Least-Squares Finite Element Methods

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1 Pavel В. Bochev Max D. Gunzburger Least-Squares Finite Element Methods Spri ringer

2 Contents Part I Survey of Variational Principles and Associated Finite Element Methods 1 Classical Variational Methods Variational Methods for Operator Equations A Taxonomy of Classical Variational Formulations Weakly Coercive Problems Strongly Coercive Problems Mixed Variational Problems Relations Between Variational Problems and Optimization Problems Approximation of Solutions of Variational Problems Weakly and Strongly Coercive Variational Problems Mixed Variational Problems Examples The Poisson Equation The Equations of Linear Elasticity The Stokes Equations The Heimholte Equation A Scalar Linear Advection-Diffusion-Reaction Equation The Navier-Stokes Equations A Comparative Summary of Classical Finite Element Methods Alternative Variational Formulations Modified Variational Principles Enhanced and Stabilized Methods for Weakly Coercive Problems Stabilized Methods for Strongly Coercive Problems Least-Squares Principles A Straightforward Least-Squares Finite Element Method Practical Least-Squares Finite Element Methods 53 XV

3 XVI Contents Norm-Equivalence Versus Practicality Some Questions and Answers 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 Least-Squares Principles for Linear Operator Equations in Hubert Spaces Problems with Zero Nullity Problems with Positive Nullity Application to Partial Differential Equations Energy Balances Continuous Least-Squares Principles General Discrete Least-Squares Principles Error Analysis The Need for Continuous Least-Squares Principles Binding Discrete Least-Squares Principles to Partial Differential Equations Transformations from Continuous to Discrete Least-Squares Principles Taxonomy of Conforming Discrete Least-Squares Principles and their Analysis Compliant Discrete Least-Squares Principles Norm-Equivalent Discrete Least-Squares Principles Quasi-Norm-Equivalent Discrete Least-Squares Principles Summary Review of Discrete Least-Squares Principles The Agmon-Douglis-Nirenberg Setting for Least-Squares Finite Element Methods Transformations to First-Order Systems Energy Balances Homogeneous Elliptic Systems Non-Homogeneous Elliptic Systems Continuous Least-Squares Principles Homogeneous Elliptic Systems Non-Homogeneous Elliptic Systems Least-Squares Finite Element Methods for Homogeneous Elliptic Systems Least-Squares Finite Element Methods for Non-Homogeneous Elliptic Systems Quasi-Norm-Equivalent Discrete Least-Squares Principles Norm-Equivalent Discrete Least-Squares Principles Concluding Remarks 129

4 Contents xvii Part III Least-Squares Finite Element Methods for Elliptic Problems 5 Scalar Elliptic Equations Applications of Scalar Poisson Equations Least-Squares Finite Element Methods for the Second-Order Poisson Equation Continuous Least-Squares Principles Discrete Least-Squares Principles First-Order System Reformulations The Div-Grad System The Extended Div-Grad System Application Examples Energy Balances Energy Balances in the Agmon-Douglis-Nirenberg Setting Energy Balances in the Vector-Operator Setting Continuous Least-Squares Principles Discrete Least-Squares Principles The Div-Grad System The Extended Div-Grad System Error Analyses Error Estimates in Solution Space Norms L 2 ( 2) Error Estimates Connections Between Compatible Least-Squares and Standard Finite Element Methods The Compatible Least-Squares Finite Element Method with a Reaction Term The Compatible Least-Squares Finite Element Method Without a Reaction Term Practicality Issues Practical Rewards of Compatibility Compatible Least-Squares Finite Element Methods on Non-Affine Grids Advantages and Disadvantages of Extended Systems A Summary of Conclusions and Recommendations Vector Elliptic Equations Applications of Vector Elliptic Equations Reformulation of Vector Elliptic Problems Div-Curl Systems Curl-Curl Systems Least-Squares Finite Element Methods for Div-Curl Systems Energy Balances Continuous Least-Squares Principles Discrete Least-Squares Principles 211

5 xviii Contents Analysis of Conforming Least-Squares Finite Element Methods Analysis of Non-Conforming Least-Squares Finite Element Methods Least-Squares Finite Element Methods for Curl-Curl Systems Energy Balances Continuous Least-Squares Principles Discrete Least-Squares Principles Error Analysis Practicality Issues Solution of Algebraic Equations Implementation of Non-Conforming Methods A Summary of Conclusions The Stokes Equations First-Order System Formulations of the Stokes Equations The Velocity-Vorticity-Pressure System The Velocity-Stress-Pressure System The Velocity Gradient-Velocity-Pressure System Energy Balances in the Agmon-Douglis-Nirenberg Setting The Velocity-Vorticity-Pressure System The Velocity-Stress-Pressure System The Velocity Gradient-Velocity-Pressure System Continuous Least-Squares Principles in the Agmon-Douglis-Nirenberg Setting The Velocity-Vorticity-Pressure System The Velocity-Stress-Pressure System The Velocity Gradient-Velocity-Pressure System Discrete Least-Squares Principles in the Agmon-Douglis-Nirenberg Setting The Velocity-Vorticity-Pressure System The Velocity-Stress-Pressure System The Velocity Gradient-Velocity-Pressure System Error Estimates in the Agmon-Douglis-Nirenberg Setting The Velocity-Vorticity-Pressure System The Velocity-Stress-Pressure System The Velocity Gradient-Velocity-Pressure System Practicality Issues in the Agmon-Douglis-Nirenberg Setting Solution of the Discrete Equations Issues Related to Non-Homogeneous Elliptic Systems Mass Conservation The Zero Mean Pressure Constraint Least-Squares Finite Element Methods in the Vector-Operator Setting Energy Balances 277

