The Virtual Element Method and its Applications

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
		11.3.1 Kinematics
		11.3.2 Stiffness Matrix
		11.3.3 Force Vector
		11.3.4 The Case k=3
	11.4 Nonlinear Inelastic Material Response
		11.4.1 Plastic Behavior
		11.4.2 Viscoelastic Behavior
		11.4.3 Shape Memory Alloy Behavior
		11.4.4 Numerical Applications
			11.4.4.1 Viscoelastic Cylinder Subjected to Internal Pressure
			11.4.4.2 Elastoplastic Plate with Circular Hole
			11.4.4.3 Shape Memory Alloy Device
	11.5 Homogenization of Long Fiber Composites
		11.5.1 Problem Formulation
		11.5.2 Computational Homogenization: Smart Use of VEM Meshing Versatility
	11.6 Fracture Mechanics
		11.6.1 Interface Model
		11.6.2 Cracking Process Through VEM Technology
		11.6.3 Numerical Applications
			11.6.3.1 Non-Symmetric Three-Point Bending Test with Topological Adaptive Mesh Refinement
			11.6.3.2 Symmetric Three-Point Bending Test and Comparison with XFEM
	11.7 Concluding Remarks
	References
12 The Virtual Element Method for the Coupled System of Magneto-Hydrodynamics
	12.1 Introduction
	12.2 Mathematical Formulation
		12.2.1 Weak Formulation
	12.3 The Virtual Element Method
		12.3.1 Mesh Notation and Regularity Assumptions
		12.3.2 The Nodal Space
			12.3.2.1 The Polynomial Reconstruction Operators
		12.3.3 The Edge Space
		12.3.4 The Cell Space
		12.3.5 The de Rham Complex
		12.3.6 Fluid Flow
	12.4 Energy Estimates
	12.5 Linearization
	12.6 Well-Posedness and Stability of the Linear Solver
	12.7 Numerical Experiments
		12.7.1 Experimental Study of Convergence
		12.7.2 Magnetic Reconnection
	12.8 Conclusions
	References
13 Virtual Element Methods for Engineering Applications
	13.1 Generic Formulation of a Nonlinear Boundary Value Problem
	13.2 Formulation of the Virtual Element Method
		13.2.1 Ansatz Functions for VEM in Two Dimensions
		13.2.2 Ansatz Functions for VEM in Three Dimensions
		13.2.3 Residual and Tangent Matrix of the Virtual Elements
		13.2.4 Stabilization of the Method
	13.3 VEM for Fracturing Solids
		13.3.1 Basic Equations of Elastic Solids
		13.3.2 Crack Propagation Based on Stress Intensity Factors
		13.3.3 Construction of the Crack Path Using SIF
		13.3.4 Phase-Field Approach for Brittle Crack Propagation
		13.3.5 Numerical Examples
			13.3.5.1 Crack Propagation using Phase-Field Approach: Bi-Material Plate
			13.3.5.2 Crack Propagation Using Stress Intensity Factors
	13.4 VEM for Contact
		13.4.1 Governing Equations for Finite Elasticity and Contact
		13.4.2 Virtual Element Method for Contact
			13.4.2.1 Standard Stabilisation
			13.4.2.2 Edge Stabilisation
			13.4.2.3 Energy Stabilisation
			13.4.2.4 Patch Test and Stabilization Test
		13.4.3 Contact for Large Deformations Including Friction
		13.4.4 Node Insertion Algorithm
			13.4.4.1 Contact Discretization: Frictionless
			13.4.4.2 Contact Discretization: Friction
		13.4.5 Numerical Examples
			13.4.5.1 Hertzian Contact Problem for Small Deformations
			13.4.5.2 Large Deformational Contact: Ironing Problem
			13.4.5.3 Wall Mounting of a Bolt
	13.5 Conclusion
	References