Sarah ALI HASSAN

188 Pages·2017·7.69 MB·English

THÈSE
présentéà
L’UNIVERSITÉ PIERRE ET MARIE CURIE-PARIS 6
École doctorale : Sciences Mathématiques de Paris Centre (ED 386)
Par
Sarah ALI HASSAN
Pourobtenirlegradede
DOCTEUR de l’UNIVERSITÉ PIERRE ET MARIE CURIE-PARIS 6
Spécialité : Mathématiques Appliquées
Estimations d’erreur a posteriori et critères d’arrêt
pour des solveurs par décomposition de domaine
et avec des pas de temps locaux
Directeur de thèse : Martin VOHRALÍK
Co-directeurs de thèse : Caroline JAPHET et Michel KERN
Soutenue le : 26 juin 2017
Devant la commissiond’examen formée de :
M. Emmanuel CREUSÉ Université Lille 1 Rapporteur
M. Jan Martin NORDBOTTEN Université de Bergen Rapporteur
M. Frédéric NATAF Université Paris VI Président du jury
M. Laurent LOTH ANDRA Examinateur
M. Pascal OMNES CEA Examinateur
Mme Caroline JAPHET Université Paris XIII Co-directeur de thèse
M. Michel KERN Inria Paris Co-directeur de thèse
M. Martin VOHRALÍK Inria Paris Directeur de thèse

DOCTORAL DISSERTATION
presentedat
UNIVERSITY OF PIERRE AND MARIE CURIE-PARIS 6
Doctoral school: Mathematical Sciences of Central Paris (ED 386)
By
Sarah ALI HASSAN
Toobtainthedegreeof
DOCTOR of UNIVERSITY PIERRE AND MARIE CURIE-PARIS 6
Speciality: Applied Mathematics
A posteriori error estimates and stopping criteria
for solvers using the domain decomposition method
and with local time stepping
Thesis advisor: Martin VOHRALÍK
Thesis co-advisors: Caroline JAPHET and Michel KERN
Defended on: June 26th, 2017
In front of the examination committee consisting of:
M. Emmanuel CREUSÉ The University of Lille 1 Reviewer
M. Jan Martin NORDBOTTEN The University of Bergen Reviewer
M. Frédéric NATAF The University of Paris VI Chairman
M. Laurent LOTH ANDRA Examiner
M. Pascal OMNES CEA Examiner
Mme Caroline JAPHET The University of Paris XIII Thesis co-advisor
M. Michel KERN Inria Paris Thesis co-advisor
M. Martin VOHRALÍK Inria Paris Thesis advisor

Résumé
Cette thèse développe des estimations d’erreur a posteriori et critères d’arrêt pour
les méthodes de décomposition de domaine avec des conditions de transmission de
Robin optimisées entrelesinterfaces. Différents problèmessontconsidérés: l’équation
de Darcy stationnaire puis l’équation de la chaleur, discrétisées par les éléments fi-
nis mixtes avec un schéma de Galerkin discontinu de plus bas degré en temps pour
le second cas. Pour l’équation de la chaleur, une méthode de décomposition de do-
maine globale en temps, avec mêmes ou différents pas de temps entre les différents
sous domaines, est utilisée. Ce travail est finalement étendu à un modèle diphasique
en utilisant une méthode de volumes finis centrés par maille en espace. Pour chaque
modèle, un problème d’interface est résolu itérativement, où chaque itération néces-
site la résolution d’un problème local dans chaque sous-domaine, et les informations
sont ensuite transmises aux sous-domaines voisins. Pour les modèles instationnaires,
les problèmes locaux dans les sous-domaines sont instationnaires et les données sont
transmisesparl’interfaceespace-temps.
L’objectifdecetravailest,pourchaquemodèle,debornerl’erreurentrelasolution
exacte et la solution approchée à chaque itération de l’algorithme de décomposition
de domaine. Différentes composantes d’erreur en jeu de la méthode sont identifiées,
dont celle de l’algorithme de décomposition de domaine, de façon à définir un
critère d’arrêt efficace pour cette méthode. En particulier, pour l’équation de Darcy
stationnaire,onborneral’erreurparunestimateurdedécompositiondedomaineainsi
qu’un estimateur de discrétisation en espace. On ajoutera à la borne de l’erreur un
estimateur de discrétisation en temps pour l’équation de la chaleur et pour le modèle
diphasique. L’estimation a posteriori répose sur des techniques de reconstructions
de pressions et de flux conformes respectivement dans les espaces H1 et H(div)
et sur la résolution de problèmes locaux de Neumann dans des bandes autour des
interfaces de chaque sous-domaine pour les flux. Ainsi, des critères pour arrêter les
itérations de l’algorithme itératif de décomposition de domaine sont développés. Des
simulations numériques pour des problèmes académiques ainsi qu’un problème plus
réaliste basé sur des données industrielles sont présentées pour illustrer l’efficacité de
ces techniques. En particulier, différents pas de temps entre les sous-domaines sont
considéréspourcetexemple.
Mots-clés : Écoulement et transport en milieu poreux, éléments finis mixtes,
décomposition de domaine en espace, décomposition de domaine globale en temps,
discrétisation conforme et non-conforme en temps, pas de temps locaux, conditions
d’interfacedeRobin,estimationd’erreuraposteriori, critèred’arrêt,problèmelocalde
Neumann

