Abstract
EPSRC (Grant EP/L022745/1) to A.C.; Laboratory Directed Research and Development program (LDRD), US Department of Energy Office of Science, Office of Fusion Energy Sciences, under the auspices of the National Nuclear Security Administration of the US Department of Energy by Los Alamos National Laboratory, operated by Los Alamos National Security LLC under contract DE-AC52- 06NA25396 to G.M.; Ph.D. Studentship from the College of Science and Engineering at the University of Leicester and an EPSRC Doctoral Training Grant to O.S.
Highlights
The Virtual Element Method (VEM) was introduced in [6] as a generalisation of the conforming finite element method (FEM), offering great flexibility in utilising meshes with arbitrary polygonal elements
Unlike the polygonal finite element method (PFEM) [41] and other conforming FEM extensions based on the Generalised Finite Element [4] framework such as the CFE method [28] and the XFEM [26], the Virtual Element Methods (VEM) handles meshes with general shaped elements in a manner that avoids the explicit evaluation of the shape functions
We have introduced a unified abstract framework for the Virtual Element Method, through which conforming and nonconforming VEMs for solving general second order elliptic convectionreaction-diffusion problems with non-constant coefficients in two and three dimensions are defined, analysed, and implemented in a largely identical manner
Summary
The Virtual Element Method (VEM) was introduced in [6] as a generalisation of the conforming finite element method (FEM), offering great flexibility in utilising meshes with (almost) arbitrary polygonal elements. In this way a family of virtual element spaces is defined from which a simple choice can be made, cf Section 8 This approach differs completely from that presented for a non-constant diffusion tensor in [11], which required the construction of a bespoke projection operator dependent on the diffusion tensor. This fact is discussed, with the conclusion that the stability and optimal accuracy of the method based on using polynomials of order up to k are unaffected by the use of a quadrature scheme to approximate the consistency terms, provided that this is of at least degree 2k − 2 We stress that this is exactly the same requirement of the finite element methods [19]. |ω| denotes the d–dimensional Hausdorff measure of ω
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