Virtual Element Method for general second-order elliptic problems on polygonal meshes

Abstract

We consider the discretization of a boundary value problem for a general linear second-order elliptic operator with smooth coefficients using the Virtual Element approach. As in [A. H. Schatz, An observation concerning Ritz–Galerkin methods with indefinite bilinear forms, Math. Comput. 28 (1974) 959–962] the problem is supposed to have a unique solution, but the associated bilinear form is not supposed to be coercive. Contrary to what was previously done for Virtual Element Methods (as for instance in [L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214]), we use here, in a systematic way, the [Formula: see text]-projection operators as designed in [B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini and A. Russo, Equivalent projectors for virtual element methods, Comput. Math. Appl. 66 (2013) 376–391]. In particular, the present method does not reduce to the original Virtual Element Method of [L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214] for simpler problems as the classical Laplace operator (apart from the lowest-order cases). Numerical experiments show the accuracy and the robustness of the method, and they show as well that a simple-minded extension of the method in [L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214] to the case of variable coefficients produces, in general, sub-optimal results.

The virtual element method in 50 lines of MATLAB

“…The ideas we present here can, however, be generalised to much more complicated situations by applying similar processes to compute the various required terms. The possible extensions of this code are endless: the implementation of higher order methods, more general elliptic operators including lower order terms and non-constant coefficients [8,14], basis functions with higher global regularity properties [10], mesh adaptation driven by a posteriori error indicators [12], or the consideration of time dependent problems [30] to name but a few.…”

The Virtual Element Method for Discrete Fracture Network Flow and Transport Simulations

“…First introduced in [4] and extended in [5,6,3,21,2,26], the Virtual Element Method allows the use of quite general non-degenerate and star-shaped polygons to mesh the spatial domain, even including the possibility of straight angles. In the present framework, we take advantage from this flexibility to easily build a mesh which, on each fracture, is locally or globally conforming to the traces.…”

“…A possible choice for the term S E in the context of the simulation of the flow in DFNs is proposed in [9] and is given by the scalar product between the vectors containing the degrees of freedom of the two arguments on the element ( [4,6]). This choice guarantees property (7) under some basic regularity assumptions on the triangulation.…”

Some Estimates for Virtual Element Methods

“…(See the treatment of the original virtual element method in [10].) The first goal of our paper is to extend some basic finite element estimates to the virtual elements introduced in [1], under the shape regularity assumptions that can be found for example in [1,4,8]. The main tool is a discrete norm for virtual element functions that plays the role of the L norm in the analysis of standard Lagrange finite element functions and which can be controlled by standard shape regularity arguments.…”

“…We will follow the standard notation for differential operators, function spaces and norms that can be found for example in [13,17], and also the notation in [8] for virtual elements.…”