Abstract
A family of Virtual Element Methods for the 2D Navier-Stokes equations is proposed and analysed. The schemes provide a discrete velocity field which is point-wise divergence-free. A rigorous error analysis is developed, showing that the methods are stable and optimally convergent. Several numerical tests are presented, confirming the theoretical predictions. A comparison with some mixed finite elements is also performed.
Highlights
The virtual element method (VEM), introduced in [11, 12], is a recent paradigm for the approximation of partial differential equation problems that shares the same variational background as the finite element methods
Among the Galerkin schemes, VEM is peculiar in that the discrete spaces consist of functions which are not known pointwise, but about which a limited set of information is available
In [15] we have introduced a new family of virtual elements for the Stokes problem on polygonal meshes
Summary
The virtual element method (VEM), introduced in [11, 12], is a recent paradigm for the approximation of partial differential equation problems that shares the same variational background as the finite element methods. Following a standard procedure in the VEM framework, we define a computable discrete local bilinear form (24). We define the global approximated bilinear form ah(\cdot , \cdot ) : Vh \times Vh \rightar \BbbR by summing the local contributions:. Due to property (46), the proposed velocity-pressure coupling turns out to be stable for the Stokes problem, and for the Darcy problem This yields an interesting advantage in complex flow problems where both equations are present: the same spaces can be used in the whole computational domain. Let us define \bfitvar := vI - wI ; for every element E \in \Omega h the following facts hold: \bulet Since vI and wI are polynomials of degree k on \partialE, by definition of D\bfV 1 and D\bfV 2, we have (50).
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