دانلود کتاب Anisotropic hp-Mesh Adaptation Methods: Theory, implementation and applications (Nečas Center Series)

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Preface
Contents
1 Introduction
1.1 Numerical Solution of Partial Differential Equations
1.2 Basic Concepts of Finite Element Discretization
1.2.1 Variational Formulations
1.2.2 Galerkin Approximations
1.2.3 Abstract Error Estimates
1.3 Domain Partition and Finite Element Spaces
1.3.1 Basic Terms and Notations
1.3.2 Approximation Properties of Finite Element Spaces
1.4 Illustrative Examples
1.4.1 Poisson Problem
1.4.2 Linear Advection-Diffusion Equation: Triple Layer Problem
1.4.3 High-Speed Flow Through a Scramjet
2 Metric Based Mesh Representation
2.1 Uniform Meshes
2.2 Uniform Meshes Under a Riemannian Metric
2.3 Space Anisotropy in Two Dimensions
2.3.1 Space Anisotropy Described by an Ellipse
2.3.2 Geometry of a Triangle
2.3.3 Equilateral Triangle with Respect to Matrix MSym
2.3.4 Steiner Ellipse
2.3.5 Summary of the Description of Space Anisotropy
2.4 Space Anisotropy in Three Dimensions
2.4.1 Space Anisotropy Described by an Ellipsoid
2.4.2 Geometry of a Tetrahedron
2.4.3 Equilateral Tetrahedra with Respect to MatrixMSym
2.5 Mesh Generation by a Metric Field
2.5.1 Setting of the Nodal Metric
2.5.2 Interpolation of the Metric
2.5.3 Mesh Optimization
3 Interpolation Error Estimates for Two Dimensions
3.1 Geometry of a Triangle
3.2 Interpolation Error Function and Its Anisotropic Bound
3.2.1 Interpolation Operator
3.2.2 Interpolation Error Function
3.2.3 s-Homogeneous Functions
3.2.4 Anisotropic Bound of Interpolation Error Function for p=1
3.2.5 Anisotropic Bound of Interpolation Error Function for p>1
3.2.5.1 First Non-guaranteed Estimate
3.2.5.2 Second Guaranteed Estimate
3.3 Interpolation Error Estimates Including the Triangle Geometry
3.3.1 Estimate in the Lq(K)-Norm, 1≤q < ∞
3.3.2 Estimate in the L∞(K)-Norm
3.3.3 Estimate in the H1(K)-Seminorm
3.3.4 Estimate in the L2(∂K)-Norm
3.3.5 Estimate in the H1(∂K)-Seminorm
3.4 Numerical Verification
4 Interpolation Error Estimates for Three Dimensions
4.1 Geometry of a Tetrahedron
4.2 Interpolation Error Function and Its Anisotropic Bound
4.2.1 Interpolation Error Function
4.2.2 Anisotropic Bound of Interpolation Error Function
4.3 Interpolation Error Estimates Including the Tetrahedra Geometry
4.3.1 Estimate in the Lq(K)-Norm, 1≤q < ∞
4.3.2 Estimate in the L2(∂K)-Norm
4.3.3 Estimate in the H1(∂K)-Seminorm
5 Anisotropic Mesh Adaptation Method, h-Variant
5.1 Optimization of the Mesh Element Anisotropy (2D)
5.2 Optimization of the Mesh Element Anisotropy (3D)
5.3 Mesh Adaptation Based on the Equidistribution Principle
5.4 Continuous Mesh Model
5.5 Continuous Mesh Optimization
5.5.1 Solution of Problem 5.24
5.5.2 Solution of Problem 5.25
5.6 Adaptive Solution of Partial Differential Equations
5.6.1 Anisotropic Mesh Adaptation Algorithms for PDEs
5.6.2 Setting of Optimal Size of Mesh Elements
5.6.2.1 A Simple Approach Based on Thresholds
