Least-Squares Finite Element Methods - GBV

Pavel В. Bochev Max D. GunzburgerLeast-Squares FiniteElement MethodsSpriringer

ContentsPart I Survey of Variational Principles and Associated Finite ElementMethods12Classical Variational Methods1.1Variational Methods for Operator Equations1.2A Taxonomy of Classical Variational Formulations1.2.1 Weakly Coercive Problems1.2.2 Strongly Coercive Problems1.2.3 Mixed Variational Problems1.2.4 Relations Between Variational Problemsand Optimization Problems1.3Approximation of Solutions of Variational Problems1.3.1 Weakly and Strongly Coercive Variational Problems1.3.2 Mixed Variational Problems1.4Examples1.4.1 The Poisson Equation1.4.2 The Equations of Linear Elasticity1.4.3 The Stokes Equations1.4.4 The Heimholte Equation1.4.5 A Scalar Linear Advection-Diffusion-Reaction Equation .1.4.6 The Navier-Stokes Equations1.5A Comparative Summary of Classical Finite Element Methods . . .3488910121515182222252628303031Alternative Variational Formulations352.1Modified Variational Principles362.1.1 Enhanced and Stabilized Methods for Weakly CoerciveProblems362.1.2 Stabilized Methods for Strongly Coercive Problems462.2Least-Squares Principles492.2.1 A Straightforward Least-Squares Finite Element Method . 512.2.2 Practical Least-Squares Finite Element Methods53XV

ContentsXVI2.32.2.3 Norm-Equivalence Versus Practicality582.2.4 Some Questions and Answers60Putting Things in Perspective and What to Expectfromthe Book . 62Part П Abstract Theory of Least-Squares Finite Element Methods3Mathematical Foundations of Least-Squares Finite Element Methods 693.1 Least-Squares Principles for Linear Operator Equationsin Hubert Spaces703.1.1 Problems with Zero Nullity713.1.2 Problems with Positive Nullity733.2 Application to Partial Differential Equations753.2.1 Energy Balances763.2.2 Continuous Least-Squares Principles773.3 General Discrete Least-Squares Principles803.3.1 Error Analysis823.3.2 The Need for Continuous Least-Squares Principles843.4 Binding Discrete Least-Squares Principles to Partial DifferentialEquations853.4.1 Transformations from Continuous to DiscreteLeast-Squares Principles863.5 Taxonomy of Conforming Discrete Least-Squares Principlesand their Analysis903.5.1 Compliant Discrete Least-Squares Principles923.5.2 Norm-Equivalent Discrete Least-Squares Principles943.5.3 Quasi-Norm-Equivalent Discrete Least-Squares Principles 963.5.4 Summary Review of Discrete Least-Squares Principles .1004The Agmon-Douglis-Nirenberg Setting for Least-Squares FiniteElement Methods4.1 Transformations to First-Order Systems4.2 Energy Balances4.2.1 Homogeneous Elliptic Systems4.2.2 Non-Homogeneous Elliptic Systems4.3 Continuous Least-Squares Principles4.3.1 Homogeneous Elliptic Systems4.3.2 Non-Homogeneous Elliptic Systems4.4 Least-Squares Finite Element Methods for HomogeneousElliptic Systems4.5 Least-Squares Finite Element Methods for Non-HomogeneousElliptic Systems4.5.1 Quasi-Norm-Equivalent Discrete Least-Squares Principles4.5.2 Norm-Equivalent Discrete Least-Squares Principles4.6 Concluding Remarks103105106107107108108110112114114124129

ContentsxviiPart III Least-Squares Finite Element Methods for Elliptic Problems56Scalar Elliptic Equations5.1Applications of Scalar Poisson Equations5.2Least-Squares Finite Element Methods for the Second-OrderPoisson Equation5.2.1 Continuous Least-Squares Principles5.2.2 Discrete Least-Squares Principles5.3First-Order System Reformulations5.3.1 The Div-Grad System5.3.2 The Extended Div-Grad System5.3.3 Application Examples5.4Energy Balances5.4.1 Energy Balances in the Agmon-Douglis-NirenbergSetting5.4.2 Energy Balances in the Vector-Operator Setting5.5Continuous Least-Squares Principles5.6Discrete Least-Squares Principles5.6.1 The Div-Grad System5.6.2 The Extended Div-Grad System5.7Error Analyses5.7.1 Error Estimates in Solution Space Norms5.7.2 L2( 2) Error Estimates5.8Connections Between Compatible Least-Squaresand Standard Finite Element Methods5.8.1 The Compatible Least-Squares Finite Element Methodwith a Reaction Term5.8.2 The Compatible Least-Squares Finite Element MethodWithout a Reaction Term5.9Practicality Issues5.9.1 Practical Rewards of Compatibility5.9.2 Compatible Least-Squares Finite Element Methodson Non-Affine Grids5.9.3 Advantages and Disadvantages of Extended Systems5.10 A Summary of Conclusions and Recommendations133135Vector Elliptic Equations6.1Applications of Vector Elliptic Equations6.2Reformulation of Vector Elliptic Problems6.2.1 Div-Curl Systems6.2.2 Curl-Curl Systems6.3Least-Squares Finite Element Methods for Div-Curl Systems6.3.1 Energy Balances6.3.2 Continuous Least-Squares Principles6.3.3 Discrete Least-Squares 184190192194

