该书共5章，分别介绍有限元和混合有限元理论基础及其应用。最精彩的是第4和第5章，详细介绍非定常偏微分方程有限元法中的有限元空间和有限元未知解系数向量的降维方法，可将含数十万乃至上千万未知量的有限元迭代方程降阶成为只有很少几个未知量的降阶方程，理论和数值例子都证明了两种降维方法的正确性和有效性。这些降维方法都是作者原创性的工作，这些方法都已经在国际重要刊物发表。该书很详细做了介绍。这些方法的推广应用，将会带动计算数学向更高度发展。

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第十届中国数学学会理事，第六届至第八届北京市计算数学学会理事

Contents
Basic Theory of Standard Finite Element Method 1
1.1 The Basic Principles of Functional Analysis 1
1.1.1 Linear Operator and Linear Functional 1
1.1.2 Orthogonal Proj ection and Riesz Representation Theorem 4
1.1.3 Smooth Approximation and Fundamental Lemma of Calculus of Variation 5
1.1.4 Generalized Derivatives and Sobolev Spaces 6
1.1.5 Imbedding and Trace Theorems of Sobolev Spaces 9
1.1.6 Equivalent Module (Norm) Theorem 11
1.1.7 Green’s Formulas, Riesz-Thorin，s Theorem,Interpolation Inequality, and Closed Range Theorem 18
1.1.8 Fixed Point Theorems 20
1.2 Well-Posedness of Partial Differential Equations 21
1.2.1 The Classification for the Partial Differential Equations 21
1.2.1.1 Physical Classification for Partial Differential Equations 21
1.2.1.2 Mathematical Classification for Partial Differential Equations 21
1.2.1.3 The Second-Order Eq (1.2.2) Does Not Change Its Fomi under the Invertible Transformation 22
1.2.1.4 The Classification According to Characteristic Line 23
1.2.1.5 The Classification for the System of Partial Differential Equations 25
1.2.2 Lax-Milgram Theorem 26
1.2.3 Examples of Application for the Lax-Milgram Theorem 29
1.2.4 Differentiability (Regularity) of Generalized Solutions 41
1.3 Basic Theories of Function Interpolations 44
1.3.1 Finite Element and Related Properties 44
1.3.2 Properties of Finite Element Space and Inverse Estimation Theorem 46
1.3.3 Function Interpolation and Properties 55
1.3.4 The Interpolation Estimates in the Sobolev Spaces 58
1.4 Function Interpolations on Triangle Elements 63
1.4.1 Lagrange Linear Interpolation on the Triangle Elements 63
1.4.2 Lagrange’s Quadratic Interpolation on the Triangle Elements 66
1.4.3 Lagrange’s Cubic Interpolation on the Triangle Elements 68
1.4.4 Restricted Lagrange Cubic Interpolation 71
1.4.5 Cubic Hermite Interpolation on the Triangle Elements 74
1.4.5.1 Complete Cubic Hermite Interpolation on the Triangle Elements 74
1.4.5.2 Restricted Hermite Cubic Interpolation on the Triangle Elements 78
1.4.6 Quintic Hermite Interpolation on the Triangle Elements 80
1.4.6.1 Quintic Hermite Interpolation with 21 Degrees of Freedom 80
1.4.6.2 Quintic Hermite Interpolation with 18 Degrees of Freedom 81
1.4.7 Clough Interpolation on the Triangular Elements 89
1.4.8 Modified Clough Interpolation on the Triangular Elements 90
1.4.9 Morley，s Interpolation on the Triangle Elements 91
1.5 Function Interpolation on the Tetrahedral Element 92
1.5.1 Lagrange Linear Interpolation on the Tetrahedral Elements 92
1.5.2 Lagrange Quadratic Interpolation on the Tetrahedrons 96
1.5.3 Lagrange Cubic Interpolation on the Tetrahedral Elements 98
1.5.4 Hermite Cubic Interpolation with 20 Degrees of Freedom on the Tetrahedral Elements 100
1.5.5 Restricted Hermite Cubic Interpolation on the Tetrahedral Elements with 16 Degrees of Freedom 105
1.6 Interpolation of Functions on Rectangular Elements 110
1.6.1 Bilinear Lagrange Interpolation on the Rectangular Element 111
1.6.2 Biquadratic Lagrange Interpolation on the Rectangles 114
1.6.3 Incomplete Biquadratic Lagrange Interpolation on the Rectangular Elements 116
1.6.4 Complete Bicubic Hermite Interpolation on Rectangles 119
1.6.5 Incomplete Bicubic Hermite Interpolation on Rectangular 122