6 Contents xix Continuous Least-Squares Principles Discrete Least-Squares Principles Stability of Discrete Least-Squares Principles Conservation of Mass and Strong Compatibility Error Estimates Connection Between Discrete Least-Squares Principles and Mixed-Galerkin Methods Practicality Issues in the Vector Operator Setting A Summary of Conclusions and Recommendations 306 Part IV Least-Squares Finite Element Methods for Other Settings 8 The Navier-Stokes Equations First-Order System Formulations of the Navier-Stokes Equations Least-Squares Principles for the Navier-Stokes Equations 314 8:2.1 Continuous Least-Squares Principles Discrete Least-Squares Principles Analysis of Least-Squares Finite Element Methods Quotation of Background Results Compliant Discrete Least-Squares Principles for the Velocity-Vorticity-Pressure System Norm-Equivalent Discrete Least-Squares Principles for the Velocity-Vorticity-Pressure System 329 ~~ Compliant Discrete Least-Squares Principles for the Velocity Gradient-Velocity-Pressure System A Norm-Equivalent Discrete Least-Squares Principle for the Velocity Gradient-Velocity-Pressure System Practicality Issues Solution of the Nonlinear Equations Implementation of Norm-Equivalent Methods The Utility of Discrete Negative Norm Least-Squares Finite Element Methods Advantages and Disadvantages of Extended Systems A Summary of Conclusions and Recommendations Parabolic Partial Differential Equations The Generalized Heat Equation Backward-Euler Least-Squares Finite Element Methods Second-Order Time Accurate Least-Squares Finite Element Methods Comparison of Finite-Difference Least-Squares Finite Element Methods Space-Time Least-Squares Principles Practical Issues The Time-Dependent Stokes Equations 396

7 xx Contents 10 Hyperbolic Partial Differential Equations Model Conservation Law Problems, Energy Balances Energy Balances in Hilbert Spaces Energy Balances in Banach Spaces Continuous Least-Squares Principles Extension to Time-Dependent Conservation Laws Least-Squares Finite Element Methods in a Hilbert Space Setting Conforming Methods Non-Conforming Methods Residual Minimization Methods in a Banach Space Setting An О (П) Minimization Method Regularized L 1 (Й) Minimization Method Least-Squares Finite Element Methods Based on Adaptively Weighted L 2 (Q) Norms An Iteratively Re-Weighted Least-Squares Finite Element Method A Feedback Least-Squares Finite Element Method Practicality Issues Approximation of Smooth Solutions...: Approximation of Discontinuous Solutions A Summary of Conclusions and Recommendations Control and Optimization Problems Quadratic Optimization and Control Problems in Hilbert Spaces with Linear Constraints Existence of Optimal States and Controls Least-Squares Formulation of the Constraint Equation Solution via Lagrange Multipliers of the Optimal Control Problem Galerkin Finite Element Methods for the Optimality System Least-Squares Finite Element Methods for the Optimality System Methods Based on Direct Penalization by the Least-Squares Functional Discretization of the Perturbed Optimality System Discretization of the Eliminated System Methods Based on Constraining by the Least-Squares Functional Discretization of the Optimality System Discretize-Then-Eliminate Approach for the Perturbed Optimality System Eliminate-Then-Discretize Approach for the Perturbed Optimality System Relative Merits of the Different Approaches Example: Optimization Problems for the Stokes Equations 461

8 Contents xxi The Optimization Problems and Galerkin Finite Element Methods Least-Squares Finite Element Methods for the Constraint Equations Least-Squares Finite Element Methods for the Optimality Systems Constraining by the Least-Squares Functional for the Constraint Equations Variations on Least-Squares Finite Element Methods Weak Enforcement of Boundary Conditions LL* Finite Element Methods Mimetic Reformulation of Least-Squares Finite Element Methods Collocation Least-Squares Finite Element Methods Restricted Least-Squares Finite Element Methods Optimization-Based Least-Squares Finite Element Methods Least-Squares Finite Element Methods for Advection-Diffusion-Reaction Problems Least-Squares Finite Element Methods for Higher-Order Problems Least-Squares Finite Element Methods for Div-Grad-Curl Systems Domain Decomposition Least-Squares Finite Element Methods Least-Squares Finite Element Methods for Multi-Physics Problems Least-Squares Finite Element Methods for Problems with Singular Solutions Treffetz Least-Squares Finite Element Methods A Posteriori Error Estimation and Adaptive Mesh Refinement Least-Squares Wavelet Methods 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

9 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

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