Finite Elements and Virtual Elements on Classical Meshes

Among Numerical Methods for PDEs, the Virtual Element Methods were introduced recently in order to allow the use of decompositions of the computational domain in polytopes (polygons or polyhedra) of very general shape. The present paper investigates the possible interest in their use (together or in alternative to Finite Element Methods) also for traditional decompositions (in triangles, tetrahedra, quadrilateral or hexahedra). In particular their use looks promising in problems related to high-order PDEs (requiring C p finite dimensional spaces with p ≥ 1), as well as problems where incompressibility conditions are needed (e.g. Stokes), or problems (like mixed formulation of elasticity problems) where several useful features (symmetry of the stress tensor, possibility to hybridize, i͡nf-sup stability condition, etc.) are requested at the same time.