Journal ArticleDOI
Basic principles of Virtual Element Methods
TLDR
This work presents, on the simplest possible case, what it considers as the very basic features of the (brand new) virtual element method, which is the ultimate evolution of the mimetic finite differences approach.Abstract:
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...read more
Citations
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Journal ArticleDOI
The Hitchhiker's Guide to the Virtual Element Method
TL;DR: The essential ingredients in the Virtual Element Method for a simple linear elliptic second-order problem are presented and its computer implementation is emphasized to enable interested readers to readily implement the method.
Journal ArticleDOI
Equivalent projectors for virtual element methods
TL;DR: A variant of the virtual element method that allows the exact computations of the L^2 projections on all polynomials of degree @?k to be presented.
Journal ArticleDOI
Virtual Elements for Linear Elasticity Problems
TL;DR: The application of virtual elements to linear elasticity problems, for both the compressible and the nearly incompressible case, is discussed.
Journal ArticleDOI
Mimetic finite difference method
TL;DR: Flexibility and extensibility of the mimetic methodology are shown by deriving higher-order approximations, enforcing discrete maximum principles for diffusion problems, and ensuring the numerical stability for saddle-point systems.
Journal ArticleDOI
On the Virtual Element Method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes
TL;DR: This work focuses on the linear elasticity equations in three-dimensions and elaborate upon the key concepts underlying the first-order VEM, and presents several numerical studies in order to verify convergence of the VEM and evaluate its performance for various types of meshes.
References
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Book
The Finite Element Method for Elliptic Problems
Philippe G. Ciarlet,J. T. Oden +1 more
TL;DR: The finite element method has been applied to a variety of nonlinear problems, e.g., Elliptic boundary value problems as discussed by the authors, plate problems, and second-order problems.
Book
Finite Element Method for Elliptic Problems
TL;DR: In this article, Ciarlet presents a self-contained book on finite element methods for analysis and functional analysis, particularly Hilbert spaces, Sobolev spaces, and differential calculus in normed vector spaces.
Book
The Mathematical Theory of Finite Element Methods
TL;DR: In this article, the construction of a finite element of space in Sobolev spaces has been studied in the context of operator-interpolation theory in n-dimensional variational problems.
Book
Mixed and Hybrid Finite Element Methods
Franco Brezzi,Michel Fortin +1 more
TL;DR: Variational Formulations and Finite Element Methods for Elliptic Problems, Incompressible Materials and Flow Problems, and Other Applications.