Lecture cancelled-Studies on generalised FEM (PUFEM and VEM) - introduction and numerical experiments

Abstract:  This talk focuses on two sets of developments in the Finite Element Method (FEM) namely Partition of Unity FEM (PUFEM) and Virtual Element Method (VEM). Whereas, PUFEM was introduced around 20 years back by Melenk and Babuska, its extension in developing numerical schemes for wave problems started only around 10 years back. VEM on the other hand is only 3 years old and is being developed rapidly.

First part of this talk focuses on introducing the PUFEM particularly for short wave problems in time-harmonic acoustics. We discuss the challenge in modeling of problems involving short wavelengths and how PUFEM effectively addresses it. An inherent problem with FEM or PUFEM applied to an exterior acoustic problem is the approximate boundary condition in the far field. We treat this problem by coupling PUFEM with its boundary element method (BEM) counterpart, namely PUBEM. Numerical results showing improvements in the accuracy over classical FEM are discussed towards the end of the first part of this talk.

We focus on a basic introduction to VEM in the second part of the talk. VEM is a generalisation of FEM with its advantage being the ability to construct high order approximation spaces on general polygonal meshes. The VEM approximation space is made of usual polynomial space in FEM along with some virtual functions and hence the name. We discuss the implementation of VEM inside Distributed Unified Numerics Environment (DUNE) as a generic interface and present some initial numerical results with the first order VEM.