Part I Survey of Variational Principles and Associated
Finite Element Methods
1 Classical Variational Methods ................................ 3
1.1 Variational Methods for Operator Equations .............. 4
1.2 A Taxonomy of Classical Variational Formulations ........ 8
1.2.1 Weakly Coercive Problems ......................... 8
1.2.2 Strongly Coercive Problems ....................... 9
1.2.3 Mixed Variational Problems ...................... 10
1.2.4 Relations Between Variational Problems and
Optimization Problems ........................... 12
1.3 Approximation of Solutions of Variational Problems ..... 15
1.3.1 Weakly and Strongly Coercive Variational
Problems ........................................ 15
1.3.2 Mixed Variational Problems ...................... 18
1.4 Examples ............................................... 22
1.4.1 The Poisson Equation ............................ 22
1.4.2 The Equations of Linear Elasticity .............. 25
1.4.3 The Stokes Equations ............................ 26
1.4.4 The Helmholtz Equation .......................... 28
1.4.5 A Scalar Linear Advection-Diffusion-Reaction
Equation ........................................ 30
1.4.6 The Navier-Stokes Equations ..................... 30
1.5 A Comparative Summary of Classical Finite Element
Methods ................................................ 31
2 Alternative Variational Formulations ........................ 35
2.1 Modified Variational Principles ........................ 36
2.1.1 Enhanced and Stabilized Methods for Weakly
Coercive Problems ............................... 36
2.1.2 Stabilized Methods for Strongly Coercive
Problems ........................................ 46
2.2 Least-Squares Principles ............................... 49
2.2.1 A Straightforward Least-Squares Finite Element
Method .......................................... 51
2.2.2 Practical Least-Squares Finite Element
Methods ......................................... 53
2.2.3 Norm-Equivalence Versus Practicality ............ 58
2.2.4 Some Questions and Answers ...................... 60
2.3 Putting Things in Perspective and What to Expect from
the Book ............................................... 62
Part II Abstract Theory of Least-Squares Finite Element Methods
3 Mathematical Foundations of Least-Squares Finite Element
Methods ..................................................... 69
3.1 Least-Squares Principles for Linear Operator Equations
in Hilbert Spaces ...................................... 70
3.1.1 Problems with Zero Nullity ...................... 71
3.1.2 Problems with Positive Nullity .................. 73
3.2 Application to Partial Differential Equations .......... 75
3.2.1 Energy Balances ................................. 76
3.2.2 Continuous Least-Squares Principles ............. 77
3.3 General Discrete Least-Squares Principles .............. 80
3.3.1 Error Analysis .................................. 82
3.3.2 The Need for Continuous Least-Squares
Principles ...................................... 84
3.4 Binding Discrete Least-Squares Principles to Partial
Differential Equations ................................. 85
3.4.1 Transformations from Continuous to Discrete
Least-Squares Principles ........................ 86
3.5 Taxonomy of Conforming Discrete Least-Squares
Principles and their Analysis .......................... 90
3.5.1 Compliant Discrete Least-Squares Principles ..... 92
3.5.2 Norm-Equivalent Discrete Least-Squares
Principles ...................................... 94
3.5.3 Quasi-Norm-Equivalent Discrete Least-Squares
Principles ...................................... 96
3.5.4 Summary Review of Discrete Least-Squares
Principles ..................................... 100
4 The Agmon-Douglis-Nirenberg Setting for Least-Squares
Finite Element Methods ..................................... 103
4.1 Transformations to First-Order Systems ................ 105
4.2 Energy Balances ....................................... 106
4.2.1 Homogeneous Elliptic Systems ................... 107
4.2.2 Non-Homogeneous Elliptic Systems ............... 107
4.3 Continuous Least-Squares Principles ................... 108
4.3.1 Homogeneous Elliptic Systems ................... 108
