Finite Element and Reduced Dimension Methods for Partial Differential Equations

Preface
	About the Finite Element and Mixed Finite Element Methods
	About the Reduced-Dimension of Finite Element and Mixed Finite Methods
	About Proper Orthogonal Decomposition Method
	The Main Content and Arrangement of the Book
	Acknowledgments
Contents
List of Abbreviations
1 Basic Theory of Standard Finite Element Method
	1.1 The Basic Principles of Functional Analysis
		1.1.1 Linear Operator and Linear Functional
		1.1.2 Orthogonal Projection and Riesz Representation Theorem
		1.1.3 Smooth Approximation and Fundamental Lemma of Calculus of Variation
		1.1.4 Generalized Derivatives and Sobolev Spaces
		1.1.5 Imbedding and Trace Theorems of Sobolev Spaces
		1.1.6 Equivalent Module (Norm) Theorem
		1.1.7 Green\'s Formulas, Riesz-Thorin\'s Theorem, Interpolation Inequality, and Closed Range Theorem
		1.1.8 Fixed Point Theorems
	1.2 Well-Posedness of Partial Differential Equations
		1.2.1 The Classification for the Partial Differential Equations
			1.2.1.1 Physical Classification for Partial Differential Equations
			1.2.1.2 Mathematical Classification for Partial Differential Equations
			1.2.1.3 The Second-Order Eq.(1.2.2) Does Not Change Its form Under the Invertible Transformation
			1.2.1.4 The Classification According to Characteristic Line
			1.2.1.5 The Classification for the System of Partial Differential Equations
		1.2.2 Lax-Milgram Theorem
		1.2.3 Examples of Application for the Lax-Milgram Theorem
		1.2.4 Differentiability (Regularity) of Generalized Solutions
	1.3 Basic Theories of Function Interpolations
		1.3.1 Finite Element and Related Properties
		1.3.2 Properties of Finite Element Space and Inverse Estimation Theorem
		1.3.3 Function Interpolation and Properties
		1.3.4 The Interpolation Estimates in the Sobolev Spaces
	1.4 Function Interpolations on Triangle Elements
		1.4.1 Lagrange Linear Interpolation on the Triangle Elements
		1.4.2 Lagrange\'s Quadratic Interpolation on the TriangleElements
		1.4.3 Lagrange\'s Cubic Interpolation on the TriangleElements
		1.4.4 Restricted Lagrange Cubic Interpolation
		1.4.5 Cubic Hermite Interpolation on the Triangle Elements
			1.4.5.1 Complete Cubic Hermite Interpolation on the Triangle Elements
			1.4.5.2 Restricted Hermite Cubic Interpolation on the Triangle Elements
		1.4.6 Quintic Hermite Interpolation on the Triangle Elements
			1.4.6.1 Quintic Hermite Interpolation with 21 Degrees of Freedom
			1.4.6.2 Quintic Hermite Interpolation with 18 Degrees of Freedom
		1.4.7 Clough Interpolation on the Triangular Elements
		1.4.8 Modified Clough Interpolation on the TriangularElements
		1.4.9 Morley\'s Interpolation on the Triangle Elements
	1.5 Function Interpolation on the Tetrahedral Element
		1.5.1 Lagrange Linear Interpolation on the TetrahedralElements
		1.5.2 Lagrange Quadratic Interpolation on the Tetrahedrons
		1.5.3 Lagrange Cubic Interpolation on the TetrahedralElements
		1.5.4 Hermite Cubic Interpolation with 20 Degrees of Freedom on the Tetrahedral Elements
		1.5.5 Restricted Hermite Cubic Interpolation on the Tetrahedral Elements with 16 Degrees of Freedom
	1.6 Interpolation of Functions on Rectangular Elements
		1.6.1 Bilinear Lagrange Interpolation on the RectangularElement
		1.6.2 Biquadratic Lagrange Interpolation on the Rectangles
		1.6.3 Incomplete Biquadratic Lagrange Interpolation on the Rectangular Elements
		1.6.4 Complete Bicubic Hermite Interpolation on Rectangles
