Pavel В. Bochev Max D. Gunzburger Least-Squares Finite Element Methods Spri ringer
Contents 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 Heimholte 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 XV
XVI Contents 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 Expectfromthe Book. 62 Part П 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 Hubert 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
Contents xvii 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 L 2 ( 2) 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
xviii Contents 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
Contents xix 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
xx Contents 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 О (П) Minimization Method 416 10.5.2 Regularized L 1 (Й) Minimization Method 418 10.6 Least-Squares Finite Element Methods Based on Adaptively Weighted L 2 (Q) 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
Contents xxi 11.6.1 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. 1 General Notations and Symbols 533 A.2 Function Spaces 535 A.2.1 The Sobolev Spaces H S (Q) 536 A.2.2 Spaces Related to the Gradient, Curl, and Divergence Operators 540 A.3 Properties of Function Spaces 547 A.3.1 Embeddings of С(й) nd(ß) 547 A.3.2 Poincare-Friedrichs Inequalities 548 A.3.3 Hodge Decompositions 550 A.3.4 Trace Theorems 551
XXII Contents 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(ß) fld(ß) 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 Poincare-Friedrichs Inequalities 576 B.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 Hubert Spaces 585 C.l Auxiliary Operator Equations 586 C.2 Energy Balances 589 D The Agmon-Douglis-Nirenberg Theory and Verifying its Assumptions 593 D.l 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