Comparison of standard and stabilization free Virtual Elements on anisotropic elliptic problems

Stabilization‐free virtual element method for 2D elastoplastic problems

“…To obtain the discrete bilinear form for the stabilization‐free VEM, we define the $$$ {L}_2 $$$ projection operator $$$ {\Pi}_{k,E}^0 $$$ of the variable field and $$$ {\boldsymbol{\Pi}}_{l,E}^0\nabla $$$ of the gradient of the variable field, which is defined as $$$$ {\Pi}_{k,E}^0:{\mathscr{H}}^1(E)\to {\mathbb{P}}_k(E), $$$$ $$$$ {\boldsymbol{\Pi}}_{l,E}^0\nabla :{\mathscr{H}}^1(E)\to {\left[{\mathbb{P}}_l(E)\right]}^2, $$$$ where $$$ l\in \mathbb{N} $$$ is a parameter defining the order of the necessary polynomial space for sufficient stabilization which depends on the order $$$ k $$$ and the number of edges $$$ {n}_E $$$. As given in Reference 48, the relationship …”

“…The basic idea is to modify the first‐order virtual element space to allow the computation of a higher‐order $$$ {L}_2 $$$ projection of the gradient 47 . The well‐posedness was proven in References 45,48 and the discrete problem can be solved without a stabilizing bilinear form. According to these developments, the stabilization‐free virtual element method has been extended to the Laplacian eigenvalue problem, 49 linear plane elasticity in References 47,50, and 3D elasticity in Reference 51.…”

A first-order stabilization-free Virtual Element Method

“…This paper follows similar lines of the above-mentioned stabilisation-free attempts [12,13,30,31], but for the Laplacian problem written in the usual H(div)−L 2 mixed formulation. In particular, we consider a VEM version of the lowest order Raviart-Thomas Finite Element Method, see [4].…”

“…In [4], a lowest-order stabilization-free scheme was proposed and analysed, proving that it is possible to define coercive bilinear forms based on polynomial projections of virtual basis functions of suitable high-degree polynomial spaces. In [5], the proposed scheme was compared to standard VEM, and results showed that the absence of a stabilization operator can reduce the error and help convergence in case of strongly anisotropic problems.…”

“…Virtual Element schemes for which no stabilisation form is required have been recently presented, in different 2D frameworks, in [12,13,30,31]. These approaches share the idea to employ a projection onto a polynomial space of higher degree than the one usually taken in standard VEM.…”