A stabilization-free Virtual Element Method based on divergence-free projections

In this paper, we propose and analyze a Stabilization Free Virtual Element Method (SFVEM), that allows the definition of bilinear forms that do not require an arbitrary stabilization term, thanks to the exploitation of higher-order polynomial projections on divergence free vectors of polynomials. The method is introduced in the lowest order formulation for the Poisson problem. We provide a sufficient condition on the polynomial projection space that implies the well-posedness, proved on particular classes of polygons, and optimal order a priori error estimates. Numerical tests on convex and non-convex polygonal meshes confirm the theoretical convergence rates and show that the method is suitable for solving problems characterized by anisotropies.