The aim of this paper is to develop a virtual element method for the two-dimensional Steklov eigenvalue problem. We propose a discretization by means of the virtual elements presented in [L. Beirão da Veiga et al., Basic principles of virtual element methods, Math. Models Methods Appl. Sci.23 (2013) 199–214]. Under standard assumptions on the computational domain, we establish that the resulting scheme provides a correct approximation of the spectrum and prove optimal-order error estimates for the eigenfunctions and a double order for the eigenvalues. We also prove higher-order error estimates for the computation of the eigensolutions on the boundary, which in some Steklov problems (computing sloshing modes, for instance) provides the quantity of main interest (the free surface of the liquid). Finally, we report some numerical tests supporting the theoretical results.

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. v ∈ V := {v ∈ H(div; Ω) : v · n = 0 on Γ } , integrating by parts, using the boundary condition and eliminating p, we arrive at the following weak formulation in which, for simplicity, we have taken the physical parameters ̺ and c equal to one and denote λ = ω 2 :Since the bilinear form on the left-hand side is not H(div; Ω)-elliptic, it is convenient to use a shift argument to rewrite this eigenvalue problem in the following equivalent form:Problem 2 Find (λ, w) ∈ R × V, w = 0, such that a(w, v) = (λ + 1) b(w, v) ∀v ∈ V,

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 scheme provides 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.

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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.Key words: virtual element method, elasticity equations, eigenvalue problem, a priori error estimates, a posteriori error analysis, polygonal meshes 2000 MSC: 65N25, 65N30, 70J30, 76M25.Cτ := 2µ S τ + λ S tr(τ )I, where λ S and µ S are the Lamé coefficients, which we assume constant.We introduce the following bounded bilinear forms:Then, the eigenvalue problem above can be rewritten as follows:Problem 2. Find (λ, w) ∈ R × V, w = 0, such that a(w, v) = λb(w, v) ∀v ∈ V.

Abstract. 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 × 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.

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