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Domain Decomposition Methods in Optimal Flow Control for High DOMAINDECOMPOSITION METHODSINOPTIMALFLOW CONTROLFORHIGH PERFORMANCECOMPUTING Zur Erlangung des akademischen Grades eines DOKTOR DER NATURWISSENSCHAFTEN von der Fakultät für Mathematik des Karlsruher Instituts für Technologie (KIT) genehmigte DISSERTATION von Dipl.-Math. techn. Eva Ketelaer aus Emmerich am Rhein Tag der mündlichen Prüfung: 08. Mai 2013 Referent: Prof. Dr. Vincent Heuveline Korreferent: Prof. Dr. Götz Alefeld Korreferent: Prof. Dr. Matthias Heinkenschloss Für Oma Kleve und meine Eltern ABSTRACT This thesis is concerned with linear and non-linear optimal flow con- trol problems which are modeled by systems of partial differential equations. The numerical treatment of such problems, especially in the context of flow problems, is often very expensive and challeng- ing. To tackle this complexity, we present parallel approaches based on non-overlapping domain decomposition methods that exploit the computational power provided by modern high performance comput- ing technologies. On the algebraic level, we derive parallel solvers and Neumann-Neumann type preconditioners which enable us to solve these sophisticated problems in an efficient way. By means of investigating the cost for the preconditioners and of studying both scalability and efficiency of the domain decomposition approaches, we analyze the developed methods in the framework of high performance computing. v ACKNOWLEDGMENTS First, I want to thank Prof. Heuveline for supporting me as my supervisor and for offering me the great opportunity to be part of the EMCL-team. Especially, I want to thank him for his guidance and encouragement in many fruitful discussions during my time at his institute. Furthermore, I am grateful to Prof. Heinkenschloss for his support, in particular for inviting me for a research stay of three weeks to the Rice University in Houston. During this stay at his department, I achieved important steps towards the completion of this thesis. I also want to thank Prof. Alefeld for his support as one of my second supervisors. I am really grateful to all my great colleagues at EMCL. The working atmosphere and team spirit have been a great backup and motivation. Besides the insightful discussions, I especially enjoyed the coffee breaks, having lunch together and all the great birthday cakes. I thank all my colleagues who helped me to correct my thesis. In particular, I want to mention Staffan and Michael. Michael is a great listener. Many times, he helped me to find solutions for my problems, often without saying a word but instead simply by listening to me. Staffan always had an answer to my C++ programming issues and especially at the end of my work, he supported me a lot. Furthermore, I want to thank my friends Britta and Katrin from my hometown Kleve. Even though they try their best to keep away from math, they always listened to me, motivated me during the lows of my PhD and shared my joys during the highs. Particularly, I am deeply grateful to my family. My parents always supported me and had faith in me. Without their support, patience and love, I never would have had the motivation and ability to write this thesis. Finally, I would like to deeply thank Manuel. He has been and still is the best support I can imagine. Even though lots of times many kilometers were inbetween us, it always felt like that he was right next to me. vii CONTENTS 1 introduction1 2 domain decomposition method for the oseen equa- tions7 2.1 General Definitions 8 2.2 Inhomogeneous Dirichlet Boundary Conditions 8 2.3 Outflow Boundary Conditions 14 2.4 Continuous Domain Decomposition 17 2.4.1 Decomposition of the Domain 17 2.4.2 Decomposition of the Velocity Space 18 2.4.3 Decomposition of the Pressure Space 19 2.4.4 Decomposition of the Bilinear Forms 20 2.4.5 Weak Formulation on Subdomains 23 2.4.6 Decoupling of Weak Formulation on Subdomains 27 2.5 Discretization Based on a Finite Element Method 33 2.5.1 Definitions for the Finite Element Methods 34 2.5.2 Decoupled Finite Element Formulation 36 2.6 Algebraic Formulation 39 2.6.1 Algebraic Formulation for the Velocity 39 2.6.2 Algebraic Formulation for the Pressure 40 2.6.3 Global Linear Systems and Schur-complement Equations 45 2.6.4 General Operator 51 3 ddm for optimal control problem constrained by oseen 53 3.1 General Definitions 53 3.2 Distributed Optimal Control Problem 53 3.2.1 Strong Interpretation of the Optimality System 56 3.3 Optimal Boundary Control Problem 57 3.3.1 Strong Interpretation of the Optimality System 60 3.4 Continuous Domain Decomposition 60 3.4.1 Definitions for the Distributed Control Case 61 3.4.2 Weak Formulation on Subdomains for Distributed Control 62 3.4.3 Decoupling of Weak Formulation on Subdomains for Distributed Control 63 3.4.4 Definitions for