		1.6.5 Incomplete Bicubic Hermite Interpolation on Rectangular
	1.7 Function Interpolation on Arbitrary Quadrilaterals
		1.7.1 Bilinear Interpolation on the Arbitrary Quadrilateral
		1.7.2 Complete Biquadratic Interpolation on the Quadrilateral
		1.7.3 Incomplete Biquadratic Interpolation on the Quadrilateral
	1.8 Function Interpolation on Hexahedron Elements
		1.8.1 Interpolation Basis Functions on the Standard Cube
			1.8.1.1 Trilinear Interpolation Basis Functions on the Standard Cube
			1.8.1.2 The Basis Functions of Incomplete Triquadratic Interpolation on the Standard Cube
		1.8.2 Function Interpolation on the Arbitrary Hexahedron
			1.8.2.1 Trilinear Interpolation on the Arbitrary Hexahedral Elements
			1.8.2.2 Incomplete Triquadratic Interpolation on the Arbitrary Hexahedral Elements
	1.9 Convergence and Error Estimates of Finite Element Solutions
		1.9.1 Projection Theorem and Galerkin Approximation
		1.9.2 Finite Element Approximation for the First Homogeneous Boundary Value Problemof Poisson Equation
			1.9.2.1 Error Estimates for the Finite Element Solution of First Homogeneous Boundary Value Problem of Poisson Equation
			1.9.2.2 L2 Projection and Its Properties
			1.9.2.3 L∞ Estimate for the FE Solution of First Homogeneous Boundary Value Problem of Poisson Equation
		1.9.3 Error Analysis of Finite Element Solution for the Biharmonic Equation
			1.9.3.1 The Conforming FE Analysis of Biharmonic Equation
			1.9.3.2 The Non-Conforming FE Analysis of Biharmonic Equation
2 Basic Theory of Mixed Finite Element Method
	2.1 Mixed Generalized Solutions of Mixed Variational Problems
		2.1.1 Mixed Generalized Solutions of Biharmonic Equation
		2.1.2 Mixed Generalized Solutions of Poisson Equation
		2.1.3 Mixed Generalized Solutions for Elastic MechanicProblem
		2.1.4 Mixed Generalized Solutions of Steady Stokes Problem
		2.1.5 Abstract Mixed Generalized Problem
	2.2 Existence and Uniqueness of Generalized Solutions for Mixed Variational Problems
		2.2.1 The Stranger Sufficient Conditions for Existing Unique Mixed Generalized Solutions
		2.2.2 The Weaker Sufficient Conditions for Existing Unique Mixed Generalized Solutions
	2.3 Examples of Existence and Uniqueness of Generalized Solutions for Mixed Variational Problems
		2.3.1 Existence and Uniqueness of Mixed Generalized Solutions for the Biharmonic Equation
		2.3.2 Existence and Uniqueness of Mixed Generalized Solutions for the First Homogeneous Boundary Value Problem of the Poisson Equation
		2.3.3 Existence and Uniqueness of Mixed Generalized Solutions for the Elasticity Mechanics Problem
		2.3.4 Existence and Uniqueness of Mixed Generalized Solutions for the Steady Stokes Problem
	2.4 Existence and Error Estimations for Mixed FiniteElement Solutions
		2.4.1 The Existence and Uniqueness for the Mixed Finite Element Solutions Under the Stranger Conditions
		2.4.2 The Error Estimate of Mixed Finite Element Solutions Under the Stranger Conditions
		2.4.3 The Existence, Uniqueness, and Error Estimates of Mixed Finite Element Solutions Under the Weaker Conditions
	2.5 Existence and Uniqueness of Mixed Finite Element Solutions for the Biharmonic Equation
		2.5.1 Ciarlet-Raviart Mixed Finite Element Format for the Fourth Order Biharmonic Equation
		2.5.2 The Hermann-Miyoshi Mixed Finite Element Format for the Biharmonic Equation
		2.5.3 The Hermann-Johnson Mixed Finite Element Formulation for the Biharmonic Equation
	2.6 Mixed Finite Element Formats of Poisson Equation
		2.6.1 Raviart-Thomas\' Mixed Finite Element Formulation for the Poisson Equation
