The virtual element method and its applications / / Paola F. Antonietti, Lourenco Beirao da Veiga, Gianmarco Manzini, editors

• Pubblicazione: Cham : , : Springer, , [2022]; ©2022
• Descrizione fisica: 1 online resource (621 pages)
• Disciplina: 519.4
• Soggetto topico: Numerical analysis.<U+0009>; Anàlisi numèrica
• Soggetto genere / forma: Llibres electrònics
• Soggetto non controllato: Mathematics
• Persona (resp. second.): AntoniettiPaola Francesca; Beirão da VeigaLourenço; ManziniGianmarco
• Nota di bibliografia: Includes bibliographical references.
• Nota di contenuto: Intro -- Preface -- Contents -- Editors and Contributors -- About the Editors -- Contributors -- 1 VEM and the Mesh -- 1.1 Introduction -- 1.2 Model Problem -- 1.3 State of the Art -- 1.3.1 Geometrical Assumptions -- 1.3.2 Convergence Results in the VEM Literature -- 1.4 Violating the Geometrical Assumptions -- 1.4.1 Datasets Definition -- 1.4.2 VEM Performance over the Datasets -- 1.5 Mesh Quality Metrics -- 1.5.1 Polygon Quality Metrics -- 1.5.2 Performance Indicators -- 1.5.3 Results -- 1.6 Mesh Quality Indicators -- 1.6.1 Definition -- 1.6.2 Results -- 1.7 PEMesh Benchmarking Tool -- References -- 2 On the Implementation of Virtual Element Method for Nonlinear Problems over Polygonal Meshes -- 2.1 Introduction -- 2.1.1 Structure of the Chapter -- 2.1.2 Basic Notation -- 2.2 Governing Equations -- 2.3 Virtual Element Framework -- 2.4 Computation of the Projection Operators and Discrete Bilinear Forms -- 2.5 Fully Discrete Scheme -- 2.6 Implementation -- 2.7 Numerical Examples -- 2.8 Conclusion -- References -- 3 Discrete Hessian Complexes in Three Dimensions -- 3.1 Introduction -- 3.2 Matrix and Vector Operations -- 3.2.1 Matrix-Vector Products -- 3.2.2 Differentiation -- 3.2.3 Matrix Decompositions -- 3.2.4 Projections to a Plane -- 3.3 Two Hilbert Complexes for Tensors -- 3.3.1 Hessian Complexes -- 3.3.2 divdiv Complexes -- 3.4 Polynomial Complexes for Tensors -- 3.4.1 De Rham and Koszul Polynomial Complexes -- 3.4.2 Hessian Polynomial Complexes -- 3.4.3 Divdiv Polynomial Complexes -- 3.5 A Conforming Virtual Element Hessian Complex -- 3.5.1 ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper H left parenthesis d i v right parenthesis) /StPNE pdfmark [/StBMC pdfmarkH(div)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark-Conforming Element for Trace-Free Tensors -- 3.5.2 H2-Conforming Virtual Element -- 3.5.3 Trace Complexes. ; 3.5.4 ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper H left parenthesis c u r l right parenthesis) /StPNE pdfmark [/StBMC pdfmarkH(curl)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark-Conforming Element for Symmetric Tensors -- 3.5.5 Discrete Conforming Hessian Complex -- 3.5.6 Discrete Poincaré Inequality -- 3.6 Discretization for the Linearized Einstein-Bianchi System -- 3.6.1 Linearized Einstein-Bianchi System -- 3.6.2 Conforming Discretization -- References -- 4 Some Virtual Element Methods for Infinitesimal ElasticityProblems -- 4.1 Introduction -- 4.2 Elasticity Formulation with Infinitesimal Strain -- 4.2.1 Primal Form -- 4.2.2 Mixed Form -- 4.3 Virtual Element Methods for Elasticity -- 4.3.1 Primal Methods Based on Virtual Work Principle -- 4.3.1.1 The Local Space -- 4.3.1.2 The Local Bilinear Form -- 4.3.1.3 The Local Loading Term -- 4.3.1.4 The Discrete Scheme -- 4.3.1.5 Mixed Methods Based on Hellinger Reissner Principle: 2D Case -- 4.3.1.6 The Local