Virtual Element Methods (VEM) can be seen as an extension of Finite Element Methods to the case of decompositions of the computational domain in polytopes (that is, polygons or polyhedra) of very general shape. A key feature of the methods is that the local spaces include functions which are not polynomials: typically, solutions of PDE problems, that however are never actually computed. Hence, the name Virtual.

Since their introduction, in 2013, they called the attention of the scientific community for their ability to deal more effectively with several types of problems where the general shape of the elements can pay off in a substantial way (complex geometries, local refining and coarsening, contact problems, fractures, moving objects, inclusions, etc.). More recently it became clear that such a more general approach can also produce better ways of solving problems on triangular or quadrilateral decompositions as well: for designing structure preserving methods, using highly regular subspaces, etc.

The course will present, at the same time, the fundamental theoretical background of Virtual Elements, as well as their use for different applications, and the basic tools and tricks for their actual implementation (together with practical coding sessions).