			2.6.1.1 Construct Function Space  on the Standard Element
			2.6.1.2 Discuss (H5) in General Triangle ElementKh
		2.6.2 Linear Mixed Finite Element Format of the PoissonEquation
		2.6.3 Convergence and a Posterior Error Estimate for Simplified Stabilization Linear Mixed Finite Element Format
			2.6.3.1 Simplified Stabilization Linear MFE Format
			2.6.3.2 Convergence for Simplified Stabilization Linear MFE Solutions
			2.6.3.3 A Posterior Error Estimate for Simplified Stabilization Linear MFE Solutions
		2.6.4 Quadratic Mixed Finite Element for the Poisson Equation
	2.7 Mixed Finite Element Formats for Elastic Mechanics
		2.7.1 Johnson-Mercier\'s Mixed Finite Element Format for Elastic Mechanic Problem
		2.7.2 Linear Mixed Finite Element Format for ElasticityProblem
		2.7.3 Convergence and a Posterior Error Estimate for Simplified Stabilization Linear Mixed Finite Element Format
			2.7.3.1 Simplified Stabilization Linear MFE Format
			2.7.3.2 Convergence and a posterior Error Estimate for the Simplified Stabilization Linear Element Solutions
		2.7.4 Quadratic Mixed Element Format for Elasticity Problem
	2.8 Mixed Finite Element Formats for Steady Stokes Equation
		2.8.1 Basic Theory of the Mixed Finite Element Method for the Steady Stokes Equation
		2.8.2 First Order Mixed Finite Element Format of Triangulation for the Steady Stokes Equation
		2.8.3 The First and Second Order Mixed Finite Element Formats Based on Bubble Function for the Steady Stokes Equation
		2.8.4 The Improved First and Second Order Mixed Finite Element Formats for the Steady Stokes Equation
		2.8.5 Simplified Stabilization First and Second Order Mixed Finite Element Formats for the Steady Stokes Problem
		2.8.6 Convergence of Simplified Stabilization Mixed Finite Element Solutions for the Steady Stokes Equation
		2.8.7 A posteriori Error Estimation for the Mixed Finite Element Solutions of Steady Stokes Equation
	2.9 Mixed Finite Element Method for Steady Boussinesq Equation
		2.9.1 Existence and Uniqueness of Generalized Solutions for the Steady Boussinesq Equation
		2.9.2 Existence of Mixed Finite Element Solutions for the Steady Boussinesq Equation
		2.9.3 Error Estimation of Mixed Finite Element Solutions for the Steady Boussinesq Equation
3 Mixed Finite Element Methods for the Unsteady Partial Differential Equations
	3.1 Mixed Finite Element Method and Numerical Simulation of the Burgers Equation
		3.1.1 Existence and Uniqueness of Mixed Generalized Solutions of the Burgers Equation
		3.1.2 Existence and Error Estimation of Semi-discretized Mixed Finite Element Solutions for the Burgers Equation
		3.1.3 Existence and Error Estimation of Fully Discretized Mixed Finite Element Solutions
		3.1.4 Numerical Simulations for the Fully Discretized Mixed Finite Element Solutions of the Burgers Equation
	3.2 Mixed Finite Element Method for RLW Equation and Numerical Simulations
		3.2.1 Existence of Mixed Generalized Solutions for RLW Equation
		3.2.2 Existence and Error Estimation of Semi-discretized Mixed Element Solutions for the RLW Equation
		3.2.3 Existence and Error Estimation of Fully Discretized Mixed Finite Element Solutions for RLW Equation
		3.2.4 A Lowest Order Difference Scheme Based on Mixed Finite Element Method and NumericalSimulations
	3.3 Mixed Finite Element Method for Unsaturated Flow Problem and Its Numerical Simulations
		3.3.1 Existence of Generalized Solutions to Unsaturated Soil Flow Problem
		3.3.2 Existence and Error Estimation of Semi-discretized Mixed Finite Element Solutions