4.3.2 Non-Homogeneous Elliptic Systems ............... 110
4.4 Least-Squares Finite Element Methods for Homogeneous
Elliptic Systems ...................................... 112
4.5 Least-Squares Finite Element Methods for Non-
Homogeneous Elliptic Systems .......................... 114
4.5.1 Quasi-Norm-Equivalent Discrete Least-Squares
Principles ..................................... 114
4.5.2 Norm-Equivalent Discrete Least-Squares
Principles ..................................... 124
4.6 Concluding Remarks .................................... 129
Part III Least-Squares Finite Element Methods for
Elliptic Problems
5 Scalar Elliptic Equations .................................. 133
5.1 Applications of Scalar Poisson Equations .............. 135
5.2 Least-Squares Finite Element Methods for the Second-
Order Poisson Equation ................................ 137
5.2.1 Continuous Least-Squares Principles ............ 138
5.2.2 Discrete Least-Squares Principles .............. 139
5.3 First-Order System Reformulations ..................... 140
5.3.1 The Div-Grad System ............................ 141
5.3.2 The Extended Div-Grad System ................... 145
5.3.3 Application Examples ........................... 146
5.4 Energy Balances ....................................... 147
5.4.1 Energy Balances in the Agmon-Douglis-
Nirenberg Setting .............................. 148
5.4.2 Energy Balances in the Vector-Operator
Setting ........................................ 152
5.5 Continuous Least-Squares Principles ................... 159
5.6 Discrete Least-Squares Principles ..................... 163
5.6.1 The Div-Grad System ............................ 163
5.6.2 The Extended Div-Grad System ................... 169
5.7 Error Analyses ........................................ 171
5.7.1 Error Estimates in Solution Space Norms ........ 171
5.7.2 L2(Ω) Error Estimates .......................... 175
5.8 Connections Between Compatible Least-Squares and
Standard Finite Element Methods ....................... 176
5.8.1 The Compatible Least-Squares Finite Element
Method with a Reaction Term .................... 177
5.8.2 The Compatible Least-Squares Finite Element
Method Without a Reaction Term ................. 181
5.9 Practicality Issues ................................... 182
5.9.1 Practical Rewards of Compatibility ............. 184
5.9.2 Compatible Least-Squares Finite Element
Methods on Non-Affine Grids .................... 190
5.9.3 Advantages and Disadvantages of Extended
Systems ........................................ 192
5.10 A Summary of Conclusions and Recommendations .......... 194
6 Vector Elliptic Equations .................................. 197
6.1 Applications of Vector Elliptic Equations ............. 200
6.2 Reformulation of Vector Elliptic Problems ............. 201
6.2.1 Div-Curl Systems ............................... 202
6.2.2 Curl-Curl Systems .............................. 203
6.3 Least-Squares Finite Element Methods for Div-Curl
Systems ............................................... 206
6.3.1 Energy Balances ................................ 206
6.3.2 Continuous Least-Squares Principles ............ 209
6.3.3 Discrete Least-Squares Principles .............. 211
6.3.4 Analysis of Conforming Least-Squares Finite
Element Methods ................................ 214
6.3.5 Analysis of Non-Conforming Least-Squares
Finite Element Methods ......................... 216
6.4 Least-Squares Finite Element Methods for Curl-Curl
Systems ............................................... 221
6.4.1 Energy Balances ................................ 221
6.4.2 Continuous Least-Squares Principles ............ 224
6.4.3 Discrete Least-Squares Principles .............. 225
6.4.4 Error Analysis ................................. 230
6.5 Practicality Issues ................................... 231
6.5.1 Solution of Algebraic Equations ................ 232
6.5.2 Implementation of Non-Conforming Methods ....... 234
6.6 A Summary of Conclusions .............................. 236
7 The Stokes Equations ....................................... 237
7.1 First-Order System Formulations of the Stokes
Equations ............................................. 238
7.1.1 The Velocity-Vorticity-Pressure System ......... 239
