Lourenço Beirão da Veiga

Lourenço Beirão da Veiga, David Mora, Gonzalo Rivera et al.

We analyze in this paper a virtual element approximation for the acoustic vibration problem. We consider a variational formulation relying only on the fluid displacement and propose a discretization by means of H(div) virtual elements with vanishing rotor. Under standard assumptions on the meshes, we show that the resulting scheme provides a correct approximation of the spectrum and prove optimal order error estimates. With this end, we prove approximation properties of the proposed virtual elements. We also report some numerical tests supporting our theoretical results.

Lourenço Beirão da Veiga, David Mora, Gonzalo Rivera

We present a virtual element method for the Reissner-Mindlin plate bending problem which uses shear strain and deflection as discrete variables without the need of any reduction operator. The proposed method is conforming in $[H^{1}(Ω)]^2 \times H^2(Ω)$ and has the advantages of using general polygonal meshes and yielding a direct approximation of the shear strains. The rotations are then obtained by a simple postprocess from the shear strain and deflection. We prove convergence estimates with involved constants that are uniform in the thickness $t$ of the plate. Finally, we report numerical experiments which allow us to assess the performance of the method.

Lorenzo Mascotto, Lourenço Beirão da Veiga, Alexey Chernov et al.

In the present work, we analyze the $hp$ version of Virtual Element methods for the 2D Poisson problem. We prove exponential convergence of the energy error employing sequences of polygonal meshes geometrically refined, thus extending the classical choices for the decomposition in the $hp$ Finite Element framework to very general decomposition of the domain. A new stabilization for the discrete bilinear form with explicit bounds in $h$ and $p$ is introduced. Numerical experiments validate the theoretical results. We also exhibit a numerical comparison between $hp$ Virtual Elements and $hp$ Finite Elements.

Lourenço Beirão da Veiga, Daniele Antonio Di Pietro, Jérôme Droniou

Polytopal methods provide a flexible framework for the numerical approximation of partial differential equations on general meshes. Their convergence analysis raises specific challenges due to their inherently non-conforming nature and, in many cases, the fully discrete nature of their solution. Two main techniques are considered: the virtual-function approach, used, e.g., in the context of Virtual Element Methods, and the fully discrete approach, which underlies, e.g., the Discrete de Rham method. We introduce here a novel framework based on the notion of conforming liftings, namely bounded and consistent mappings from the discrete space into the continuous space. This approach bridges the virtual and fully discrete viewpoints, clarifies the role of norm equivalence for virtual functions, and leads to a decomposition of the consistency error usable for polytopal methods. The three approaches are demonstrated on a model problem, which provides the opportunity to discuss relevant technical points. Bridges with the convergence properties of discrete differential complexes are also built.