Mathematics

Finite Element Methods (FEM) are generally constructed based on so

called finite element triples (K,B,L). Here K is an element in the grid  
(e.g., a triangle), B is the basis of some finite dimensional space  
(e.g., a set of polynomials), and L is a set of functionals with |L| = |B| 
(e.g., the evaluation of functions on a set of Lagrange points). 

The set L are called local degrees of freedom (dofs). Often the aim in the 
construction of finite element spaces V_h is to achieve some level of  
conformity, i.e., guaranteeing that V_h is a subset of some function space 
V. Typical spaces are conforming with V=H^1. Achieving conformity with  
V=H^2 or V=H(div) is a lot more challenging. For example, the lowest order  
finite element on triangles which leads to V_h being H^2 conforming  
requires the use of polynomials of order 5 locally (without using a  
piecewise definition). Also changing K from triangles to quadrilaterals 
requires defining a complete new set of basis function B and dofs L. 
Often a suitable choice for L is not the problem but defining a suitable B 
can be challenging. 

The Virtual Element Method (VEM) is a recent approach to define a wide  
range of spaces on general element shapes including general polygons. 
In this talk we will provide a description of the virtual element spaces  
which shows that it can be considered to be a direct extension of the FEM  
constructing approach. The approach uses a fixed B on each element  
independent of the choice of L thus avoiding the problem described above. 

We introduce a VEM tuple and describe how that can 
be used to define the local spaces. We will focus our presentation on spaces 
which can be used to solve forth order problems but will also demonstrate 
how the approach can be used to construct other spaces, i.e., divergence  
free spaces for fluid dynamics problems. 

We will show how this approach simplifies the implementation of VEM methods 
within existing FEM codes and discuss a-priori error analysis and numerical 
experiments for linear forth order problems with varying coefficients.