Virtual element method: who are the virtual functions?

Resumo

The Virtual Element Method (VEM) is a generalization of the Finite Element Method (FEM). It
proposes approximations to fields of interest on elements of almost any shape, by employing a function space
containing a full polynomial space of arbitrary degree, for its convergence properties, along with additional
suitable functions. The degrees of freedom of the element are designed such that these additional functions can
remain unknown while solving the discretized version of the problem. A term containing the projection of the
approximate solution onto the polynomial functions space guarantees the method’s consistency, while a term with
projection’s residual has to be approximated to ensure stability. The approximation for the latter term is still an
open problem for many applications. The objective of this paper is to show a procedure to solve for these virtual
functions, their projection onto the polynomial space and its residual. This is proposed for the 2D Poisson problem,
by presenting the functions as solutions, and solving for them via FEM. Then, finding their polynomial projection
and the residual of the projection. From this, the stabilization term can be approximated up to the error of the FEM
solution. This procedure may be a clue to visualize the virtual functions behavior and, in future, to help on
proposing stabilization schemes.