In this paper we initiate the investigation of Virtual Elements with curved faces. We consider the case of a fixed curved boundary in two dimensions, as it happens in the approximation of problems posed on a curved domain or with a curved interface. While an approximation of the domain with polygons leads, for degree of accuracy k≥2, to a sub-optimal rate of convergence, we show (both theoretically and numerically) that the proposed curved VEM lead to an optimal rate of convergence.

The virtual element method (VEM) is a recent technology that can make use of very general polygonal/polyhedral meshes without the need to integrate complex nonpolynomial functions on the elements and preserving an optimal order of convergence. In this article, we develop for the first time, the VEM for parabolic problems on polygonal meshes, considering time-dependent diffusion as our model problem. After presenting the scheme, we develop a theoretical analysis and show the practical behavior of the proposed method through a large array of numerical tests. I. INTRODUCTIONIn the engineering and numerical analysis literature, there has been a recent growth of interest in developing numerical methods that can make use of general polygonal and polyhedral meshes, as opposed to more standard triangular/quadrilateral (tetrahedral/hexahedral) grids. Indeed, making use of polygonal meshes brings forth a range of advantages, including for instance better domain meshing capabilities, automatic use of nonconforming grids, more efficient approximation of geometric data features, more efficient and easier adaptivity, more robustness to mesh deformation, and others (see for instance, the book [1] for more details). This interest in the literature is also reflected in commercial codes, such as CD-Adapco, that have recently included polytopal meshes.We refer to the recent papers [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] as an example of the increasing list of technologies that make use of polygonal/polyhedral meshes. We name here in particular the polygonal finite elements, that generalize finite elements to polygons/polyhedrons by making use of generalized nonpolynomial shape functions, and the mimetic discretization schemes, that combine ideas from the finite difference and finite element methods.The virtual element method (in short, VEM) was introduced in [21] as a generalization of the finite element method. The main idea behind VEM is to use approximated discrete bilinear forms that require only integration of polynomials on the (polytopal) element in order to be computed.Using an integration by parts [21], it is easy to check that, for any v ∈ V k |K , the values of the linear operators (D) above are sufficient in order to compute ∇ K,k (v). As a consequence, the projection operator ∇ K,k depends only on the values of the operators (D). We are now ready to define our virtual local spaceswhere the symbol P k /P k−2 (K) denotes the polynomials of degree k living on K that are L 2 − orthogonal to all polynomials of degree k − 2 on K. We observe that, since W k |K ⊂ V k |K , the operator ∇ K,k is well defined on W k |K and computable only on the basis of the values of the operators (D). Moreover, the space W k |K has three fundamental properties.

The focus of the present paper is on developing a Virtual Element Method for Darcy and Brinkman equations. In [15] we presented a family of Virtual Elements for Stokes equations and we defined a new Virtual Element space of velocities such that the associated discrete kernel is pointwise divergence-free. We use a slightly different Virtual Element space having two fundamental properties: the L 2 -projection onto P k is exactly computable on the basis of the degrees of freedom, and the associated discrete kernel is still pointwise divergence-free. The resulting numerical scheme for the Darcy equation has optimal order of convergence and H 1 conforming velocity solution. We can apply the same approach to develop a robust virtual element method for the Brinkman equation that is stable for both the Stokes and Darcy limit case. We provide a rigorous error analysis of the method and several numerical tests.The focus of this paper is on developing a new Virtual Element Method for the Darcy equation that is suitable for a robust extension to the (more complex) Brinkman problem. For such a problem, other VEM numerical schemes have been proposed, see for example [23,8].In [15] the authors developed a new Virtual Element Method for Stokes problems by exploiting the flexibility of the Virtual Element construction in a new way. In particular, they define a new Virtual Element space of velocities carefully designed to solve the Stokes problem. In connection with a suitable pressure space, the new Virtual Element space leads to an exactly divergence-free discrete velocity, a favorable property when more complex problems, such as the Navier-Stokes problem, are considered. We highlight that this feature is not shared by the method defined in [6] or by most of the standard mixed Finite Element methods, where the divergence-free constraint is imposed only in a weak (relaxed) sense.In the present contribution we develop the Virtual Element Method for Darcy equations by introducing a slightly different virtual space for the velocities such that the local L 2 orthogonal projection onto the space of polynomials of degree less or equal than k (where k is the polynomial degree of accuracy of the method) can be computed using the local degrees of freedom. The resulting Virtual Elements family inherits the advantages on the scheme proposed in [15], in particular it yields an exactly divergence-free discrete kernel. Thus we obtain a stable Darcy element that is also uniformly stable for the Stokes problem. A sample of uniformly stable methods for Darcy-Stokes model is for instance [39,53,36,49].The last part of the paper deals with the analysis of a new mixed finite element method for Brinkman equations that stems from the above scheme for the Darcy problem. Mathematically, the Brinkman problem resembles both the Stokes problem for fluid flow and the Darcy problem for flow in porous media (see [35,2,34]). Constructing finite element methods to solve the Brinkman equation that are robust for both (Stokes and Darcy) limits is challenging. We ...

In the present paper we develop the Virtual Element Method for hyperbolic problems on polygonal meshes, considering the linear wave equations as our model problem. After presenting the semi-discrete scheme, we derive the convergence estimates in H 1 semi-norm and L 2 norm. Moreover we develop a theoretical analysis on the stability for the fully discrete problem by comparing the Newmark method and the Bathe method. Finally we show the practical behaviour of the proposed method through a large array of numerical tests.

We study the virtual element (VEM) approximation of elliptic eigenvalue problems. The main result of the paper states that VEM provides an optimal order approximation of the eigenmodes. A wide set of numerical tests confirm the theoretical analysis.