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The virtual element method and its applications. (English) Zbl 1497.65005

SEMA SIMAI Springer Series 31. Cham: Birkhäuser (ISBN 978-3-030-95318-8/hbk; 978-3-030-95321-8/pbk; 978-3-030-95319-5/ebook). xxiv, 605 p. (2022).

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Publisher’s description: 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.
The articles of this volume will be reviewed individually.
Indexed articles:
Sorgente, Tommaso; Prada, Daniele; Cabiddu, Daniela; Biasotti, Silvia; Patanè, Giuseppe; Pennacchio, Micol; Bertoluzza, Silvia; Manzini, Gianmarco; Spagnuolo, Michela, VEM and the mesh, 1-57 [Zbl 1503.65301]
Adak, Dibyendu; Arrutselvi, M.; Natarajan, E.; Natarajan, S., On the implementation of virtual element method for nonlinear problems over polygonal meshes, 59-91 [Zbl 1503.65214]
Chen, Long; Huang, Xuehai, Discrete Hessian complexes in three dimensions, 93-135 [Zbl 1503.65291]
Artioli, Edoardo; de Miranda, Stefano; Lovadina, Carlo; Patruno, Luca; Visinoni, Michele, Some virtual element methods for infinitesimal elasticity problems, 137-183 [Zbl 1503.65285]
Beirão da Veiga, Lourenço; Vacca, Giuseppe, An introduction to second order divergence-free VEM for fluidodynamics, 185-225 [Zbl 1503.65286]
Gatica, Gabriel N.; Márquez, Antonio; Meddahi, Salim, A virtual marriage à la mode: some recent results on the coupling of VEM and BEM, 227-274 [Zbl 1503.65294]
Boffi, Daniele; Gardini, Francesca; Gastaldi, Lucia, Virtual element approximation of eigenvalue problems, 275-320 [Zbl 1503.65287]
Mora, David; Silgado, Alberth, Virtual element methods for a stream-function formulation of the Oseen equations, 321-361 [Zbl 1503.65300]
Mascotto, Lorenzo; Perugia, Ilaria; Pichler, Alexander, The nonconforming Trefftz virtual element method: general setting, applications, and dispersion analysis for the Helmholtz equation, 363-410 [Zbl 1503.65299]
Antonietti, Paola F.; Manzini, Gianmarco; Mazzieri, Ilario; Scacchi, Simone; Verani, Marco, The conforming virtual element method for polyharmonic and elastodynamics problems: a review, 411-451 [Zbl 1503.65215]
Artioli, Edoardo; Marfia, Sonia; Sacco, Elio, The virtual element method in nonlinear and fracture solid mechanics, 453-498 [Zbl 1503.65217]
Naranjo Alvarez, Sebastian; Bokil, Vrushali A.; Gyrya, Vitaliy; Manzini, Gianmarco, The virtual element method for the coupled system of magneto-hydrodynamics, 499-556 [Zbl 1503.65241]
Wriggers, Peter; Aldakheel, Fadi; Hudobivnik, Blaž, Virtual element methods for engineering applications, 557-605 [Zbl 1503.65251]

MSC:

65-06 Proceedings, conferences, collections, etc. pertaining to numerical analysis
00B25 Proceedings of conferences of miscellaneous specific interest
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74Sxx Numerical and other methods in solid mechanics
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