1.7 Function Interpolation on Arbitrary Quadrilaterals 126
1.7.1 Bilinear Interpolation on the Arbitrary Quadrilateral 132
1.7.2 Complete Biquadratic Interpolation on the Quadrilateral 133
1.7.3 Incomplete Biquadratic Interpolation on the Quadrilateral 135
1 _ 8 Function Interpolation on Hexahedron Elements 136
1.8.1 Interpolation Basis Functions on the Standard Cube 137
1.8.1.1 Trilinear Interpolation Basis Functions on the Standard Cube 137
1.8.1.2 The Basis Functions of Incomplete Triquadratic Interpolation on the Standard Cube 138
1.8.2 Function Interpolation on the Arbitrary Hexahedron 139
1.8.2.1 Trilinear Interpolation on the Arbitrary Hexahedral Elements 146
1.8.2.2 Incomplete Triquadratic Interpolation on the Arbitrary Hexahedral Elements 146
1.9 Convergence and Error Estimates of Finite Element Solutions 148
1.9.1 Projection Theorem and Galerkin Approximation 148
1.9.2 Finite Element Approximation for the First Homogeneous Boundary Value Problem of Poisson Equation 150
1.9.2.1 Error Estimates for the Finite Element Solution of First Homogeneous Boundary Value Problem of Poisson Equation 150
1.9.2.2 L2 Projection and Its Properties 153
1.9.2.3 L°° Estimate for the FE Solution of First Homogeneous Boundary Value Problem of Poisson Equation 157
1.9.3 Error Analysis of Finite Element Solution for the Biharmonic Equation 158
1.9.3.1 The Conforming FE Analysis of Biharmonic Equation 158
1.9.3.2 The Non-Conforming FE Analysis of Biharmonic Equation 162
2 Basic Theory of Mixed Finite Element Method 171
2.1 Mixed Generalized Solutions of Mixed Variational Problems 172
2.1.1 Mixed Generalized Solutions of Biharmonic Equation 172
2.1.2 Mixed Generalized Solutions of Poisson Equation 173
2.1.3 Mixed Generalized Solutions for Elastic Mechanic Problem 175
2.1.4 Mixed Generalized Solutions of Steady Stokes Problem 177
2.1.5 Abstract Mixed Generalized Problem 178
2.2 Existence and Uniqueness of Generalized Solutions for Mixed Variational Problems 180
2.2.1 The Stranger Sufficient Conditions for Existing Unique Mixed Generalized Solutions 180
2.2.2 The Weaker Sufficient Conditions for Existing Unique Mixed Generalized Solutions 185
2.3 Examples of Existence and Uniqueness of Generalized Solutions for Mixed Variational Problems 191
2.3.1 Existence and Uniqueness of Mixed Generalized Solutions for the Biharmonic Equation 191
2.3.2 Existence and Uniqueness of Mixed Generalized Solutions for the First Homogeneous Boundary Value Problem of the Poisson Equation 193
2.3.3 Existence and Uniqueness of Mixed Generalized Solutions for the Elasticity Mechanics Problem 195
2.3.4 Existence and Uniqueness of Mixed Generalized Solutions for the Steady Stokes Problem 197
2.4 Existence and Error Estimations for Mixed Finite Element Solutions 199
2.4.1 The Existence and Uniqueness for the Mixed Finite Element Solutions Under the Stranger Conditions 199
2.4.2 The Error Estimate of Mixed Finite Element Solutions Under the Stranger Conditions 202
2.4.3 The Existence, Uniqueness, and Error Estimates of Mixed Finite Element Solutions Under the Weaker Conditions 206
2.5 Existence and Uniqueness of Mixed Finite Element Solutions for the Biharmonic Equation 211
2.5.1 Ciarlet-Raviart Mixed Finite Element Format for the Fourth Order Biharmonic Equation 212
2.5.2 The Hermann-Miyoshi Mixed Finite Element Format for the Biharmonic Equation 218
2.5.3 The Hermann-Johnson Mixed Finite Element Formulation for the Biharmonic Equation 221
2.6 Mixed Finite Element Formats of Poisson Equation 223
2.6.1 Raviart-Thomas5 Mixed Finite Element Formulation for the Poisson Equation 224
