A lowest-order Stabilization-free Virtual Element Method for 3D problems

This work aims at presenting a novel Stabilization-free Virtual Element method for 3D problems. In particular, we focus on the discretization of the linear elastic equation, but the new projection operator introduced can be applied to derive a self-stabilized formulation also for general scalar elliptic equations. The method is introduced in its lowest order formulation and for the analysis we consider the class of polyhedra with triangular faces, typically called deltahedra. We provide a sufficient condition on the polynomial projection space that implies the well-posedness. Several numerical tests assess the robustness of the method and confirm the theoretical convergence rates. Furthermore, we test the proposed method on a nonlinear elasticity problem, to show the robustness of the proposed self-stabilized formulation for solving nonlinear problems. • Virtual Element method without an arbitrary stabilization. • operator-preserving bilinear form defined on polyhedra with triangular faces. • Well-posedness analysis and algorithm to compute stable polynomial projector. • Numerical tests assessing local coercivity. • Robustness with respect to nonlinearities.