xviiiContents6.3.46.46.56.67Analysis of Conforming Least-Squares Finite ElementMethods6.3.5 Analysis of Non-Conforming Least-Squares FiniteElement MethodsLeast-Squares Finite Element Methods for Curl-Curl Systems6.4.1 Energy Balances6.4.2 Continuous Least-Squares Principles6.4.3 Discrete Least-Squares Principles6.4.4 Error AnalysisPracticality Issues6.5.1 Solution of Algebraic Equations6.5.2 Implementation of Non-Conforming MethodsA Summary of ConclusionsThe Stokes Equations7.1First-Order System Formulations of the Stokes Equations7.1.1 The Velocity-Vorticity-Pressure System7.1.2 The Velocity-Stress-Pressure System7.1.3 The Velocity Gradient-Velocity-Pressure System7.2Energy Balances in the Agmon-Douglis-Nirenberg Setting7.2.1 The Velocity-Vorticity-Pressure System7.2.2 The Velocity-Stress-Pressure System7.2.3 The Velocity Gradient-Velocity-Pressure System7.3Continuous Least-Squares Principlesin the Agmon-Douglis-Nirenberg Setting7.3.1 The Velocity-Vorticity-Pressure System7.3.2 The Velocity-Stress-Pressure System7.3.3 The Velocity Gradient-Velocity-Pressure System7.4Discrete Least-Squares Principlesin the Agmon-Douglis-Nirenberg Setting7.4.1 The Velocity-Vorticity-Pressure System7.4.2 The Velocity-Stress-Pressure System7.4.3 The Velocity Gradient-Velocity-Pressure System7.5Error Estimates in the Agmon-Douglis-Nirenberg Setting7.5.1 The Velocity-Vorticity-Pressure System7.5.2 The Velocity-Stress-Pressure System7.5.3 The Velocity Gradient-Velocity-Pressure System7.6Practicality Issues in the Agmon-Douglis-Nirenberg Setting7.6.1 Solution of the Discrete Equations7.6.2 Issues Related to Non-Homogeneous Elliptic Systems . . . .7.6.3 Mass Conservation7.6.4 The Zero Mean Pressure Constraint7.7Least-Squares Finite Element Methodsin the Vector-Operator Setting7.7.1 Energy 3264264265266271274277277

uous Least-Squares PrinciplesDiscrete Least-Squares PrinciplesStability of Discrete Least-Squares PrinciplesConservation of Mass and Strong CompatibilityError EstimatesConnection Between Discrete Least-Squares Principlesand Mixed-Galerkin Methods7.7.8 Practicality Issues in the Vector Operator SettingA Summary of Conclusions and Recommendations281281284287293302304306Part IV Least-Squares Finite Element Methods for Other Settings89The Navier-Stokes Equations8.1 First-Order System Formulations of the Navier-Stokes Equations8.2 Least-Squares Principles for the Navier-Stokes Equations8:2.1 Continuous Least-Squares Principles8.2.2 Discrete Least-Squares Principles8.3 Analysis of Least-Squares Finite Element Methods8.3.1 Quotation of Background Results8.3.2 Compliant Discrete Least-Squares Principlesfor the Velocity-Vorticity-Pressure System8.3.3 Norm-Equivalent Discrete Least-Squares Principlesfor the Velocity-Vorticity-Pressure System 8.3.4 Compliant Discrete Least-Squares Principlesfor the Velocity Gradient-Velocity-Pressure System8.3.5 A Norm-Equivalent Discrete Least-Squares Principlefor the Velocity Gradient-Velocity-Pressure System8.4 Practicality Issues8.4.1 Solution of the Nonlinear Equations8.4.2 Implementation of Norm-Equivalent Methods8.4.3 The Utility of Discrete Negative Norm Least-SquaresFinite Element Methods8.4.4 Advantages and Disadvantages of Extended Systems8.5 A Summary of Conclusions and 46348351354359364Parabolic Partial Differential Equations3679.1 The Generalized Heat Equation3689.1.1 Backward-Euler Least-Squares Finite Element Methods . 3699.1.2 Second-Order Time Accurate Least-Squares FiniteElement Methods3829.1.3 Comparison of Finite-Difference Least-Squares FiniteElement Methods3899.1.4 Space-Time Least-Squares Principles3919.1.5 Practical Issues3959.2 The Time-Dependent Stokes Equations396