7.1.2 The Velocity-Stress-Pressure System ............ 242
7.1.3 The Velocity Gradient-Velocity-Pressure
System ......................................... 243
7.2 Energy Balances in the Agmon-Douglis-Nirenberg
Setting ............................................... 246
7.2.1 The Velocity-Vorticity-Pressure System ......... 247
7.2.2 The Velocity-Stress-Pressure System ............ 250
7.2.3 The Velocity Gradient-Velocity-Pressure
System ......................................... 251
7.3 Continuous Least-Squares Principles in the Agmon-
Douglis-Nirenberg Setting ............................. 253
7.3.1 The Velocity-Vorticity-Pressure System ......... 253
7.3.2 The Velocity-Stress-Pressure System ............ 256
7.3.3 The Velocity Gradient-Velocity-Pressure
System ......................................... 256
7.4 Discrete Least-Squares Principles in the Agmon-
Douglis-Nirenberg Setting ............................. 257
7.4.1 The Velocity-Vorticity-Pressure System ......... 258
7.4.2 The Velocity-Stress-Pressure System ............ 260
7.4.3 The Velocity Gradient-Velocity-Pressure
System ......................................... 260
7.5 Error Estimates in the Agmon-Douglis-Nirenberg
Setting ............................................... 261
7.5.1 The Velocity-Vorticity-Pressure System ......... 261
7.5.2 The Velocity-Stress-Pressure System ............ 263
7.5.3 The Velocity Gradient-Velocity-Pressure
System ......................................... 264
7.6 Practicality Issues in the Agmon-Douglis-Nirenberg
Setting ............................................... 264
7.6.1 Solution of the Discrete Equations ............. 265
7.6.2 Issues Related to Non-Homogeneous Elliptic
Systems ........................................ 266
7.6.3 Mass Conservation .............................. 271
7.6.4 The Zero Mean Pressure Constraint .............. 274
7.7 Least-Squares Finite Element Methods in the Vector-
Operator Setting ...................................... 277
7.7.1 Energy Balances ................................ 277
7.7.2 Continuous Least-Squares Principles ............ 281
7.7.3 Discrete Least-Squares Principles .............. 281
7.7.4 Stability of Discrete Least-Squares
Principles ..................................... 284
7.7.5 Conservation of Mass and Strong
Compatibility .................................. 287
7.7.6 Error Estimates ................................ 293
7.7.7 Connection Between Discrete Least-Squares
Principles and Mixed-Galerkin Methods .......... 302
7.7.8 Practicality Issues in the Vector Operator
Setting ........................................ 304
7.8 A Summary of Conclusions and Recommendations .......... 306
Part IV Least-Squares Finite Element Methods for Other Settings
8 The Navier-Stokes Equations ................................ 311
8.1 First-Order System Formulations of the Navier-Stokes
Equations ............................................. 313
8.2 Least-Squares Principles for the Navier-Stokes
Equations ............................................. 314
8.2.1 Continuous Least-Squares Principles ............ 315
8.2.2 Discrete Least-Squares Principles .............. 316
8.3 Analysis of Least-Squares Finite Element Methods ...... 317
8.3.1 Quotation of Background Results ................ 318
8.3.2 Compliant Discrete Least-Squares Principles
for the Velocity-Vorticity-Pressure System ..... 321
8.3.3 Norm-Equivalent Discrete Least-Squares
Principles for the Velocity-Vorticity-
Pressure System ................................ 329
8.3.4 Compliant Discrete Least-Squares Principles
for the Velocity Gradient-Velocity-Pressure
System ......................................... 340
8.3.5 A Norm-Equivalent Discrete Least-Squares
Principle for the Velocity Gradient-Velocity-
Pressure System ................................ 344
8.4 Practicality Issues ................................... 346
8.4.1 Solution of the Nonlinear Equations ............ 348
8.4.2 Implementation of Norm-Equivalent Methods ...... 351
8.4.3 The Utility of Discrete Negative Norm Least-
Squares Finite Element Methods ................. 354
8.4.4 Advantages and Disadvantages of Extended
Systems ........................................ 359