		3.3.3 Existence and Error Estimations of Fully Discretized Mixed Finite Element Solutions
		3.3.4 Numerical Simulation of the Unsaturated Flow Problem
	3.4 The Mixed Finite Element Method for the Unsteady Boussinesq Equation
		3.4.1 Existence and Uniqueness of Generalized Solutions for the Unsteady Boussinesq Equation
		3.4.2 Existence and Error Estimation of Semi-discretized Mixed Finite Element Solutions of Boussinesq Equation
			3.4.2.1 Existence and Uniqueness of the SDMFESolutions
			3.4.2.2 Error Analysis of the SDMFE Solutions
		3.4.3 Existence and Error Estimations of Fully Discretized Mixed Finite Element Solutions for theBoussinesq Equation
		3.4.4 A Difference Scheme Based on Mixed Finite Element Method and Numerical Testof Boussinesq Equation
	3.5 The Mixed Finite Element Method of the Improved System of Time-Domain Maxwell\'s Equations
		3.5.1 Establish the Improved System of Maxwell\'s Equations
		3.5.2 Existence and Uniqueness of Generalized Solutions for the Improved System of Maxwell\'s Equations
		3.5.3 The Time Semi-discretized Method for the Improved System of Time-Domain Maxwell\'s Equations
		3.5.4 The Fully Discretized Crank–Nicolson Mixed Finite Element Method for the System of Maxwell\'s Equations
		3.5.5 Two Set of Numerical Simulations of Maxwell\'sEquations
		3.5.6 The Crank–Nicolson Mixed Finite Element Method for the 3D Improved System of Maxwell\'s Equations
4 The Reduced Dimension Methods of Finite Element Subspaces for Unsteady Partial Differential Equations
	4.1 The Basis Theory of Reduced-Dimension for Finite Element Subspaces
		4.1.1 The Brief of Finite Element or Mixed Finite Methods for the Unsteady Partial Differential Equations
		4.1.2 Continuous Proper Orthogonal Decomposition Method
	4.2 The Reduced-Dimension Method of Finite Element Subspace for Viscoelastic Wave Equation
		4.2.1 Generalized Solution for the Viscoelastic Wave Equation
		4.2.2 Semi-Discretized Crank–Nicolson Formulation About Time for the Viscoelastic Wave Equation
		4.2.3 Classical Fully Discretized Crank–Nicolson Finite Element Method for the Viscoelastic Wave Equation
		4.2.4 The Reduced-Dimension Method of Finite Element Subspace for the Viscoelastic Wave Equation
		4.2.5 Error Estimates of the Reduced-Dimension Solutions for the Viscoelastic Wave Equation
		4.2.6 The Flowchart for Finding the Reduced-Dimension Solutions to the Viscoelastic Wave Equation
		4.2.7 A Numerical Example for the ViscoelasticWave Equation
	4.3 The Reduced-Dimension Method of Finite Element Subspace for the Unsteady Burgers Equation
		4.3.1 Generalized Solution for the Unsteady Burgers Equation
		4.3.2 Semi-Discretized Crank–Nicolson Formulation with Respect to Time for the Unsteady Burgers Equation
		4.3.3 Fully Discretized Crank–Nicolson Finite Element Method for the Unsteady Burgers Equation
		4.3.4 The Reduced-Dimension Method of Finite Element Subspace for the Unsteady Burgers Equation
		4.3.5 The Existence, Stability, and Error Estimates of Reduced-Dimension Solutions to the Unsteady Burgers Equation
		4.3.6 The Flowchart for Finding the Reduced-Dimension Solutions for the Unsteady Burgers Equation
		4.3.7 Numerical Simulations for the UnsteadyBurgers Equation
	4.4 Reduced-Dimension Method of Finite Element Subspace for the Unsteady Navier-Stokes Equations
		4.4.1 The Mixed Finite Element Method for the Unsteady Navier-Stokes Equations
		4.4.2 The Reduced-Dimension Mixed Finite Method for the Unsteady Navier–Stokes Equations
