We analyze the virtual element methods (VEM) on a simple elliptic model problem, allowing for more general meshes than the one typically considered in the VEM literature. For instance, meshes with arbitrarily small edges (with respect to the parent element diameter) can be dealt with. Our general approach applies to different choices of the stability form, including, for example, the "classical" one introduced in Ref. 4, and a recent one presented in Ref. 34. Finally, we show that the stabilization term can be simplified by dropping the contribution of the internal-to-the-element degrees of freedom. The resulting stabilization form, involving only the boundary degrees of freedom, can * Corresponding author

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 present work deals with the formulation of a Virtual Element Method (VEM) for two dimensional structural problems. The contribution is split in two parts: in part I, the elastic problem is discussed, while in part II [[3]] the method is extended to material nonlinearity, considering different inelastic responses of the material. In particular, in part I a standardized procedure for the construction of all the terms required for the implementation of the method in a code is explained. The procedure is initially illustrated for the simplest case of quadrilateral virtual elements with linear approximation of displacement variables on the boundary of the element. Then, the case of polygonal elements with quadratic and, even, higher order interpolation is considered. The construction of the method is detailed, deriving the approximation of the consistent term, the required stabilization term and the loading term for all the considered virtual elements. A wide numerical investigation is performed to assess the performances of the developed virtual elements, considering different number of edges describing the elements and different order of approximations of the unknown field. Numerical

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Abstract. This paper deals with the a posteriori error analysis of mixed finite element methods for second order elliptic equations. It is shown that a reliable and efficient error estimator can be constructed using a postprocessed solution of the method. The analysis is performed in two different ways: under a saturation assumption and using a Helmholtz decomposition for vector fields.

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The numerical approximation of 2D elasticity problems is considered, in the framework of the small strain theory and in connection with the mixed Hellinger-Reissner variational formulation. A low-order Virtual Element Method (VEM) with a-priori symmetric stresses is proposed. Several numerical tests are provided, along with a rigorous stability and convergence analysis.we consider the (mixed) Hellinger-Reissner functional (see, for instance, [12,14]) as the starting point of the discretization procedure. Thus, the numerical scheme approximates both the stress and the displacement fields.It is well-known that in the Finite Element practice, designing a stable and accurate element for the Hellinger-Reissner functional, is not at all a trivial task. Essentially, one is led either to consider quite cumbersome schemes, or to relax the symmetry of the Cauchy stress field, or to employ composite elements (a discussion about this issue can be found in [12], for instance). We here exploit the flexibility of the VEM approach to propose and study a low-order scheme, with a-priori symmetric Cauchy stresses, that can be used for general polygons, from triangular shapes on. Furthermore, the method is robust with respect to the compressibility parameter, and therefore can be used for nearly incompressible situations. Our scheme approximates the stress field by using traction degrees of freedom (three per each edge), while the displacement field inside each polygon is essentially a rigid body motion. The VEM concept is then applied essentially for the stress field. We also remark that the construction of the discrete stress field is somehow similar to the construction of the discrete velocity field used for the Stokes problem in [10]. Instead, the displacement field is modelled with polynomial functions, in accordance with the classical Finite Element procedure.An outline of the paper is as follows. In Section 2 we briefly introduce the Hellinger-Reissner variational formulation of the elasticity problem. Section 3 concerns with the discrete problem: all the bilinear and linear forms are introduced and detailed. Numerical experiments are reported in Section 4, where suitable error measures are considered. These numerical tests are supported by the stability and convergence analysis developed in Section 5. Finally, Section 6 draws some conclusions, including possible future extensions of the present study.Throughout the paper, given two quantities a and b, we use the notation a b to mean: there exists a constant C, independent of the mesh-size, such that a ≤ C b. Moreover, we use standard notations for Sobolev spaces, norms and semi-norms (cf.[27], for example). The elasticity problem in mixed formIn this section we briefly present the elasticity problem as it stems from the Hellinger-Reissner principle. More details can be found in [12,14].

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