8.5 A Summary of Conclusions and Recommendations .......... 364
9 Parabolic Partial Differential Equations ................... 367
9.1 The Generalized Heat Equation ......................... 368
9.1.1 Backward-Euler Least-Squares Finite Element
Methods ........................................ 369
9.1.2 Second-Order Time Accurate Least-Squares
Finite Element Methods ......................... 382
9.1.3 Comparison of Finite-Difference Least-Squares
Finite Element Methods ......................... 389
9.1.4 Space-Time Least-Squares Principles ............ 391
9.1.5 Practical Issues ............................... 395
9.2 The Time-Dependent Stokes Equations ................... 396
10 Hyperbolic Partial Differential Equations .................. 403
10.1 Model Conservation Law Problems ....................... 404
10.2 Energy Balances ....................................... 406
10.2.1 Energy Balances in Hilbert Spaces .............. 407
10.2.2 Energy Balances in Banach Spaces ............... 409
10.3 Continuous Least-Squares Principles ................... 410
10.3.1 Extension to Time-Dependent Conservation
Laws ........................................... 412
10.4 Least-Squares Finite Element Methods in a Hilbert
Space Setting ......................................... 413
10.4.1 Conforming Methods ............................. 413
10.4.2 Non-Conforming Methods ......................... 414
10.5 Residual Minimization Methods in a Banach Space
Setting ............................................... 416
10.5.1 An L1(Ω) Minimization Method ................... 416
10.5.2 Regularized L1(Ω) Minimization Method .......... 418
10.6 Least-Squares Finite Element Methods Based on
Adaptively Weighted L2(Ω) Norms ....................... 419
10.6.1 An Iteratively Re-Weighted Least-Squares
Finite Element Method .......................... 419
10.6.2 A Feedback Least-Squares Finite Element
Method ......................................... 420
10.7 Practicality Issues .............................. 422
10.7.1 Approximation of Smooth Solutions .............. 422
10.7.2 Approximation of Discontinuous Solutions ....... 423
10.8 A Summary of Conclusions and Recommendations ..... 427
11 Control and Optimization Problems .......................... 429
11.1 Quadratic Optimization and Control Problems in
Hilbert Spaces with Linear Constraints ................ 431
11.1.1 Existence of Optimal States and Controls ....... 432
11.1.2 Least-Squares Formulation of the Constraint
Equation ....................................... 435
11.2 Solution via Lagrange Multipliers of the Optimal
Control Problem ....................................... 438
11.2.1 Galerkin Finite Element Methods for the
Optimality System .............................. 439
11.2.2 Least-Squares Finite Element Methods for the
Optimality System .............................. 442
11.3 Methods Based on Direct Penalization by the Least-
Squares Functional .................................... 447
11.3.1 Discretization of the Perturbed Optimality
System ......................................... 450
11.3.2 Discretization of the Eliminated System ........ 453
11.4 Methods Based on Constraining by the Least-Squares
Functional ............................................ 455
11.4.1 Discretization of the Optimality System ........ 457
11.4.2 Discretize-Then-Eliminate Approach for the
Perturbed Optimality System .................... 457
11.4.3 Eliminate-Then-Discretize Approach for the
Perturbed Optimality System .................... 459
11.5 Relative Merits of the Different Approaches ........... 460
11.6 Example: Optimization Problems for the Stokes
Equations ............................................. 461
11.5 The Optimization Problems and Galerkin Finite
Element Methods ....................................... 463
11.6.2 Least-Squares Finite Element Methods for the
Constraint Equations ........................... 467
11.6.3 Least-Squares Finite Element Methods for
the Optimality Systems ......................... 468
11.6.4 Constraining by the Least-Squares Functional
for the Constraint Equations ................... 471
12 Variations on Least-Squares Finite Element Methods ......... 475
12.1 Weak Enforcement of Boundary Conditions ............... 475
