We present, on the simplest possible case, what we consider as the very basic features of the (brand new) virtual element method. As the readers will easily recognize, the virtual element method could easily be regarded as the ultimate evolution of the mimetic finite differences approach. However, in their last step they became so close to the traditional finite elements that we decided to use a different perspective and a different name. Now the virtual element spaces are just like the usual finite element spaces with the addition of suitable non-polynomial functions. This is far from being a new idea. See for instance the very early approach of E. Wachspress [A Rational Finite Element Basic (Academic Press, 1975)] or the more recent overview of T.-P. Fries and T. Belytschko [The extended/generalized finite element method: An overview of the method and its applications, Int. J. Numer. Methods Engrg.84 (2010) 253–304]. The novelty here is to take the spaces and the degrees of freedom in such a way that th...

Basic principles of Virtual Element Methods

This paper provides a tutorial and survey of methods for parameterizing surfaces with a view to applications in geometric modelling and computer graphics. We gather various concepts from differential geometry which are relevant to surface mapping and use them to understand the strengths and weaknesses of the many methods for parameterizing piecewise linear surfaces and their relationship to one another.

/pdf/surface-parameterization-a-tutorial-and-survey-3u3e0o47hm.pdf

Surface Parameterization: a Tutorial and Survey

Constructing a function that interpolates a set of values defined at vertices of a mesh is a fundamental operation in computer graphics. Such an interpolant has many uses in applications such as shading, parameterization and deformation. For closed polygons, mean value coordinates have been proven to be an excellent method for constructing such an interpolant. In this paper, we generalize mean value coordinates from closed 2D polygons to closed triangular meshes. Given such a mesh P, we show that these coordinates are continuous everywhere and smooth on the interior of P. The coordinates are linear on the triangles of P and can reproduce linear functions on the interior of P. To illustrate their usefulness, we conclude by considering several interesting applications including constructing volumetric textures and surface deformation.

/pdf/mean-value-coordinates-for-closed-triangular-meshes-3qnm5vwa9c.pdf

Mean value coordinates for closed triangular meshes

We present the essential ingredients in the Virtual Element Method for a simple linear elliptic second-order problem. We emphasize its computer implementation, which will enable interested readers to readily implement the method.

http://arturo.imati.cnr.it/brezzi/papers/hitchhikers-preprint.pdf

The Hitchhiker's Guide to the Virtual Element Method

In the original virtual element space with degree of accuracy k, projector operators in the H^1-seminorm onto polynomials of degree @?k can be easily computed. On the other hand, projections in the L^2 norm are available only on polynomials of degree @?k-2 (directly from the degrees of freedom). Here, we present a variant of the virtual element method that allows the exact computations of the L^2 projections on all polynomials of degree @?k. The interest of this construction is illustrated with some simple examples, including the construction of three-dimensional virtual elements, the treatment of lower-order terms, the treatment of the right-hand side, and the L^2 error estimates.

http://arturo.imati.cnr.it/brezzi/papers/projectors.pdf

Eugene L. Wachspress

Papers

A Rational Finite Element Basis