A residual a posteriori error estimate for the Stabilization-free Virtual Element Method

In this work, we present the a posteriori error analysis of Stabilization-Free Virtual Element Methods for the 2D Poisson equation. The abscence of a stabilizing bilinear form in the scheme allows to prove the equivalence between a suitably defined error measure and standard residual error estimators, which is not obtained in general for stabilized virtual elements. Several numerical experiments are carried out, confirming the expected behaviour of the estimator in the presence of different mesh types, and robustness with respect to jumps of the diffusion term.

💡 Research Summary

This paper addresses the a‑posteriori error estimation for a stabilization‑free Virtual Element Method (VEM) applied to the two‑dimensional Poisson problem with homogeneous Dirichlet boundary conditions. Classical VEM formulations introduce a stabilization bilinear form to control the kernel of the polynomial projection operators; however, this additional term complicates the derivation of reliable and efficient error estimators. The authors consider a recent class of VEMs that completely omit the stabilization operator, relying instead on high‑order polynomial projections of the gradients of the virtual basis functions.

The analysis begins by defining a conforming polygonal mesh (\mathcal{M}h) satisfying standard star‑shaped and edge‑length regularity assumptions. For each element (E) a local virtual space (V{E}^{h,k}) is introduced, characterized by the conditions (\Delta v \in \mathbb{P}_k(E)), edge traces belonging to (\mathbb{P}_k(e)), and a vanishing average of the higher‑order moments. Degrees of freedom consist of vertex values, interior Gauss‑Lobatto points (for (k>1)), and internal moments.

A key ingredient is the computable gradient projection (\Pi^{P\nabla}{0,E}) mapping (H^1(E)) onto a polynomial space (\mathbb{P}{k,\ell_E}(E)). Because the projection can be evaluated using only the degrees of freedom, the discrete bilinear form is defined as
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