Basic principles of mixed Virtual Element Methods Brezzi, F.; Falk, Richard S.; Donatella Marini, L.
ESAIM. Mathematical modelling and numerical analysis,
07/2014, Letnik:
48, Številka:
4
Journal Article
Recenzirano
Odprti dostop
The aim of this paper is to give a simple, introductory presentation of the extension of the Virtual Element Method to the discretization of H(div)-conforming vector fields (or, more generally, of (n ...− 1) − Cochains). As we shall see, the methods presented here can be seen as extensions of the so-called BDM family to deal with more general element geometries (such as polygons with an almost arbitrary geometry). For the sake of simplicity, we limit ourselves to the 2-dimensional case, with the aim of making the basic philosophy clear. However, we consider an arbitrary degree of accuracy k (the Virtual Element analogue of dealing with polynomials of arbitrary order in the Finite Element Framework).
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.
► We apply the new-born Virtual Element method to linear plate bending problems, in the Kirchhoff–Love formulation. ► We show how easily C1-elements can be designed. ► Basic idea: choose first d.o.f. ...at the interelement boundaries to guarantee C1-continuity, then add a suitable amount of internal d.o.f.
We discuss the application of Virtual Elements to linear plate bending problems, in the Kirchhoff–Love formulation. As we shall see, in the Virtual Element environment the treatment of the C1-continuity condition is much easier than for traditional Finite Elements. The main difference consists in the fact that traditional Finite Elements, for every element K and for every given set of degrees of freedom, require the use of a space of polynomials (or piecewise polynomials for composite elements) for which the given set of degrees of freedom is unisolvent. For Virtual Elements instead we only need unisolvence for a space of smooth functions that contains a subset made of polynomials (whose degree determines the accuracy). As we shall see the non-polynomial part of our local spaces does not need to be known in detail, and therefore the construction of the local stiffness matrix is simple, and can be done for much more general geometries.
The virtual element method Beirão Da Veiga, Lourenço; Brezzi, Franco; Marini, L. Donatella ...
Acta numerica,
05/2023, Letnik:
32
Journal Article
Recenzirano
Odprti dostop
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.
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 Virtual Element Method (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.
We analyse the family of C1-Virtual Elements introduced in Brezzi and Marini (2013) for fourth-order problems and prove optimal estimates in L2 and in H1 via classical duality arguments.
We apply the weighted-residual approach recently introduced in F. Brezzi et al., Comput. Methods Appl. Mech. Engrg., 195 (2006), pp. 3293–3310 to derive discontinuous Galerkin formulations for ...advection-diffusion-reaction problems. We devise the basic ingredients to ensure stability and optimal error estimates in suitable norms, and propose two new methods.
Bubble stabilization of discontinuous Galerkin methods Antonietti, Paola F.; Brezzi, Franco; Marini, L. Donatella
Computer methods in applied mechanics and engineering,
05/2009, Letnik:
198, Številka:
21
Journal Article
Recenzirano
We analyze the stabilizing effect of the introduction of suitable bubble functions in DG formulations for linear second order elliptic problems, working, for the sake of simplicity, on Laplace ...operator. In particular we analyze the non-symmetric formulation of Baumann–Oden on rather general decompositions, and we show that the piecewise linear discontinuous approximation, without jump stabilization, can be used if suitable bubbles are added to the local spaces.
In a recent paper of Arnold et al. D.N. Arnold, F. Brezzi, L.D. Marini, A family of discontinuous Galerkin finite elements for the Reissner–Mindlin plate, J. Sci. Comput. 22 (2005) 25–45, the ideas ...of discontinuous Galerkin methods were used to obtain and analyze two new families of locking free finite element methods for the approximation of the Reissner–Mindlin plate problem. By following their basic approach, but making different choices of finite element spaces, we develop and analyze other families of locking free finite elements that eliminate the need for the introduction of a reduction operator, which has been a central feature of many locking-free methods. For
k
⩾
2
, all the methods use piecewise polynomials of degree
k to approximate the transverse displacement and (possibly subsets) of piecewise polynomials of degree
k
−
1 to approximate both the rotation and shear stress vectors. The approximation spaces for the rotation and the shear stress are always identical. The methods vary in the amount of interelement continuity required. In terms of smallest number of degrees of freedom, the simplest method approximates the transverse displacement with continuous, piecewise quadratics and both the rotation and shear stress with rotated linear Brezzi–Douglas–Marini elements.
PDF
