A convergent adaptive Uzawa finite element method for the nonlinear Stokes problem [Elektronische Ressource] / vorgelegt von Christian Kreuzer

Sujets

• Mathematics

Informations

• Publié par: universitat_augsburg
• Publié le: 01 janvier 2008
• Nombre de lectures: 15
• Langue: Deutsch

Extrait

A Convergent Adaptive Uzawa
Finite Element Method for the
Nonlinear Stokes Problem
Dissertation zur Erlangung des Doktorgrades der
Mathematisch-Naturwissenschaftlichen Fakult¨at der
Universit¨at Augsburg
vorgelegt von Christian Kreuzer
April 2008Erster Gutachter: Prof. Dr. K. G. Siebert, Augsburg, Deutschland
Zweiter Gutachter: Prof. Dr. R. H. Nochetto, College Park, USA
Dritter Gutachter: Prof. Dr. A. Veeser, Mailand, Italien
Mu¨ndliche Pru¨fung: 23. Juli, 2008iii
Danksagung
ObgleichichdieseArbeitselbstverfasstundmichkeinerfremdenHilfebedient
habe, gibt es doch einige Menschen, die zur Entstehung der vorliegenden Seiten
beigetragen haben.
In erster Linie m¨ochte ich mich bei meinem Doktorvater Kunibert G. Siebert
bedanken,dermireinerseitsvieleFreir¨aumegelassenhat,andererseitsbeiProble-
men immer zur Stelle war. Auch bedanken m¨ochte ich mich bei ihm fu¨r die tolle
Zusammenarbeit und die vielen fruchtbaren Kontakte zu anderen Forschungs-
gruppen, die er mir erm¨oglicht hat.
Weiterhin danke ich allen Kollegen, vor allem Christian M¨oller, der aus-
dauernd als Korrektor fungiert hat, und falls n¨otig (und das war es oft), mit
Kaffee zur Stelle war. Außerdem m¨ochte ich noch Carina Lorenzen danken, die
es auf sich genommen hat, das Englisch der Arbeit zu verbessern wo sie es ver-
standen hat.
Dankgebu¨hrtauch demProjekt C.1derDFG-Research-Unit“NonlinearPar-
tial Differential Equations” Generalized Newtonian fluids and electrorheological
fluids, dem ich es zu verdanken habe, dass ich nicht Hunger leiden musste.
Abschließend m¨ochte ich noch meiner Familie und meinen Freunden danken,
die in der letzten Zeit doch sehr zuru¨ckstecken mussten und trotzdem immer
wenn es n¨otig war — und sei es fu¨r ein Bier oder mehrere — zur Stelle waren.
Yuri, danke fu¨r die Zigaretten.
iiiiv
ivContents
1 Introduction 1
1.1 Quasi-Newtonian Flows. . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Outline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Analytical Background 5
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Orlicz and Orlicz-Sobolev Spaces . . . . . . . . . . . . . . . . . . 8
2.2.1 N-functions . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.3 Orlicz-Sobolev Spaces . . . . . . . . . . . . . . . . . . . . 21
3 Adaptive Finite Elements for the Nonlinear Poisson Problem 25
3.1 Nonlinear Poisson Equation . . . . . . . . . . . . . . . . . . . . . 25
3.1.1 Stating the Problem . . . . . . . . . . . . . . . . . . . . . 25
3.1.2 Existence and Uniqueness of Solutions . . . . . . . . . . . 27
3.1.3 The Energy Functional . . . . . . . . . . . . . . . . . . . . 33
3.2 Concept of Distance . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.1 Shifted N-functions . . . . . . . . . . . . . . . . . . . . . . 35
3.2.2 Quasi-Norm . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Finite Element Approach . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.1 Triangulation and Refinement Framework . . . . . . . . . 49
3.3.2 Finite Element Space and Discrete Problem . . . . . . . . 52
3.3.3 Modular Interpolation Estimates . . . . . . . . . . . . . . 53
3.4 A Posteriori Error Estimators . . . . . . . . . . . . . . . . . . . . 55
3.4.1 Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4.2 Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.5 Adaptive Finite Elements . . . . . . . . . . . . . . . . . . . . . . 70
3.5.1 Adaptive Finite Element Method (AFEM) . . . . . . . . . 70
3.5.2 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . 72
3.5.3 Contraction of AFEM . . . . . . . . . . . . . . . . . . . . 79
vvi CONTENTS
4 Adaptive Uzawa FEM for the nonlinear Stokes Problem 87
4.1 Nonlinear Stationary Stokes Equations . . . . . . . . . . . . . . . 87
4.1.1 Stating the Problem . . . . . . . . . . . . . . . . . . . . . 87
4.1.2 Existence and Uniqueness of Solutions . . . . . . . . . . . 89
4.1.3 The Lagrangian Function . . . . . . . . . . . . . . . . . . 93
4.2 Generalized Uzawa Algorithm . . . . . . . . . . . . . . . . . . . . 101
4.2.1 Quasi-Steepest Descent Direction . . . . . . . . . . . . . . 101
4.2.2 Convergent Generalized Uzawa Algorithm (GUA) . . . . . 103
4.3 Adaptive Uzawa Finite Element Method . . . . . . . . . . . . . . 115
4.3.1 Approximation of the Quasi-Steepest Descent Direction . . 115
4.3.2 Interpolation of Discrete Functions . . . . . . . . . . . . . 118
