Séance Séminaire

Colloquium de Mathématiques

An introduction to Virtual Elements: basic theory and applications

The Virtual Element Method (VEM) is a recent technology for the numerical solution of boundary value problems for Partial Differential Equations, which could be seen as a generalization of the Finite Element Method (FEM). VEM responds to the recent interest in using decompositions of the computational domain into polygons/polyhedra of very general shape in the approximation of problems of practical interest, overcoming the limit of using (conforming) triangles and quads/tetrahedra and hexahedra as done with FEM. Indeed, the possibility of using general polytopal meshes opens up a new range of opportunities in terms of accuracy, efficiency and flexibility. This is for instance reflected by the fact that various commercial and free codesrecently have included and are continuing to develop polytopal meshes, showing in selected applications an improved computational efficiency with respect to standard tetrahedral or hexahedral grids. In this talk we will describe the basic ideas of conforming VEM on a simple model problem, and then show some numerical results on more general problems in two and three dimension. We will also give hints on the Serendipity version, which allows to decrease significantly the number of degrees of freedom, that is, to reduce the dimension of the final linear system.