Finite Element Methods (FEM) are generally constructed based on socalled 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.

## A General Approach for Implementing Virtual Element Schemes

Research Group:

Andreas Dedner

Institution:

University of Warwick

Schedule:

Wednesday, December 7, 2022 - 14:00

Location:

A-137

Abstract: