Virtual Element and Discontinuous Galerkin Methods
Virtual element methods (VEM) are the latest evolution of the Mimetic Finite Difference Method and can be considered to be more close to the Finite Element approach. They combine the ductility of mimetic finite differences for dealing with rather weird el
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n. The aim of this paper is to present a possible way to introduce the virtual element method (VEM) in the discontinuous Galerkin (DG) framework. From several points of view VEM can be considered as the natural extension of Finite Element Methods to more general geometries and continuity requirements. Apparently, their extension to the Discontinuous Galerkin world could be seen as useless, as DG methods can already deal with rather general geometries. However, in a certain number of their applications there is some need of a conforming interpolant that for general geometries or for higher order continuity (as for plate problems, among others) will not be easily available within the usual DG framework. Here, however, to start with, we will deal with the simplest possible case, that is the discretization of the Poisson problem in two dimensions. The idea is to start understanding what are the most convenient ways to deal with Discontinuous Virtual Elements. We shall see that a direct application of the DG technology cannot be done, but some simple variants are available that still ensure uniqueness, stability, and convergence with optimal error bounds. As a first step we will recall the basic concepts of VEM. This will be done with some details, taking into account that the introduction of VEM is quite recent, and we cannot expect many readers to be familiar with them. In the next section we will present the basic assumptions (on the element geometry, on the discrete spaces) and recall an abstract convergence result. ∗ IUSS-Pavia, Pavia, Italy and King Abdulaziz University, Jeddah, Saudi Arabia, [email protected] † Universit` a di Pavia, Pavia, Italy, [email protected]
X. Feng et al. (eds.), Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations, The IMA Volumes in Mathematics and its Applications 157, DOI 10.1007/978-3-319-01818-8 9, © Springer International Publishing Switzerland 2014
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Franco Brezzi and L. Donatella Marini
Then we will recall the general way to construct the discrete bilinear form, in Sect. 3, and the discrete right-hand side, in Sect. 4. In Sect. 5 we will recall the classical instruments and concepts of DG formulations (in a much less detailed way, this time). The novelty of the paper will appear in Sect. 6, where VEM will be adapted to DG formulations, and in Sect. 7 where optimal error bounds will be proved. Throughout the paper, we will follow the usual notation for Sobolev spaces and norms (see, e.g., [6]). In particular, for an open bounded domain D, we will use | · |s,D and · s,D to denote seminorm and norm, respectively, in the Sobolev space H s (D), while (·, ·)0,D will denote the L2 (D) inner product. Often the subscript will be omitted when D is the computational domain Ω. For a nonnegative integer k, the space of polynomials of degree less than or equal to k will be denoted by Pk . Following a common convention, we will also use P−1 := {0}. Finally, C will be a generic constant independent of the decomposition that could
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