2.6.1.1 Construct Function Space Q on the Standard Element K 226
2.6.1.2 Discuss (H5) in General Triangle Element K eh 230
2.6.2 Linear Mixed Finite Element Format of the Poisson Equation 239
2.6.3 Convergence and a Posterior Error Estimate for Simplified Stabilization Linear Mixed Finite Element Format 245
2.6.3.1 Simplified Stabilization Linear MFE Format 245
2.6.3.2 Convergence for Simplified Stabilization Linear MFE Solutions 248
2.6.3.3 A Posterior Error Estimate for Simplified Stabilization Linear MFE Solutions 252
2.6.4 Quadratic Mixed Finite Element for the Poisson Equation 255
2.7 Mixed Finite Element Formats for Elastic Mechanics 260
2.7.1 Johnson-Mercier5s Mixed Finite Element Format for Elastic Mechanic Problem 261
2.7.2 Linear Mixed Finite Element Format for Elasticity Problem 275
2.7.3 Convergence and a Posterior Error Estimate for Simplified Stabilization Linear Mixed Finite Element Format 281
2.7.3.1 Simplified Stabilization Linear MFE Format 281
2.7.3.2 Convergence and a Posterior Error Estimate for the Simplified Stabilization Linear Element Solutions 287
2.7.4 Quadratic Mixed Element Format for Elasticity Problem 289
2.8 Mixed Finite Element Formats for Steady Stokes Equation 295
2.8.1 Basic Theory of the Mixed Finite Element Method for the Steady Stokes Equation 295
2.8.2 First Order Mixed Finite Element Format of Triangulation for the Steady Stokes Equation 298
2.8.3 The First and Second Order Mixed Finite Element Formats Based on Bubble Function for the Steady Stokes Equation 301
2.8.4 The Improved First and Second Order Mixed Finite Element Formats for the Steady Stokes Equation 305
2.8.5 Simplified Stabilization First and Second Order Mixed Finite Element Formats for the Steady Stokes Problem 307
2.8.6 Convergence of Simplified Stabilization Mixed Finite Element Solutions for the Steady Stokes Equation 310
2.8.7 A posteriori Error Estimation for the Mixed Finite Element Solutions of Steady Stokes Equation 312
2.9 Mixed Finite Element Method for Steady Boussinesq Equation 316
2.9.1 Existence and Uniqueness of Generalized Solutions for the Steady Boussinesq Equation 316
2.9.2 Existence of Mixed Finite Element Solutions for the Steady Boussinesq Equation 326
2.9.3 Error Estimation of Mixed Finite Element Solutions for the Steady Boussinesq Equation 333
3 Mixed Finite Element Methods for the Unsteady Partial Differential Equations 337
3.1 Mixed Finite Element Method and Numerical Simulation of the Burgers Equation 337
3.1.1 Existence and Uniqueness of Mixed Generalized Solutions of the Burgers Equation 338
3.1.2 Existence and Error Estimation of Semi-discretized Mixed Finite Element Solutions for the Burgers Equation 341
3.1.3 Existence and Error Estimation of Fully Discretized Mixed Finite Element Solutions 347
3.1.4 Numerical Simulations for the Fully Discretized Mixed Finite Element Solutions of the Burgers Equation 351
3.2 Mixed Finite Element Method for RLW Equation and Numerical Simulations 355
3.2.1 Existence of Mixed Generalized Solutions for RLW Equation 355
3.2.2 Existence and Error Estimation of Semi-discretized Mixed Element Solutions for the RLW Equation 357
3.2.3 Existence and Error Estimation of Fully Discretized Mixed Finite Element Solutions for RLW Equation 362
3.2.4 A Lowest Order Difference Scheme Based on Mixed Finite Element Method and Numerical Simulations 366
3.3 Mixed Finite Element Method for Unsaturated Flow Problem and Its Numerical Simulations 371
3.3.1 Existence of Generalized Solutions to Unsaturated Soil Flow Problem 371
