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Textbook in PDF format

The purpose of this book is to present the current state of the art of the Virtual Element Method (VEM) by collecting contributions from many of the most active researchers in this field and covering a broad range of topics: from the mathematical foundation to real life computational applications.
The book is naturally divided into three parts. The first part of the book presents recent advances in theoretical and computational aspects of VEMs, discussing the generality of the meshes suitable to the VEM, the implementation of the VEM for linear and nonlinear PDEs, and the construction of discrete hessian complexes. The second part of the volume discusses Virtual Element discretization of paradigmatic linear and non-linear partial differential problems from computational mechanics, fluid dynamics, and wave propagation phenomena. Finally, the third part contains challenging applications such as the modeling of materials with fractures, magneto-hydrodynamics phenomena and contact solid mechanics.
The book is intended for graduate students and researchers in mathematics and engineering fields, interested in learning novel numerical techniques for the solution of partial differential equations. It may as well serve as useful reference material for numerical analysts practitioners of the field.
VEM and the Mesh
On the Implementation of Virtual Element Method for Nonlinear Problems over Polygonal Meshes
Discrete Hessian Complexes in Three Dimensions
Some Virtual Element Methods for Infinitesimal Elasticity Problems
An Introduction to Second Order Divergence-Free VEM for Fluidodynamics
A Virtual Marriage à la Mode: Some Recent Results on the Coupling of VEM and BEM
Virtual Element Approximation of Eigenvalue Problems
Virtual Element Methods for a Stream-Function Formulation of the Oseen Equations
The Nonconforming Trefftz Virtual Element Method: General Setting, Applications, and Dispersion Analysis for the Helmholtz Equation
The Conforming Virtual Element Method for Polyharmonic and Elastodynamics Problems: A Review
The Virtual Element Method in Nonlinear and Fracture Solid Mechanics
The Virtual Element Method for the Coupled System of Magneto-Hydrodynamics
Virtual Element Methods for Engineering Applications