Stefano Berrone: Stabilization-free Virtual Element Method: motivation, construction and a posteriori error estimates

Speaker
Stefano Berrone, Politecnico di Torino

Abstract
In this presentation, we introduce and analyze a stabilization-free Virtual Element Method (VEM) for solving second-order elliptic equations. The distinctive feature of this method is the use of novel polynomial projections that enable the construction of structure-preserving schemes, thereby eliminating the need for traditional stabilization techniques.

Within the framework of VEM discretization, stabilization has long been a central topic of research, as highlighted by numerous recent studies. Stabilization-free VEMs are now attracting growing interest, particularly for applications involving anisotropic diffusion operators, nonlinear elasticity, elastoplasticity, and a posteriori error analysis.

In this talk, we will present a detailed overview of the construction and implementation of the stabilization-free VEM scheme. We will also emphasize its robustness, especially in dealing with anisotropic problems and in deriving a posteriori error estimates that avoid the cumbersome presence of stabilization terms in both the lower and upper bounds of the error.

Finally, several numerical experiments will be shown to illustrate the stability and effectiveness of the method in solving complex problems characterized by strong anisotropies.