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Algebraic Multigrid Methods for the Numerical Solution of the Incompressible Navier-Stokes Equations

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Algebraic Multigrid Methods for the Numerical Solution of the Incompressible Navier-Stokes Equations J O H A N N E S K E P L E R U N I V E R S I T A¨ T L I N Z N e t z w e r k f u¨ r F o r s c h u n g , L e h r e u n d P r a x i s Algebraic Multigrid Methods for the Numerical Solution of the Incompressible Navier-Stokes Equations Dissertation zur Erlangung des akademischen Grades Doktor der Technischen Wissenschaften Angefertigt am Institut fur¨ Numerische Mathematik Begutachter: A.Univ.Prof. Dipl.-Ing. Dr. Walter Zulehner Prof. Dr. Arnold Reusken Eingereicht von: Dipl.-Ing. Markus Wabro Mitbetreuung: O.Univ.Prof. Dipl.-Ing. Dr. Ulrich Langer Linz, August 2003 Johannes Kepler Universit¨at A-4040 Linz Altenbergerstraße 69 Internet: http://www.uni-linz.ac.at DVR 0093696 · · · 2 Eidesstattliche Erkl¨arung Ich erkl¨are an Eides statt, dass ich die vorliegende Dissertation selbstst¨andig und ohne fremde Hilfe verfasst, andere als die angegebenen Quellen und Hilfsmittel nicht benutzt bzw. die w¨ortlich oder sinngem¨aß entnommenen Stellen als solche kenntlich gemacht habe. Diese Dissertation wurde bisher weder im In- noch im Ausland als Prufungsarb¨ eit vorgelegt. Dipl.-Ing. Markus Wabro Linz, August 2003 3 Abstract If the Navier-Stokes equations for incompressible fluids are linearized using fixed point iterations, the Oseen equations arise. In this thesis we provide concepts for the coupled algebraic multigrid (AMG) solution of this saddle point system, where `coupled' here is meant in contrast to methods, where pressure and velocity equations are iteratively decou- pled, and `standard' AMG is used for the solution of the resulting scalar problems. We show how the coarse levels can be constructed (where their stability is an important issue) and which smoothers (known from geometric multigrid methods for saddle point systems) can be used. To prove the efficiency of our methods experimentally, we apply them to finite element discretizations of various problems (model problems and also more complex industrial settings) and compare them with classical approaches. Zusammenfassung Durch die Fixpunkt-Linearisierung der Navier-Stokes Gleichungen fur¨ inkompressible Flu- ide erh¨alt man die sogenannten Oseen Gleichungen. In der vorliegenden Arbeit entwick- eln wir Konzepte fur¨ die numerische L¨osung dieses Sattelpunktsystems durch gekoppelte algebraische Mehrgittermethoden (AMG), wobei \gekoppelt" im Gegensatz zu Vefahren steht, bei denen Druck- und Geschwindigkeitsgleichungen iterativ entkoppelt werden und `Standard'-AMG zur L¨osung der entstehenden skalaren Probleme angewandt wird. Wir pr¨asentieren M¨oglichkeiten der Konstruktion der Grobgittersysteme (wobei ins- besondere auf deren Stabilit¨at geachtet wird) und der Anwendung von Gl¨attern, welche von geometrischen Mehrgittermethoden fur¨ Sattelpunktgleichungen her bekannt sind. Die Effizienz der entwickelten Methoden wird schließlich experimentell gezeigt, indem sie sowohl fur¨ einfachere Modellprobleme als auch fur¨ durchaus komplexe industrielle An- wendungen getestet und mit den klassischen Methoden verglichen werden. 4 Acknowledgements First I owe a great debt of gratitude to Walter Zulehner, not only for advising my thesis but for arousing my interest and deepening my understanding in the scientific field I am currently working in. I would also like to thank Ulrich Langer who, as leader of the project and head of the institute I am employed in, provided me with scientific and financial background, and who made constructive suggestions on many details of this thesis, and I want to thank Arnold Reusken for co-refereeing this dissertation. Special thanks go to Ferdinand Kickinger who supplied me with industrial test cases and with whom I had some very fruitful discussions, and to all other people who are responsible for some aspects of this thesis by \asking the right questions". This work has been supported by the Austrian Science Foundation \Fonds zur F¨orderung der wissenschaftlichen Forschung (FWF)" under the grant P14953 \Robust Algebraic Multigrid Methods and their Parallelization". Contents Notation 7 1 Introduction 9 2 Preliminaries 12 2.1 Navier-Stokes Equations . 12 2.1.1 Analysis of the Associated Stokes Problem | the Inf-Sup Condition 14 2.2 Finite Element Discretization . 16 2.2.1 Examples of Mixed Elements . 18 2.2.2 Multi-Element Meshes . 21 2.3 The Non-Stationary Problem . 22 2.4 The Convective Term . 24 2.4.1 Instability . 24 2.4.2 Nonlinearity . 25 2.5 Iterative Solvers . 