We present a Virtual Element Method (VEM) for possibly nonlinear elastic and inelastic problems, mainly focusing on a small deformation regime. The numerical scheme is based on a low-order approximation of the displacement field, as well as a suitable treatment of the displacement gradient. The proposed method allows for general polygonal and polyhedral meshes, it is efficient in terms of number of applications of the constitutive law, and it can make use of any standard black-box constitutive law algorithm. Some theoretical results have been developed for the elastic case. Several numerical results within the 2D setting are presented, and a brief discussion on the extension to large deformation problems is included.

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.

Abstract. In this paper we propose and analyze a novel stream formulation of the virtual element method (VEM) for the solution of the Stokes problem. The new formulation hinges upon the introduction of a suitable stream function space (characterizing the divergence free subspace of discrete velocities) and it is equivalent to the velocity-pressure (inf-sup stable) mimetic scheme presented in [L. Beirão da Veiga et al., J. Comput. Phys., 228 (2009), pp. 7215-7232] (up to a suitable reformulation into the VEM framework). Both schemes are thus stable and linearly convergent but the new method results to be more desirable as it employs much less degrees of freedom and it is based on a positive definite algebraic problem. Several numerical experiments assess the convergence properties of the new method and show its computational advantages with respect to the mimetic one.

Abstract. The aim of this paper is to analyze the linear elasticity eigenvalue problem formulated in terms of the stress tensor and the rotation. This is achieved by considering a mixed variational formulation in which the symmetry of the stress tensor is imposed weakly. We show that a discretization of the mixed eigenvalue elasticity problem with reduced symmetry based on the lowest order Arnold-Falk-Winther element provides a correct approximation of the spectrum and prove quasi-optimal error estimates. Finally, we report some numerical experiments.Key words. Mixed elasticity equations, eigenvalue problem, finite elements, error estimates.AMS subject classifications. 65N12, 65N15, 65N25, 65N30, 74B051. Introduction. We analyze in this paper a mixed finite element approximation of an eigenvalue problem arising in linear elasticity. The use of mixed methods for the numerical solution of elasticity problems may be motivated by the need of obtaining direct finite element approximations of stresses ensuring the equilibrium condition. It is also well-known that mixed methods are suitable to deal safely with nearly incompressible materials since they are free from the locking phenomenon.The preservation of the stress tensor symmetry represents the more complicated issue in the construction of mixed finite elements in continuum mechanics. During the last decade, stable mixed finite element methods for linear elasticity, including strong and weakly imposed symmetry for the stresses, have been derived using mathematical tools based on the finite element exterior calculus (cf. [3,4,5,6]). The first mixed finite elements known to be stable for the symmetric stress-displacement two-dimensional formulation is provided in [6]. A three-dimensional analogue of this element was proposed in [1]. We are interested here in mixed methods in which the symmetry of the stress tensor is imposed weakly by means of a suitable Lagrange multiplier. In spite of the introduction of an additional variable, these methods produce mixed finite elements with less degrees of freedom. One of the oldest methods in this category was introduced in [2]; it is based on the so called PEERS element. Recently, further stable elements with a weak symmetry condition for the stresses have been constructed in [3] and [5]. Proofs employing more classical techniques are given in [11] for some of the main results obtained in [3] and [5]. We illustrate here our spectral approximation theory for the mixed formulation of the elasticity problem by employing the lowest-order Arnold-Falk-Winther (AFW) element. It consists of piecewise linear approximations for the stress and piecewise constant functions for the rotation (as well as for the displacement, which will not appear as an unknown in our problem). We point out that we could as well have chosen other finite element

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.

This paper deals with the analysis of new mixed finite element methods for the Brinkman equations formulated in terms of velocity, vorticity and pressure. Employing the Babuška-Brezzi theory, it is proved that the resulting continuous and discrete variational formulations are well-posed. In particular, we show that RaviartThomas elements of order k ≥ 0 for the approximation of the velocity field, piecewise continuous polynomials of degree k + 1 for the vorticity, and piecewise polynomials of degree k for the pressure, yield unique solvability of the discrete problem. On the other hand, we also show that families of finite elements based on Brezzi-DouglasMarini elements of order k +1 for the approximation of velocity, piecewise continuous polynomials of degree k +2 for the vorticity, and piecewise polynomials of degree k for the pressure ensure the well-posedness of the associated Galerkin scheme. We note that these families provide exactly divergence-free approximations of the velocity field. We establish a priori error estimates in the natural norms with constants independent B Ricardo Ruiz-Baier

No abstract