Lorenzo Mascotto: Thursday 29 April at 11:00 am
ABSTRACT:
Solutions to elliptic partial differential equations (PDEs) on smooth domains with smooth data are smooth. However, when solving a PDE on a Lipschitz polygon, its solution is singular at the vertices of the domain. The singular behaviour is known a priori: the solution can be split as a sum of a smooth term, plus series of singular terms that belong to the kernel of the differential operator appearing in the PDE.
The virtual element method (VEM) is a generalization of the finite element method (FEM) to polygonal/polyhedral meshes and is based on approximation spaces consisting of solutions to local problems mimicking the target PDE and has been recently generalized to the extended VEM (XVEM). Here, the approximation spaces are enriched with suitable singular functions. A partition of unity (PU) is used to patch local spaces. The approach of the XVEM is close to that of the extended FEM.
In this talk, we present a new paradigm for enriching virtual element (VE) spaces. Instead of adding special functions to the global space and eventually patch local spaces with a PU, we modify the definition of the local spaces by tuning the boundary conditions of local problems. By doing this, local VE spaces contain the desired singular functions, but there is no need to patch them with a PU. This results in an effective and slight modification of the already existing implementation of VE codes, as well as in a natural extension for existing theoretical results.
