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The stability and convergence properties of the mimetic finite difference method for diffusion-type problems on polyhedral meshes are analyzed. The optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved.

We provide a framework for the analysis of a large class of discontinuous methods for second-order elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed over the past three decades for the numerical treatment of elliptic problems.

The present review paper has several objectives. Its primary aim is to give an idea of the general features of virtual element methods (VEMs), which were introduced about a decade ago in the field of numerical methods for partial differential equations, in order to allow decompositions of the computational domain into polygons or polyhedra of a very general shape. Nonetheless, the paper is also addressed to readers who have already heard (and possibly read) about VEMs and are interested in gaining more precise information, in particular concerning their application in specific subfields such as ${C}^1$ -approximations of plate bending problems or approximations to problems in solid and fluid mechanics.

In this paper we construct Discontinuous Galerkin approximations of the Stokes problem where the velocity field is H ( div , Ω ) -conforming. This implies that the velocity solution is divergence-free in the whole domain. This property can be exploited to design a simple and effective preconditioner for the final linear system.

We consider the 9-node shell element referred to as the MITC9 shell element in plate bending solutions and present a simplified mathematical analysis. The element uses bi-quadratic interpolations of the rotations and transverse displacement, and the “rotated Raviart-Thomas” interpolations for the transverse shear stresses. A rigorous mathematical analysis of the element is still lacking, even for the simplified case of plate solutions (that is, flat shells), although the numerical evidence suggests a good and reliable behavior. Here we start such an analysis by considering a very simple particular case; namely, a rectangular plate, clamped all around the boundary, and solved with a uniform decomposition. Moreover, we consider only the so-called limit case , corresponding to the limit equations that are obtained for the thickness t going to zero. While the mathematical analysis of the limit case is simpler, such analysis, in general, gives an excellent indication of whether shear locking is present in the real case t  > 0. We detail that the element in the setting considered shows indeed optimal behavior.

In the approximation of linear elliptic operators in mixed form, it is well known that the so-called inf-sup and ellipticity in the kernel properties are sufficient (and, in a sense to be made precise, necessary) in order to have good approximation properties and optimal error bounds. One might think, in the spirit of Mercier-Osborn-Rappaz-Raviart and in consideration of the good behavior of commonly used mixed elements (like Raviart–Thomas or Brezzi–Douglas–Marini elements), that these conditions are also sufficient to ensure good convergence properties for eigenvalues. In this paper we show that this is not the case. In particular we present examples of mixed finite element approximations that satisfy the above properties but exhibit spurious eigenvalues. Such bad behavior is proved analytically and demonstrated in numerical experiments. We also present additional assumptions (fulfilled by the commonly used mixed methods already mentioned) which guarantee optimal error bounds for eigenvalue approximations as well.

We develop a family of locking-free elements for the Reissner–Mindlin plate using Discontinuous Galerkin (DG) techniques, one for each odd degree, and prove optimal error estimates. A second family uses conforming elements for the rotations and nonconforming elements for the transverse displacement, generalizing the element of Arnold and Falk to higher degree.

In this paper we prove convergence and error estimates for the so-called 3-field formulation using piecewise linear finite elements stabilized with boundary bubbles. Optimal error bounds are proved in L2L^2 and in the broken H1H^1 norm for the internal variable uu, and in suitable weighted L2L^2 norms for the other variables λ\\lambda and ψ\\psi.

We revisit classical Virtual Element approximations on polygonal and polyhedral decompositions. We also recall the treatment proposed for dealing with decompositions into polygons with curved edges. In the second part of the paper we introduce a couple of new ideas for the construction of VEM-approximations on domains with curved boundary, both in two and three dimensions. The new approach looks promising, although sound numerical tests should be made to validate the efficiency of the method.

The stability and convergence properties of the mimetic finite difference method for diffusion-type problems on polyhedral meshes are analyzed. The optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved.