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The Mimetic Finite Difference Method for Elliptic Problems / / by Lourenco Beirao da Veiga, Konstantin Lipnikov, Gianmarco Manzini
| The Mimetic Finite Difference Method for Elliptic Problems / / by Lourenco Beirao da Veiga, Konstantin Lipnikov, Gianmarco Manzini |
| Autore | Beirao da Veiga Lourenco |
| Edizione | [1st ed. 2014.] |
| Pubbl/distr/stampa | Cham : , : Springer International Publishing : , : Imprint : Springer, , 2014 |
| Descrizione fisica | 1 online resource (399 p.) |
| Disciplina | 515.353 |
| Collana | MS&A, Modeling, Simulation and Applications |
| Soggetto topico |
Computer mathematics
Mathematical physics Partial differential equations Applied mathematics Engineering mathematics Computational Mathematics and Numerical Analysis Mathematical Applications in the Physical Sciences Partial Differential Equations Mathematical and Computational Engineering |
| ISBN | 3-319-02663-1 |
| Formato | Materiale a stampa  |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto | 1 Model elliptic problems -- 2 Foundations of mimetic finite difference method -- 3 Mimetic inner products and reconstruction operators -- 4 Mimetic discretization of bilinear forms -- 5 The diffusion problem in mixed form -- 6 The diffusion problem in primal form -- 7 Maxwells equations. 8. The Stokes problem. 9 Elasticity and plates -- 10 Other linear and nonlinear mimetic schemes -- 11 Analysis of parameters and maximum principles -- 12 Diffusion problem on generalized polyhedral meshes. |
| Record Nr. | UNINA-9910300158803321 |
Beirao da Veiga Lourenco
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| Cham : , : Springer International Publishing : , : Imprint : Springer, , 2014 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
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The virtual element method and its applications / / Paola F. Antonietti, Lourenco Beirao da Veiga, Gianmarco Manzini, editors
| The virtual element method and its applications / / Paola F. Antonietti, Lourenco Beirao da Veiga, Gianmarco Manzini, editors |
| Pubbl/distr/stampa | Cham : , : Springer, , [2022] |
| Descrizione fisica | 1 online resource (621 pages) |
| Disciplina | 519.4 |
| Collana | SEMA SIMAI Springer series |
| Soggetto topico |
Numerical analysis. Anàlisi numèrica |
| Soggetto genere / forma | Llibres electrònics |
| Soggetto non controllato | Mathematics |
| ISBN | 3-030-95319-X |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| 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. |
| Record Nr. | UNINA-9910616364903321 |
| Cham : , : Springer, , [2022] | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
The virtual element method and its applications / / Paola F. Antonietti, Lourenco Beirao da Veiga, Gianmarco Manzini, editors
| The virtual element method and its applications / / Paola F. Antonietti, Lourenco Beirao da Veiga, Gianmarco Manzini, editors |
| Pubbl/distr/stampa | Cham : , : Springer, , [2022] |
| Descrizione fisica | 1 online resource (621 pages) |
| Disciplina | 519.4 |
| Collana | SEMA SIMAI Springer series |
| Soggetto topico |
Numerical analysis. Anàlisi numèrica |
| Soggetto genere / forma | Llibres electrònics |
| Soggetto non controllato | Mathematics |
| ISBN | 3-030-95319-X |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| 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. |
| Record Nr. | UNISA-996495170203316 |
| Cham : , : Springer, , [2022] | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. di Salerno | ||
| ||