Abstract
We analyze the virtual element methods (VEM) on a simple elliptic model problem, allowing for more general meshes than the one typically considered in the VEM literature. For instance, meshes with arbitrarily small edges (with respect to the parent element diameter) can be dealt with. Our general approach applies to different choices of the stability form, including, for example, the “classical” one introduced in Ref. 4, and a recent one presented in Ref. 34. Finally, we show that the stabilization term can be simplified by dropping the contribution of the internal-to-the-element degrees of freedom. The resulting stabilization form, involving only the boundary degrees of freedom, can be used in the VEM scheme without affecting the stability and convergence properties. The numerical tests are in accordance with the theoretical predictions.
Highlights
The virtual element method (VEM) has been introduced recently in Refs. 4, 5, 16 and 1 as a generalization of the finite element method that allows to make use of general polygonal/polyhedral meshes
A VEM scheme may be seen as a Galerkin method built by means of two parts: (1) a first term strongly consistent on polynomials, which guarantees the accuracy; (2) a stabilization term sE(·, ·), involving a suitably designed bilinear form
The stability analysis is more involved if one allows for more general mesh assumptions
Summary
The virtual element method (VEM) has been introduced recently in Refs. 4, 5, 16 and 1 as a generalization of the finite element method that allows to make use of general polygonal/polyhedral meshes. The virtual element method (VEM) has been introduced recently in Refs. 4, 5, 16 and 1 as a generalization of the finite element method that allows to make use of general polygonal/polyhedral meshes. The VEM, that enjoyed an increasing interest in the recent literature, has been developed in many aspects and applied to many different problems; we here cite only a few works 8, 6, 7, 11, 12, 15, 24, 17, 27, 28, 10, 34 and 33) in addition to the ones above, without pretending to be exhaustive. We note that VEM is not the only recent method that can make use of polytopal meshes: we refer, again as a minimal sample list of papers, to Refs. We note that VEM is not the only recent method that can make use of polytopal meshes: we refer, again as a minimal sample list of papers, to Refs. 18, 19, 20, 23, 29, 31 and 32
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