Spaces -- 4.3.1.7 The Local Bilinear Forms -- 4.3.1.8 The Local Loading Term -- 4.3.1.9 The Discrete Scheme -- 4.3.2 Mixed Methods Based on Hellinger Reissner Principle: 3D Case -- 4.3.2.1 The Local Spaces -- 4.3.2.2 The Local Forms -- 4.3.2.3 The Local Loading Term -- 4.3.2.4 The Discrete Scheme -- 4.4 Numerical Results -- 4.4.1 2D Numerical Tests -- 4.4.1.1 Primal Formulation -- 4.4.1.2 Hellinger-Reissner Mixed Formulation -- 4.4.2 3D Numerical Results -- 4.5 Conclusions -- References -- 5 An Introduction to Second Order Divergence-Free VEM for Fluidodynamics -- 5.1 Introduction -- 5.2 The Navier-Stokes Equation -- 5.3 Notations and Preliminaries -- 5.4 Virtual Element Spaces in 2D -- 5.4.1 Virtual Elements for Stokes -- 5.4.2 Enhanced Virtual Elements for Navier-Stokes -- 5.5 Virtual Elements on Curved Polygons -- 5.6 Virtual Element Spaces in 3D -- 5.6.1 Face Spaces. ; 5.6.2 Virtual Elements for Stokes -- 5.6.3 Enhanced Virtual Elements for Navier-Stokes -- 5.7 Virtual Element Problem -- 5.7.1 Global Spaces -- 5.7.2 Discrete Forms -- 5.7.3 Divergence-Free Velocity Solution -- 5.8 Convergence Results and Exploring the Divergence-FreeProperty -- 5.8.1 Convergence Results -- 5.8.2 Reduced Virtual Elements -- 5.8.3 Stokes Complex and curl Formulation -- 5.8.4 Stability in the Darcy Limit and Brinkman Equation -- 5.9 Numerical Tests -- 5.10 Conclusions -- References -- 6 A Virtual Marriage à la Mode: Some Recent Results on the Coupling of VEM and BEM -- 6.1 Introduction -- 6.2 The Coupling Procedures -- 6.2.1 BIEM for Laplace and Helmholtz -- 6.2.2 The Costabel & -- Han Coupling -- 6.2.3 The Modified Costabel & -- Han Coupling -- 6.2.4 Solvability Analysis -- 6.3 The Costabel & -- Han VEM/BEM Schemes in 2D -- 6.3.1 Preliminaries -- 6.3.2 The Costabel & -- Han VEM/BEM Schemefor Poisson -- 6.3.2.1 The Discrete Setting -- 6.3.2.2 Solvability and a Priori Error Analyses -- 6.3.3 The Costabel & -- Han VEM/BEM Schemefor Helmholtz -- 6.3.3.1 The Discrete Setting -- 6.3.3.2 Solvability and a Priori Error Analyses -- 6.4 The Modified Costabel & -- Han VEM/BEM Schemes in 3D -- 6.4.1 Preliminaries -- 6.4.2 The Discrete Setting -- 6.4.3 Solvability and a Priori Error Analyses -- 6.5 Numerical Results -- 6.5.1 Convergence Tests for the Poisson Model -- 6.5.2 Convergence Tests for the Helmholtz Model -- References -- 7 Virtual Element Approximation of Eigenvalue Problems -- 7.1 Introduction -- 7.2 Abstract Setting -- 7.2.1 Model Problem -- 7.3 Virtual Element Approximation of the Laplace Eigenvalue Problem -- 7.3.1 Virtual Element Method -- 7.3.2 The VEM Discretization of the LaplaceEigenproblem -- 7.3.3 Convergence Analysis -- 7.3.4 Numerical Results -- 7.4 Extension to Nonconforming and hp Version of VEM. ; 7.4.1 Nonconforming VEM -- 7.4.2 hp Version of VEM -- 7.5 The Choice of the Stabilization Parameters -- 7.5.1 A Simplified Setting -- 7.5.2 The Role of the VEM Stabilization Parameters -- 7.6 Applications -- 7.6.1 The Mixed Laplace Eigenvalue Problem -- 7.6.2 The Steklov Eigenvalue Problem -- 7.6.3 An Acoustic Vibration Problem -- 7.6.4 Eigenvalue Problems Related to Plate Models -- 7.6.5 Eigenvalue Problems Related to Linear ElasticityModels -- References -- 8 Virtual Element Methods for a Stream-Function Formulation of the Oseen Equations -- 8.1 Introduction -- 8.2 Model Problem -- 8.3 Virtual Element Methods -- 8.3.1 Virtual Spaces and Polynomial Projections Operator -- 8.3.2 Construction of