the Boundary Control Case 66 3.4.5 Weak Formulation on Subdomains for Boundary Control 67 3.4.6 Decoupling of Weak Formulation on Subdomains for Boundary Control 69 3.5 Discretization Based on FEM for Distributed Control 73 3.5.1 Decoupled Finite Element Formulation 73 ix x contents 3.6 Discretization Based on FEM for Boundary Control 75 3.6.1 Decoupled Finite Element Formulation 76 3.7 Algebraic Formulation 77 3.7.1 Definitions for the Distributed Control Case 78 3.7.2 Definitions for the Boundary Control Case 80 3.7.3 Global Linear System and Schur-complement Equation for the Distributed Control Case 80 3.7.4 General Operator for the Distributed Control Case 86 3.7.5 Global Linear System and Schur-complement Equation for the Boundary Control Case 87 4 ddm for the navier-stokes equations 93 4.1 Navier-Stokes Equations with Outflow Boundary Con- ditions 93 4.2 Continuous Domain Decomposition 95 4.2.1 Decomposition of the Bilinear Forms 96 4.2.2 Decomposition of the Residuals 98 4.2.3 Weak Formulation on Subdomains 98 4.2.4 Decoupling of Weak Formulation on Subdomains 99 4.3 Discretization Based on a Finite Element Method 101 4.3.1 Decoupled Finite Element Formulation 101 4.4 Algebraic Formulation 102 4.4.1 Global Linear System and Schur-complement Equation 103 4.4.2 Solution Algorithm 106 4.4.3 General Operator 106 5 ddm for an optimal control problem s.t. the navier- stokes eq 109 5.1 Distributed Optimal Control Problem 109 5.2 Continuous Domain Decomposition 112 5.2.1 Decomposition of the Residuals 112 5.2.2 Weak Formulation on Subdomains 113 5.2.3 Decoupling of Weak Formulation on Subdomains 115 5.3 Discretization Based on a Finite Element Method 118 5.3.1 Decoupled Finite Element Formulation 119 5.4 Algebraic Formulation 120 5.4.1 General Operator 127 6 preconditioned solving routines based on ddms 129 6.1 Generalization of the Schur-complement Equation 130 6.2 Preconditioners for the Schur-complement equation 132 6.2.1 Neumann-Neumann Preconditioner 133 6.2.2 Balancing Neumann-Neumann Preconditioner 135 6.3 Comparison of the Neumann-Neumann and the balanc- ing Neumann-Neumann Preconditioner 140 contents xi 6.4 Solution Algorithm for the Schur-complement equa- tion 142 6.5 Parallelization of Global System via Domain Decompo- sition 144 6.6 Global Schur-complement Preconditioner 145 6.6.1 Subdomain Preconditioner PII 146 6.6.2 Skeleton Preconditioner PG 147 6.7 Alg. for the Global Schur-complement Preconditioner 148 7 numerical experiments 151 7.1 HiFlow3-Software 151 7.2 Overview and Structure of the Numerical Experiments 152 7.3 Verification of both Methods and their Implementa- tion 157 7.4 Simulation Results in Backward Facing Step Geome- try 159 7.5 Analysis of the Preconditioners 159 7.6 Scalability Study 173 8 conclusion 179 bibliography 183 LISTOFFIGURES Figure 1 Chapter overview. 4 Figure 2 Step 1 of the derivation of the domain decom- position method for the Oseen equations with inhomogeneous boundary conditions. 8 Figure 3 Step 1 of the derivation of the domain decompo- sition method for Oseen or Navier-Stokes equa- tions with mixed outflow and Dirichlet bound- ary conditions. 15 Figure 4 Step 2b of the derivation of the domain decompo- sition method for the Oseen equations with inho- mogeneous Dirichlet boundary conditions. 17 Figure 5 Step 2a of the derivation of the domain decompo- sition method for the Oseen equations with inho- mogeneous Dirichlet boundary conditions. 23 Figure 6 Step 2a of the derivation of the domain decom- position method for the Oseen or Navier-Stokes equations with mixed outflow and Dirichlet bound- ary conditions. 26 Figure 7 Step 2b of the derivation of the domain decompo- sition method for the Oseen equations with inho- mogeneous Dirichlet boundary conditions. 30 Figure 8 Step 3 of the derivation of the domain decompo- sition method for the Oseen equations with inho- mogeneous Dirichlet boundary conditions. 34 Figure 9 Step 3 of the derivation of the domain decom- position method for the Oseen or Navier-Stokes equations with mixed outflow and Dirichlet bound- ary conditions. 38 Figure 10 Step 1 of the derivation of the domain decompo- sition method for the optimal control problem assuming distributed control constrained by the Oseen or Navier-Stokes equations. 54 Figure 11 Step 1 of the derivation of the domain decompo- sition method for the optimal boundary con- trol problem constrained by the Oseen equa- tions. 57 Figure 12 Step 2a of the derivation of the domain decom- position method for the optimal control problem assuming distributed control constrained by the Oseen or Navier-Stokes equations. 61 xii List of Figures xiii Figure 13 Step 2b of the derivation of the domain decom- position method for the optimal control problem assuming distributed control constrained by the Oseen or Navier-Stokes equations.
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