5.6.2.2 Approaches Based on Error Equidistribution
5.7 Numerical Experiments
5.7.1 Boundary Layer Problem
5.7.2 Multiple Difficulties Problem
5.7.3 Summary of the Numerical Examples
6 Anisotropic Mesh Adaptation Method, hp-Variant
6.1 Continuous Mesh Model
6.2 Semi-analytical Optimization
6.3 Anisotropic hp-Mesh Adaptation Algorithm
6.4 Numerical Experiments
6.4.1 L2-Projection of Piecewise Polynomial Function
6.4.2 Boundary Layers Problem
6.4.3 Multiple Difficulties Problem
6.4.4 Mixed Hyperbolic-Elliptic Problem
6.4.5 Convection-Dominated Problem
7 Framework of the Goal-Oriented Error Estimates
7.1 Goal-Oriented Error Estimates for Linear PDEs
7.1.1 Primal Problem
7.1.2 Quantity of Interest and the Adjoint Problem
7.1.3 Abstract Goal-Oriented Error Estimates
7.1.4 Computable Goal-Oriented Error Estimates
7.2 Goal-Oriented Error Estimates for Nonlinear PDEs
7.2.1 Primal Problem
7.2.2 Quantity of Interest and Adjoint Problem Based on Differentiation
7.2.3 Quantity of Interest and Adjoint Problem Based on Linearization
7.2.4 Goal-Oriented Error Estimates
7.3 Error Estimates for Linear Convection–Diffusion Equation
7.3.1 Problem Formulation
7.3.2 Triangulation and Finite Element Spaces
7.3.3 Discretization of the Primal Problem
7.3.4 Consistency
7.3.5 Quantity of Interest and Adjoint Problem
7.3.6 Adjoint Consistency
7.3.7 Goal-Oriented Error Estimates
7.3.7.1 Error Estimate of Type I
7.3.7.2 Localization of the Error Estimate
7.3.7.3 Error Estimate of Type II
7.4 Error Estimates for Nonlinear Convection–DiffusionEquations
7.4.1 Problem Formulation
7.4.2 Target Functional and Adjoint Problem
7.4.3 Goal-Oriented Error Estimates
7.5 Compressible Euler Equations
8 Goal-Oriented Anisotropic Mesh Adaptation
8.1 Goal-Oriented Estimates Including the Geometry of Elements
8.1.1 Estimates Including the Geometry of Elements for 2D
8.1.2 Goal-Oriented Optimization of the Anisotropy of Triangles
8.1.3 Estimates Including the Geometry of Elements for 3D
8.1.4 Goal-Oriented Optimization of the Anisotropy of Tetrahedra
8.2 Goal-Oriented Anisotropic hp-Mesh Adaptive Algorithm
8.2.1 Setting of Element Size
8.2.2 Setting of Polynomial Approximation Degree and Element Shape
8.3 Numerical Experiments
8.3.1 Elliptic Problem on a ``Cross\'\' Domain
8.3.2 Mixed Hyperbolic-Elliptic Problem
8.3.3 Convection-Dominated Problem
9 Implementation Aspects
9.1 Higher-Order Reconstruction Techniques
9.1.1 Weighted Least-Square Reconstruction
9.1.2 Reconstruction Based on the Solutionof Local Problems
9.2 Anisotropic Mesh Adaptation for Time-Dependent Problems
9.2.1 Space-Time Discontinuous Galerkin Method
9.2.2 Interpolation Error Estimates for Time-Dependent Problems
9.2.3 Setting of the Time Step
9.2.4 Adaptive Solution of Time-Dependent Problem
10 Applications
10.1 Steady-State Inviscid Compressible Flow
10.1.1 Subsonic Flow
10.1.2 Transonic Flow
10.2 Viscous Shock-Vortex Interaction
10.3 Transient Flow Through a Nonhomogeneous Landfill Dam
Conclusion
References
Index