3.3.2 Existence and Error Estimation of Semi-discretized Mixed Finite Element Solutions 377
3.3.3 Existence and Error Estimations of Fully Discretized Mixed Finite Element Solutions 381
3.3.4 Numerical Simulation of the Unsaturated Flow Problem 384
3.4 The Mixed Finite Element Method for the Unsteady Boussinesq Equation 388
3.4.1 Existence and Uniqueness of Generalized Solutions for the Unsteady Boussinesq Equation 389
3.4.2 Existence and Error Estimation of Semi-discretized Mixed Finite Element Solutions of Boussinesq Equation 398
3.4.2.1 Existence and Uniqueness of the SDMFE Solutions 398
3A.2.2 Error Analysis of the SDMFE Solutions 403
3.4.3 Existence and Error Estimations of Fully Discretized Mixed Finite Element Solutions for the Boussinesq Equation 409
3.4.4 A Difference Scheme Based on Mixed Finite Element Method and Numerical Test of Boussinesq Equation 420
3.5 The Mixed Finite Element Method of the Improved System of Time-Domain Maxwell’s Equations 432
3.5.1 Establish the Improved System of Maxwell’s Equations 434
3.5.2 Existence and Uniqueness of Generalized Solutions for the Improved System of Maxwell’s Equations 436
3.5.3 The Time Semi-discretized Method for the Improved System of Time-Domain Maxwell’s Equations 438
3.5.4 The Fully Discretized Crank-Nicolson Mixed Finite Element Method for the System of Maxwell’s Equations 449
3.5.5 Two Set of Numerical Simulations of Maxwell’s Equations 457
3.5.6 The Crank-Nicolson Mixed Finite Element Method for the 3D Improved System of Maxwell’s Equations 459
4 The Reduced Dimension Methods of Finite Element Subspaces for Unsteady Partial Differential Equations 465
4.1 The Basis Theory of Reduced-Dimension for Finite Element Subspaces 466
4.1.1 The Brief of Finite Element or Mixed Finite Methods for the Unsteady Partial Differential Equations 466
4.1.2 Continuous Proper Orthogonal Decomposition Method 468
4.2 The Reduced-Dimension Method of Finite Element Subspace for "Viscoelastic Wave Equation 477
4.2.1 Generalized Solution for the Viscoelastic Wave Equation 477
4.2.2 Semi-Discretized Crank-Nicolson Formulation About Time for the "Viscoelastic Wave Equation 479
4.2.3 Classical Fully Discretized Crank-Nicolson Finite Element Method for the Viscoelastic Wave Equation 482
4.2.4 The Reduced-Dimension Method of Finite Element Subspace for the Viscoelastic Wave Equation 485
4.2.5 Error Estimates of the Reduced-Dimension Solutions for the Viscoelastic Wave Equation 488
4.2.6 The Flowchart for Finding the Reduced-Dimension Solutions to the Viscoelastic Wave Equation 492
4.2.7 A Numerical Example for the Viscoelastic Wave Equation 493
4.3 The Reduced-Dimension Method of Finite Element Subspace for the Unsteady Burgers Equation 495
4.3.1 Generalized Solution for the Unsteady Burgers Equation 496
4.3.2 Semi-Discretized Crank-Nicolson Formulation with Respect to Time for the Unsteady Burgers Equation 500
4.3.3 Fully Discretized Crank-Nicolson Finite Element Method for the Unsteady Burgers Equation 502
4.3.4 The Reduced-Dimension Method of Finite Element Subspace for the Unsteady Burgers Equation 507
4.3.5 The Existence, Stability, and Error Estimates of Reduced-Dimension Solutions to the Unsteady Burgers Equation 510
4.3.6 The Flowchart for Finding the Reduced-Dimension Solutions for the Unsteady Burgers Equation 513
4.3.7 Numerical Simulations for the Unsteady Burgers Equation 514
4.4 Reduced-Dimension Method of Finite Element Subspace for the Unsteady Navier-Stokes Equations 516
4.4.1 The Mixed Finite Element Method for the Unsteady Navier-Stokes Equations 516
4.4.2 The Reduced-Dimension Mixed Finite Method for the Unsteady Navier-Stokes Equations 520