		4.4.3 Existence, Stability, and Error Estimates for Dimensional Reduction Mixed Finite Element Solutions to the Unsteady Navier-Stokes Equations
		4.4.4 The Flowchart for Finding Reduced-Dimension Solutions of the Unsteady Navier-Stokes Equations
		4.4.5 Some Numerical Simulations of the Unsteady Navier-Stokes Equations
	4.5 Summary of Reduced-Dimension Methods for the Finite Element Subspaces
5 The Reduced Dimension of Finite Element Solution Coefficient Vectors for Unsteady Partial Differential Equations
	5.1 The Reduced-Dimension Basic Theory About Finite Element Solution Coefficient Vectors
		5.1.1 The Review for the Finite Element or Mixed Finite Element Methods of the Unsteady Partial Differential Equations
		5.1.2 Discrete Proper Orthogonal Decomposition Method
		5.1.3 The Reduced-Dimension Method for Unknown Finite Element Solution Coefficient Vectors
		5.1.4 Some Useful Matrix Properties
	5.2 The Reduced-Dimension Method of Finite Element Solution Coefficient Vectors for Parabolic Equation
		5.2.1 The Crank–Nicolson Finite Element Method of the Parabolic Equation
		5.2.2 The Reduced-Dimension Method of Crank–Nicolson Finite Element Solution Coefficient Vectors of Parabolic Equation
			5.2.2.1 Construction of POD Basis Vectors
			5.2.2.2 Formulation of Matrix-Form RDRCNFE Model
			5.2.2.3 The Stability and Error Estimates of the RDRCNFE Solutions
		5.2.3 Some Numerical Simulations for Parabolic Equation
	5.3 The Reduced-Dimension Method of Finite Element Solution Coefficient Vectors for Sobolev Equation
		5.3.1 The Crank–Nicolson Finite Element Method for the Sobolev Equation
		5.3.2 The Reduced-Dimension of Crank–Nicolson Finite Element Solution Coefficient Vectors for the Sobolev Equation
			5.3.2.1 Generation of POD Bases
			5.3.2.2 Establishment of the RDRCNFE Method in the Matrix-Form
			5.3.2.3 The Stability and Error Estimates to the RDRCNFE Solutions
		5.3.3 Two Numerical Examples for the Sobolev Equation
			5.3.3.1 The First Numerical Example
			5.3.3.2 The Second Numerical Example
	5.4 The Reduced-Dimension Method for Mixed Finite Element Solution Coefficient Vectors of Unsteady Stokes Equation
		5.4.1 The Crank–Nicolson Mixed Finite Element Method of the 2D Unsteady Stokes Equation
		5.4.2 The Reduced-Dimension Recursive Crank–Nicolson Mixed Finite Element Method for the Unsteady Stokes Equation
			5.4.2.1 Construction of POD Basis
			5.4.2.2 Creation of the RDRCNMFE Format
			5.4.2.3 The Stability and Convergence for the RDRCNMFE Solutions
		5.4.3 Some Numerical Simulations
	5.5 The Dimensional Reduction Method of Mixed Finite Element Solution Coefficient Vectors for Unsteady Boussinesq Equation
		5.5.1 The Crank–Nicolson Mixed Finite Element Method for the Unsteady Boussinesq Equation
			5.5.1.1 The Time Semi-Discretized CN Scheme
			5.5.1.2 The Fully Discretized CNMFE Format
		5.5.2 The Reduced-Dimension Recursive Crank–Nicolson Mixed Finite Element Method for the Boussinesq Equation
			5.5.2.1 Production of POD Bases
			5.5.2.2 Establishment of the RDRCNMFE Format
			5.5.2.3 The Stability and Convergence of the RDRCNMFE Solutions
			5.5.2.4 The Flowchart for Finding the RDRCNMFE Solutions
		5.5.3 Some Numerical Simulations for the BoussinesqEquation
			5.5.3.1 The Back-Step Flow
			5.5.3.2 Flow Around Airfoil Problem
	5.6 The Summary of Reduced-Dimension Methods for Finite Element Solution Coefficient Vectors
Postscript and Author\'s Own Statement
	Summary of This Book
	The Author\'s Own Statement
	The Author\'s True Wishes
Bibliography
Index