12.2 LL* Finite Element Methods ............................ 480
12.3 Mimetic Reformulation of Least-Squares Finite
Element Methods ....................................... 483
12.4 Collocation Least-Squares Finite Element Methods ...... 488
12.5 Restricted Least-Squares Finite Element Methods ....... 490
12.6 Optimization-Based Least-Squares Finite Element
Methods ............................................... 492
12.7 Least-Squares Finite Element Methods for Advection-
Diffusion-Reaction Problems ........................... 494
12.8 Least-Squares Finite Element Methods for Higher-
Order Problems ........................................ 503
12.9 Least-Squares Finite Element Methods for Div-Grad-
Curl Systems .......................................... 505
12.10 Domain Decomposition Least-Squares Finite Element
Methods ............................................... 507
12.11 Least-Squares Finite Element Methods for Multi-
Physics Problems ...................................... 513
12.12 Least-Squares Finite Element Methods for Problems
with Singular Solutions ............................... 517
12.13 Treffetz Least-Squares Finite Element Methods ........ 521
12.14 A Posteriori Error Estimation and Adaptive Mesh
Refinement ............................................ 523
12.15 Least-Squares Wavelet Methods ........................ 526
12.16 Meshless Least-Squares Methods ....................... 528
Part V Supplementary Material
A Analysis Tools ............................................. 533
A.l General Notations and Symbols ......................... 533
A.2 Function Spaces ....................................... 535
A.2.1 The Sobolev Spaces HS(Ω) ....................... 536
A.2.2 Spaces Related to the Gradient, Curl, and
Divergence Operators ........................... 540
A.3 Properties of Function Spaces ......................... 547
A.3.1 Embeddingsof C(Ω) ∩ D(Ω) ....................... 547
A.3.2 Poincare-Friedrichs Inequalities ............... 548
A.3.3 Hodge Decompositions ........................... 550
A.3.4 Trace Theorems ................................. 551
В Compatible Finite Element Spaces ........................... 553
B.l Formal Definition and Properties of Finite Element
Spaces ................................................ 554
B.2 Finite Element Approximation of the De Rham Complex ... 557
B.2.1 Examples of Compatible Finite Element Spaces ... 559
B.2.2 Approximation of C(Ω) ∩ D(Ω) ................... 567
B.2.3 Exact Sequences of Finite Element Spaces ....... 569
B.3 Properties of Compatible Finite Element Spaces ........ 571
B.3.1 Discrete Operators ............................. 571
B.3.2 Discrete Poincaré-Friedrichs Inequalities ...... 576
В.3.3 Discrete Hodge Decompositions .................. 577
B.3.4 Inverse Inequalities ........................... 580
B.4 Norm Approximations ................................... 581
B.4.1 Quasi-Norm-Equivalent Approximations ........... 581
B.4.2 Norm-Equivalent Approximations ................. 582
С Linear Operator Equations in Hiibert Spaces ................ 585
C.1 Auxiliary Operator Equations .......................... 586
C.2 Energy Balances ....................................... 589
D The Agmon-Douglis-Nirenberg Theory and Verifying
its Assumptions ............................................ 593
D.1 The Agmon-Douglis-Nirenberg Theory .................... 593
D.2 Verifying the Assumptions of the Agmon-Douglis-
Nirenberg Theory ...................................... 597
D.2.1 Div-Grad Systems ............................... 598
D.2.2 Div-Grad-Curl Systems .......................... 602
D.2.3 Div-Curl Systems ............................... 606
D.2.4 The Velocity-Vorticity-Pressure Formulation
of the Stokes System ........................... 608
D.2.5 The Velocity-Stress-Pressure Formulation of
the Stokes System .............................. 622
References .................................................... 625
Acronyms ...................................................... 641
Glossary ...................................................... 643
Index ......................................................... 647
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