Gonzalo Rivera

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.

David Mora, Gonzalo Rivera, Iván Velásquez

The aim of this paper is to develop a virtual element method (VEM) for the vibration problem of thin plates on polygonal meshes. We consider a variational formulation relying only on the transverse displacement of the plate and propose an $H^2(Ω)$ conforming discretization by means of the VEM which is simple in terms of degrees of freedom and coding aspects. Under standard assumptions on the computational domain, we establish that the resulting schemeprovides a correct approximation of the spectrum and prove optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. The analysis restricts to simply connected polygonal clamped plates, not necessarily convex. Finally, we report several numerical experiments illustrating the behaviour of the proposed scheme and confirming our theoretical results on different families of meshes. Additional examples of cases not covered by our theory are also presented.

David Mora, Gonzalo Rivera

We present a priori and a posteriori error analysis of a Virtual Element Method (VEM) to approximate the vibration frequencies and modes of an elastic solid. We analyze a variational formulation relying only on the solid displacement and propose an $H^1(Ω)$-conforming discretization by means of VEM. Under standard assumptions on the computational domain, we show that the resulting scheme provides a correct approximation of the spectrum and prove an optimal order error estimate for the eigenfunctions and a double order for the eigenvalues. Since, the VEM has the advantage of using general polygonal meshes, which allows implementing efficiently mesh refinement strategies, we also introduce a residual-type a posteriori error estimator and prove its reliability and efficiency. We use the corresponding error estimator to drive an adaptive scheme. Finally, we report the results of a couple of numerical tests that allow us to assess the performance of this approach.

Felipe Lepe, Gonzalo Rivera

In this paper, we propose and analyze a mixed virtual element method for the approximation of the eigenvalues and eigenfunctions of the two-dimensional elasticity eigenvalue problem. Under standard assumptions on polygonal meshes, we prove the convergence of the discrete solution operator to its continuous counterpart as the mesh size tends to zero. Relying on the spectral theory of compact operators, we establish the spectral correctness of the method and derive error estimates for both eigenvalues and eigenfunctions. A series of numerical experiments is presented to support the theoretical analysis. The results confirm the predicted convergence rates and show that the method is locking-free and able to approximate the spectrum accurately, independently of the shape of the polygonal elements in the mesh.

David Mora, Gonzalo Rivera, Rodolfo Rodríguez

The paper deals with the a posteriori error analysis of a virtual element method for the Steklov eigenvalue problem. The virtual element method has the advantage of using general polygonal meshes, which allows implementing very efficiently mesh refinement strategies. We introduce a residual type a posteriori error estimator and prove its reliability and efficiency. We use the corresponding error estimator to drive an adaptive scheme. Finally, we report the results of a couple of numerical tests, that allow us to assess the performance of this approach.