4.3.3 Convergent Adaptive Uzawa Algorithm (AUA). . . . . . . 126
4.4 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . 138
A Bibliography 141
B Notation Index 149
viChapter 1
Introduction
Partial differential equations like the stationary Stokes problem arise in numer-
ous physical models, particularly in the modeling of Quasi-Newtonian fluids; see
section 1.1. We know the formulation of the stationary Stokes equations to be
−divA(∇u)+∇p=f in Ω,
(1) divu= 0 in Ω,
u= 0 on∂Ω,
with A being a vector-field, which in general is nonlinear.
The main objective of this dissertation is the formulation of a convergent
adaptive Uzawa algorithm (AUA) for the numerical solution of the nonlinear
stationary Stokes problem. For this purpose, we reformulate the system (1) into
asaddle-point problem, which isequivalent to minimizing afunctionalF relative
tothepressure. ThebasicideabehindAUAisthemethodofthesteepest descent
[18, 24], which is equivalent to the Uzawa method in the linear case [64, 6].
It turns out that the derivative of F for the pressure q is the divergence of
the solution to the nonlinear elliptic equation
−divA(∇u ) =f−∇q in Ω,q
(2)
u =0 on∂Ω.q
Hence,d is a descent direction ofF in q if and only ifZ
DF(q)(d)= ddivu < 0,q
Ω
where DF is the Fr´echet derivative of F. We compute a numerical solution of
(2) using an adaptive finite element method (AFEM) proposed in [27]. Adaptive
finite element methods are a powerful and efficient toolfor solving elliptic partial
differential equations. Usually they consist of the loop
(AFEM) Solve→Estimate→Mark→Refine
12 Introduction
and their convergence has been analyzed in [57, 58, 74, 55, 28, 19, 61, 60]. In
particular, our AFEM, based on the quasi-norm error concept introduced in [8],
converges to the true solution in a linear fashion.
This motivatestheuse ofthe quasi-norm techniques inthe AUA aswell. Asa
consequence we define a so called quasi-steepest descent direction. Then starting
from an initial guess Q of the pressure p, the AUA consists of a loop0
(AUA) Q :=Q +D ,j+1 j j
where≥0 and we instrumentalize the AFEM to compute a reasonable approx-
imation D to the quasi-steepest descent direction in the jth step. The mainj
result shows convergence of the AUA for a fixed step-size .
1.1 Quasi-Newtonian Flows
The viscosity ν of a fluid describes its resistance to flow. It is defined to be the
proportionality constant between the shear stress τ and the shear rate, i.e., the
1 tsymmetric part of the velocity gradient E(u)= (∇u+∇u )
2
τ =νE(u).
Newton’s law of viscosity states that the viscosityν does not change with the
shear rate, i.e., ν is constant.
However, many fluids do not obey Newton’s hypothesis, i.e., the viscosity de-
pendsontheshearrate: Whenpaintisshearedwithabrush,itflowscomfortably,
but when the shear stress is removed, its viscosity increases so that it no longer
flows easily.
Wespeakofapseudo-plasticorashearthinningfluid,iftheviscositydecreases
with increasing shear rate. Examples of shear thinning fluids are polymer melts,
polymersolutionsandsomepaints. Theoppositebehaviorcalleddilatantorshear
thickening is found in corn starch, clay slurries, and some surfactants. Fluids of
this kind are called quasi-Newtonian fluids.
The traditional engineering model for quasi-Newtonian fluids is the so-called
power law
r−2
ν(|E(u)|)=ν |E(u)| ,0
where ν > 0. Thereby pseudo-plastic fluids correspond to r ∈ (1,2) whereas0
dilatant fluids correspond to r > 2. It seems to work well for dilatant fluids,
but seems to be rather inconvenient for pseudo plastic ones since the power
r−2 becomes negative. Moreover, many shear-thinning fluids exhibit Newtonian
behavior at extreme shear, both lowandhigh. These difficulties can be overcome
by the Carreau law
r−222
2ν(|E(u)|) =ν +(ν −ν )(κ +|E(u)| ) ,∞ 0 ∞
21.2 Outline 3
whereκ> 0 andν >ν ≥0. In the case of pseudo-plastic fluids, i.e., whenr∈0 ∞
2(1,2), for|E(u)|≪κ, the fluid is almost Newtonian withν ≈ν +(ν −ν )κ .∞ 0 ∞
And for|E(u)|≫κ the fluid is again Newtonian withν≈ν . In most polymers∞
ν is zero.∞
The steady state of afluid can be modeled by the stationaryStokes equations−div νE(u) +∇p=f in Ω,
(1) divu= 0 in Ω,
u= 0 on∂Ω,
where u is the velocity and p the kinematic pressure of a fluid inside a domain
Ω due to an external body force f. Thereby the definition of the viscosity ν has
to be chosen according to the Newtonian, pseudo-plastic, or dilatant behavior of
the fluid.
Fortheeaseofexpositionwedecidedtoformulatethethesisforthegradientof
thevelocityinsteadofitssymmetric gradient; see(1). However, thankstoKorn’s
inequality all