27 2.5.1 Krylov Space Methods . 27 2.5.2 SIMPLE . 29 2.5.3 Inexact Uzawa Methods . 30 3 Multigrid Methods 35 3.1 A General Algebraic Multigrid Method . 35 3.1.1 Basic Convergence Analysis . 37 3.2 Examples in the Scalar Elliptic Case . 39 3.2.1 Geometric Multigrid . 39 3.2.2 Algebraic Multigrid . 40 4 AMG Methods for the Mixed Problem 48 4.1 Construction of the Coarse Level Systems . 49 4.1.1 The P1isoP2-P1 Hierarchy . 51 4.1.2 The P1-P1-stab Hierarchy . 53 nc 4.1.3 The P1 -P0 Hierarchy | Applying AMGe . 58 4.2 Smoothers . 64 4.2.1 Standard Smoothers for the Squared System . 64 5 CONTENTS 6 4.2.2 Transforming Smoothers . 64 4.2.3 Braess-Sarazin Smoother . 65 4.2.4 Vanka Smoothers . 68 5 Software and Numerical Studies 71 5.1 The Software Package AMuSE . 71 5.1.1 Structure . 72 5.1.2 Matrices . 73 5.2 Numerical results . 74 5.2.1 2D Test Cases . 74 5.2.2 3D Problems . 85 5.3 Conclusions and Outlook . 89 Notation We generally use standard characters for scalar values and scalar functions (p, q,. ) and boldface characters for vectors and vector valued functions (u, v,. ). We will use the underline notation (which will be introduced in detail in section 2.2) for finite-element vectors associated to scalar or vector valued functions (p, q,. , resp. u, v, . ). The T components of vectors are denoted by (u1; : : : ; un) = u. For matrices we use capital letters (A,. ), or component-notation A = (aij)i;j. is an open, connected subset of Rd with space dimension d (generally d = 2 or 3), @ Gits boundary. G Operators u v = d u v (scalar product). · i=1 i i u v = (uivj)i;j=1;:::;d (tensor product). ⊗ P@p @jp = (partial derivative of p with respect to xj). @xj @ju = (@jui)i=1;:::;d. @p @tp = @t (partial derivative of p with respect to t) . @tu = (@tui)i=1;:::;d. p = (@ p) (gradient of p). r i i=1;:::;d u = (@iuj)i;j=1;:::;d. r d div u = i=1 @iui (divergence of u). (u )' = d u @ '. · r Pj=1 j j (u )v = ( d u @ v ) . · r P j=1 j j i i=1;:::;d P Function spaces C( ) space of continuous functions on . CkG( ) space of functions with continuousG k-th derivative on . G G C1( ) space of infinitely smooth functions with compact support in . 0 G G C1( ¯) space of infinitely smooth functions on ¯. G G 1 p p p L ( ) Lebesgue space of measurable functions q with finite norm q 0;p = q . G k k G j j R 7 NOTATION 8 k p Wp ( ) Sobolev space of functions with k-th derivatives in L ( ). k G k G H ( ) = W2 ( ). 1 G G 1 H0 ( ) the closure of C01( ) in H ( ). 1G G1 G H− ( ) the dual space of H ( ). G 0 G N The natural numbers. Z The integer numbers. R The real numbers. Norms 2 q 0 = q 0;2 for q L ( ). k k k k 2 1G q 1 = q 0 for q H ( ). j j kr k 2 2 2 G 1 q 1 = q 0 + q 0 for q H ( ). k k kT k kr k n 2 G n n v X = pv Xv for v R and a symmetric positive definite matrix X R × . k k p 2 2 p T n v `2 = v v for v R . k k Y2v α n n k k Y α = sup0=v n v for Y R × k k 6 2 α 2 (consistent matrixk k norm to the vector norm : ). k kα n 2 n n Y F = yij for Y R × (Frobenius norm). k k i;j=1 2 qP Often used Indices, etc. d Space dimension. L Total number of multigrid levels. l Index indicating a certain multigrid level (l 1; : : : ; L ). 2 f g D Index indicating a diffusive term or Laplacian. C Index indicating a convective term. R Index indicating a reaction term. S Index indicating a stabilization term. s Index used if we want to emphasize that some operator is scalar. Chapter 1 Introduction A very important set of partial differential equations in the field of computational fluid dynamics are the Navier-Stokes equations. They are capable of describing various phenom- ena of (in our case incompressible) Newtonian fluid flow, but give rise to many nontrivial mathematical problems despite of their relatively simple outer form. So, for example, the existence and smoothness of solutions of their non-stationary form are currently the topic of one of the prominent one-million-dollar-problems [Fef00, Dic00]. This thesis will un- fortunately make no contribution to that aspect (in all probability), but to an efficient numerical solution of the equations. After deciding which kind of nonlinear iteration to use (in our case fixed point iteration, which leads to the Oseen equations) and which discretization to choose (in our case the finite element method) one obtains an (indefinite) saddle point problem, which has to be solved. Classical iterative methods for that are variants of SIMPLE schemes (as introduced by Patankar and Spalding [PS72]) or Uzawa's algorithm [AHU58], having in common an iterative decoupling of the saddle point system into separate equations for pressure and velocity, which then can be solved with methods known for the solution of positive definite systems.
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