schemes are particularly useful when approximating functions that naturally present singular/discontinuous solutions, e.g., in presence of crack propagation.Time permitting, we discuss more recent approaches that extend the above gospel with a new twist to the case of more general meshes (Enriched nonconforming Virtual Elements and Hybrid High-Order Methods).We shall try to only sketch the ideas of the above schemes and reduce the technical aspects as much as possible. Required prerequisites are the knowledge of the finite element method and basic PDE theory.