xxContents10 Hyperbolic Partial Differential Equations10.1 Model Conservation Law Problems,10.2 Energy Balances10.2.1 Energy Balances in Hilbert Spaces10.2.2 Energy Balances in Banach Spaces10.3 Continuous Least-Squares Principles10.3.1 Extension to Time-Dependent Conservation Laws10.4 Least-Squares Finite Element Methods in a Hilbert Space Setting10.4.1 Conforming Methods10.4.2 Non-Conforming Methods10.5 Residual Minimization Methods in a Banach Space Setting10.5.1 An О (П) Minimization Method10.5.2 Regularized L1 (Й) Minimization Method10.6 Least-Squares Finite Element Methods Based on AdaptivelyWeighted L2(Q) Norms10.6.1 An Iteratively Re-Weighted Least-Squares FiniteElement Method10.6.2 A Feedback Least-Squares Finite Element Method10.7 Practicality Issues10.7.1 Approximation of Smooth Solutions . . . :10.7.2 Approximation of Discontinuous Solutions10.8 A Summary of Conclusions and 1641841941942042242242342711 Control and Optimization Problems42911.1 Quadratic Optimization and Control Problems in Hilbert Spaceswith Linear Constraints43111.1.1 Existence of Optimal States and Controls43211.1.2 Least-Squares Formulation of the Constraint Equation43511.2 Solution via Lagrange Multipliers of the Optimal ControlProblem43811.2.1 Galerkin Finite Element Methods for the OptimalitySystem43911.2.2 Least-Squares Finite Element Methods for the OptimalitySystem44211.3 Methods Based on Direct Penalization by the Least-SquaresFunctional44711.3.1 Discretization of the Perturbed Optimality System45011.3.2 Discretization of the Eliminated System45311.4 Methods Based on Constraining by the Least-Squares Functional . 45511.4.1 Discretization of the Optimality System45711.4.2 Discretize-Then-Eliminate Approach for the PerturbedOptimality System45711.4.3 Eliminate-Then-Discretize Approach for the PerturbedOptimality System45911.5 Relative Merits of the Different Approaches46011.6 Example: Optimization Problems for the Stokes Equations461

Contentsxxi11.6.1 The Optimization Problems and Galerkin Finite ElementMethods11.6.2 Least-Squares Finite Element Methods for the ConstraintEquations11.6.3 Least-Squares Finite Element Methods for the OptimalitySystems11.6.4 Constraining by the Least-Squares Functionalfor the Constraint Equations46346746847112 Variations on Least-Squares Finite Element Methods47512.1 Weak Enforcement of Boundary Conditions47512.2 LL* Finite Element Methods48012.3 Mimetic Reformulation of Least-Squares Finite Element Methods. 48312.4 Collocation Least-Squares Finite Element Methods48812.5 Restricted Least-Squares Finite Element Methods49012.6 Optimization-Based Least-Squares Finite Element Methods49212.7 Least-Squares Finite Element Methodsfor Advection-Diffusion-Reaction Problems49412.8 Least-Squares Finite Element Methods for Higher-OrderProblems50312.9 Least-Squares Finite Element Methods for Div-Grad-CurlSystems50512.10 Domain Decomposition Least-Squares Finite Element Methods. 50712.11 Least-Squares Finite Element Methods for Multi-PhysicsProblems51312.12 Least-Squares Finite Element Methods for Problemswith Singular Solutions51712.13 Treffetz Least-Squares Finite Element Methods52112.14 A Posteriori Error Estimation and Adaptive Mesh Refinement52312.15 Least-Squares Wavelet Methods52612.16 Meshless Least-Squares Methods528Part V Supplementary MaterialAAnalysis ToolsA. 1 General Notations and SymbolsA.2 Function SpacesA.2.1 The Sobolev Spaces HS(Q)A.2.2 Spaces Related to the Gradient, Curl, and DivergenceOperatorsA.3 Properties of Function SpacesA.3.1 Embeddings of С(й) nD(ß)A.3.2 Poincare-Friedrichs InequalitiesA.3.3 Hodge DecompositionsA.3.4 Trace Theorems533533535536540547547548550551