Regularization and Simulation of Constrained Partial Differential Equations vorgelegt von Diplom-Mathematiker Robert Altmann aus Berlin Von der Fakult¨atII - Mathematik und Naturwissenschaften der Technischen Universit¨atBerlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften Dr. rer. nat genehmigte Dissertation Promotionsausschuss: Vorsitzende: Prof. Dr. Noemi Kurt Berichter: Prof. Dr. Volker Mehrmann Berichterin: Prof. Dr. Caren Tischendorf Berichter: Prof. Dr. Alexander Ostermann Tag der wissenschaftlichen Aussprache: 29. 05. 2015 Berlin 2015 Contents Zusammenfassung . v Abstract . vii Published Papers. ix 1. Introduction . 1 Part A Preliminaries 5 2. Differential-algebraic Equations (DAEs) . 6 2.1. Index Concepts . 6 2.1.1. Differentiation Index. 7 2.1.2. Further Index Concepts. 8 2.2. High-index DAEs . 8 2.3. Index Reduction Techniques. 9 2.3.1. Index Reduction by Differentiation . 9 2.3.2. Minimal Extension . 9 3. Functional Analytic Tools . 11 3.1. Fundamentals . 11 3.1.1. Dual Operators and Riesz Representation Theorem . 11 3.1.2. Test Functions and Distributions . 12 3.1.3. Sobolev Spaces . 13 3.1.4. Traces . 14 3.1.5. Poincar´eInequality and Negative Norms . 15 3.1.6. Weak Convergence and Compactness . 17 3.2. Bochner Spaces . 17 3.3. Sobolev-Bochner Spaces . 20 3.3.1. Gelfand Triples . 20 3.3.2. Definition and Embeddings . 21 4. Abstract Differential Equations . 23 4.1. Nemytskii Mapping . 23 4.2. Operator ODEs . 24 4.2.1. First-order Equations . 25 4.2.2. Second-order Equations. 26 4.3. Operator DAEs . 27 i 5. Discretization Schemes . 29 5.1. Spatial Discretization . 29 5.1.1. Finite Element Spaces. 29 5.1.2. Finite Element Discretization . 31 5.1.3. Stability for Saddle Point Problems . 33 5.2. Time Integration . 34 5.2.1. Implicit Euler Scheme. 35 5.2.2. Schemes for Second-order Systems. 35 5.3. Discretization of Time-dependent PDEs . 36 5.3.1. Method of Lines. 36 5.3.2. Rothe Method . 37 Part B Regularization of Operator DAEs 41 6. Regularization of First-order Operator DAEs . 42 6.1. Linear Constraints . 43 6.1.1. Assumptions on B ................................ 43 6.1.2. Regularization . 45 6.1.3. Influence of Perturbations . 48 6.2. Nonlinear Constraints . 50 6.2.1. Assumptions on B ................................ 50 6.2.2. Regularization . 52 6.2.3. Influence of Perturbations . 53 6.3. Applications . 54 6.3.1. Navier-Stokes Equations . 55 6.3.2. Optimal Control of Fluid Flows . 56 6.3.3. Regularized Stefan Problem . 56 7. Regularization of Second-order Operator DAEs . 59 7.1. Equations of Motion in Elastodynamics . 59 7.1.1. Principle of Virtual Work . 59 7.1.2. Dirichlet Boundary Conditions . 61 7.1.3. Formulation as Operator DAE . 63 7.2. Extension and Regularization . 64 7.3. Existence Results and Well-posedness . 65 7.3.1. Homogeneous Problem . 65 7.3.2. Existence of the Lagrange Multiplier . 66 7.3.3. Well-posedness of the Saddle Point Problem . 67 7.4. Influence of Perturbations . 69 7.5. Applications in Flexible Multibody Dynamics . 71 Part C The Method of Lines 73 8. The Method of Lines for First-order Systems . 74 8.1. Preliminaries and Notation . 74 8.2. Linear Constraints . 75 8.2.1. Conforming Discretization . 76 ii 8.2.2. Nonconforming Discretization. 76 8.3. Nonlinear Constraints . 77 8.4. Application to Flow Equations . 78 8.4.1. Decomposition for Crouzeix-Raviart Elements . 80 8.4.2. Decomposition for Bernardi-Raugel Elements . 82 8.4.3. Further Elements . 83 8.4.4. Numerical Example . 84 9. The Method of Lines for Second-order Systems . 86 9.1. Recap and Notation . 86 9.2. Determination of the Index . 87 9.3. Commutativity . 88 Part D The Rothe Method 91 10. Convergence for First-order Systems . 92 10.1. Setting . 92 10.2. Temporal Discretization . 94 10.2.1. Existence of Solutions. 94 10.2.2. A Priori Estimates . 95 10.3. Global Approximations and Convergence. 97 10.3.1. Definition of U1,τ , U2,τ , and V2,τ ..................... 97 10.3.2. Definition of Λτ ................................ 98 10.3.3. Convergence Results. 99 10.4. Influence of Perturbations. 102 10.4.1. Error Analysis. 102 10.4.2. Spatial Discretization as Perturbation . 104 10.5. Nonlinear Constraints . 105 10.5.1. Temporal Discretization . 107 10.5.2. Convergence Results. 108 10.5.3. Influence of Perturbations . 111 11. Convergence for Second-order Systems . 113 11.1. Setting and Discretization. 113 11.2. Stability and Convergence. 115 11.2.1. Stability Estimate . 115 11.2.2. Definition of Global Approximations . 117 11.2.3. Passing to the Limit. 119 11.2.4. Lagrange Multiplier . 122 11.3. Influence of Perturbations. ..