the Local and Global Discrete Forms -- 8.3.3 Discrete Formulation -- 8.4 Error Analysis -- 8.4.1 Preliminary Results -- 8.4.2 A Priori Error Estimates -- 8.5 Recovering the Velocity, Vorticity and Pressure Fields -- 8.5.1 Computing the Velocity Field -- 8.5.2 Computing the Fluid Vorticity -- 8.5.3 Computing the Fluid Pressure -- 8.6 Numerical Results -- 8.6.1 Test 1: Smooth Solution -- 8.6.2 Test 2: Solution with Boundary Layer -- 8.6.3 Test 3: Solution with Non Homogeneous Dirichlet Boundary Conditions -- References -- 9 The Nonconforming Trefftz Virtual Element Method: General Setting, Applications, and Dispersion Analysis for the Helmholtz Equation -- 9.1 Introduction -- 9.2 Polygonal Meshes and Broken Sobolev Spaces -- 9.3 The Nonconforming Trefftz Virtual Element Method for the Laplace Problem -- 9.4 General Structure of Nonconforming Trefftz Virtual Element Methods -- 9.5 The Nonconforming Trefftz Virtual Element Method for the Helmholtz Problem -- 9.6 Stability and Dispersion Analysis for the Nonconforming Trefftz VEM for the Helmholtz Equation -- 9.6.1 Abstract Dispersion Analysis -- 9.6.2 Minimal Generating Subspaces -- 9.6.3 Numerical Results. ; 9.6.3.1 Dependence of Dispersion and Dissipation on the Bloch Wave Angle -- 9.6.3.2 Exponential Convergence of the Dispersion Error Against the Effective Degree q -- 9.6.3.3 Algebraic Convergence of the Dispersion Error Against the Wave Number k -- References -- 10 The Conforming Virtual Element Method for Polyharmonic and Elastodynamics Problems: A Review -- 10.1 Introduction -- 10.1.1 Paradigmatic Examples -- 10.1.1.1 Cahn-Hilliard Equation -- 10.1.1.2 Anisotropic Cahn-Hilliard Equation -- 10.1.1.3 A High Order Phase Field Model for Brittle Fracture -- 10.1.2 Notation and Technicalities -- 10.1.3 Mesh Assumptions -- 10.2 The Virtual Element Method for the Polyharmonic Problem -- 10.2.1 The Continuous Problem -- 10.2.2 The Conforming Virtual Element Approximation -- 10.2.2.1 Virtual Element Spaces -- 10.2.2.2 Modified Lowest Order Virtual Element Spaces -- 10.2.2.3 Discrete Bilinear Form -- 10.2.2.4 Discrete Load Term -- 10.2.2.5 VEM Spaces with Arbitrary Degree of Continuity -- 10.2.2.6 Convergence Results -- 10.3 The Virtual Element Method for the Cahn-Hilliard Problem -- 10.3.1 The Continuous Problem -- 10.3.2 The Conforming Virtual Element Approximation -- 10.3.2.1 A C1 Virtual Element Space -- 10.3.2.2 Virtual Element Bilinear Forms -- 10.3.2.3 The Discrete Problem -- 10.3.3 Numerical Results -- 10.4 The Virtual Element Method for the Elastodynamics Problem -- 10.4.1 The Continuous Problem -- 10.4.2 The Conforming Virtual Element Approximation -- 10.4.2.1 Virtual Element Spaces -- 10.4.2.2 Discrete Bilinear Forms -- 10.4.2.3 Discrete Load Term -- 10.4.2.4 The Discrete Problem -- 10.4.2.5 Stability and Convergence Analysis for the Semi-Discrete Problem -- 10.4.3 Numerical Results -- References -- 11 The Virtual Element Method in Nonlinear and Fracture Solid Mechanics -- 11.1 Introduction -- 11.2 Position of the Problem. ; 11.3 Basis of the VEM in 2D Solid Mechanics.
• Sommario/riassunto: 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 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.
• Titolo autorizzato: The virtual element method and its applications
• ISBN: 3-030-95319-X
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910616364903321
• Lo trovi qui: Univ. Federico II
• Serie: SEMA SIMAI Springer series ; ; Volume 31.