4.4.3 Existence, Stability, and Error Estimates for Dimensional Reduction Mixed Finite Element Solutions to the Unsteady Navier-Stokes Equations 523
4.4.4 The Flowchart for Finding Reduced-Dimension Solutions of the Unsteady Navier-Stokes Equations 529
4.4.5 Some Numerical Simulations of the Unsteady Navier-Stokes Equations 530
4.5 Summary of Reduced-Dimension Methods for the Finite Element Subspaces 535
5 The Reduced Dimension of Finite Element Solution Coefficient Vectors for Unsteady Partial Differential Equations 539
5.1 The Reduced-Dimension Basic Theory About Finite Element Solution Coefficient Vectors 540
5.1.1 The Review for the Finite Element or Mixed Finite Element Methods of the Unsteady Partial Differential Equations 540
5.1.2 Discrete Proper Orthogonal Decomposition Method 542
5.1.3 The Reduced-Dimension Method for Unknown Finite Element Solution Coefficient Vectors 545
5.1.4 Some Useful Matrix Properties 546
5.2 The Reduced-Dimension Method of Finite Element Solution Coefficient Vectors for Parabolic Equation 547
5.2.1 The Crank-Nicolson Finite Element Method of the Parabolic Equation 548
5.2.2 The Reduced-Dimension Method of Crank-Nicolson Finite Element Solution Coefficient Vectors of Parabolic Equation 551
5.2.2.1 Construction of POD Basis Vectors 551
5.2.2.2 Formulation of Matrix-Form RDRCNFE Model 551
5.2.2.3 The Stability and Error Estimates of the RDRCNFE Solutions 552
5.2.3 Some Numerical Simulations for Parabolic Equation 555
5.3 The Reduced-Dimension Method of Finite Element Solution Coefficient Vectors for Sobolev Equation 557
5.3.1 The Crank-Nicolson Finite Element Method for the Sobolev Equation 558
5.3.2 The Reduced-Dimension of Crank-Nicolson Finite Element Solution Coefficient Vectors for the Sobolev Equation 561
5.3.2.1 Generation of POD Bases 561
5.3.2.2 Establishment of the RDRCNFE Method in the Matrix-Form 561
5.3.2.3 The Stability and Error Estimates to the RDRCNFE Solutions 562
5.3.3 Two Numerical Examples for the Sobolev Equation 562
5.3.3.1 The First Numerical Example 565
5.3.3.2 The Second Numerical Example 567
5.4 The Reduced-Dimension Method for Mixed Finite Element Solution Coefficient Vectors of Unsteady Stokes Equation 570
5.4.1 The Crank-Nicolson Mixed Finite Element Method of the 2D Unsteady Stokes Equation 571
5.4.2 The Reduced-Dimension Recursive Crank-Nicolson Mixed Finite Element Method for the Unsteady Stokes Equation 574
5.4.2.1 Construction of POD Basis 574
5.4.2.2 Creation of the RDRCNMFE Format 575
5.4.2.3 The Stability and Convergence for the RDRCNMFE Solutions 576
5.4.3 Some Numerical Simulations 579
5.5 The Dimensional Reduction Method of Mixed Finite Element Solution Coefficient Vectors for Unsteady Boussinesq Equation 582
5.5.1 The Crank-Nicolson Mixed Finite Element Method for the Unsteady Boussinesq Equation 583
5.5.1.1 The Time Semi-Discretized CN Scheme 583
5.5.1.2 The Fully Discretized CNMFE Format 586
5.5.2 The Reduced-Dimension Recursive Crank-Nicolson Mixed Finite Element Method for the Boussinesq Equation 588
5.5.2.1 Production of POD Bases 588
5.5.2.2 Establishment of the RDRCNMFE Format 589
5.5.2.3 The Stability and Convergence of the RDRCNMFE Solutions 590
5.5.2.4 The Flowchart for Finding the RDRCNMFE Solutions 595
5.5.3 Some Numerical Simulations for the Boussinesq Equation 596
5.5.3.1 The Back-Step Flow 596
5.5.3.2 Flow Around Airfoil Problem 613
5.6 The Summary of Reduced-Dimension Methods for Finite Element Solution Coefficient Vectors Postscript and Author’s Own Statement 631
Bibliography 637
Index 647