Abstract
This work contributes to the developpement of a posteriori error estimates and stop-
ping criteria for domain decomposition methods with optimized Robin transmission
conditions on the interface between subdomains. We study several problems. First,
we tacklethe steadydiffusion equation usingthe mixedfinite elementsubdomaindis-
cretization. Then the heat equation using the mixed finite element method in space
andthediscontinuousGalerkinschemeoflowestorderin timeis investigated. Forthe
heat equation, a global-in-time domain decomposition method is used for both con-
forming and nonconforming time grids allowing for different time steps in different
subdomains. This work is then extended to a two-phase flowmodel using a finite vol-
ume scheme in space. For each model, the multidomain formulation can be rewritten
as an interface problem which is solved iteratively. Here at each iteration, local sub-
domain problems are solved, and information is then transferred to the neighboring
subdomains. Forunsteadyproblems,thesubdomainproblemsaretime-dependentand
information istransferred viaaspace-time interface.
The aim of this work is to bound the error between the exact solution and the
approximatesolutionateachiteration ofthedomaindecompositionalgorithm. Differ-
enterrorcomponents,suchas the domain decomposition error, are identified in order
to define efficient stopping criteria for the domain decomposition algorithm. More
precisely, for the steady diffusion problem, the error of the domain decomposition
methodandthatofthediscretizationinspaceareestimatedseparately. Inaddition,the
timeerrorfortheunsteadyproblemsisidentified. Ouraposterioriestimatesarebased
onthereconstructiontechniquesforpressuresandfluxesrespectivelyinthespaces H1
and H(div). For the fluxes, local Neumann problems in bands arround the interfaces
extracted from the subdomains are solved. Consequently, an effective criterion to
stop the domain decomposition iterations is developed. Numerical experiments, both
academic andmore realistic with industrial data, are shown to illustrate the efficiency
of these techniques. In particular, different time steps in different subdomains for the
industrial exampleare used.
Keywords: Flow and transport in porous media, mixed finite element method,
domain decomposition in space, global-in-time domain decomposition, conforming
andnonconformingtimegrids,localtimesteps,Robininterfaceconditions,aposteriori
errorestimate,stoppingcriteria, localNeumannproblem

Acknowledgements
First and foremost I would like to thank God, for providing me with the strength to
carry out this thesis, and without whose blessings I could never have completed it.
Thank you for everything and the fact that I am now here to acknowledge the people
towhomIamdeeplyindebtedfortheirdirectorindirectcontributions tothis work.
I would like to say a big thank you to Luís Neves de Almeida, the director of the
MathematicsappliedtoBiologicalandMedicalSciencesMaster’scourseatParis6,and
who was my supervisor during my master internship. He is the one who encouraged
metocontactMartinVohralík todoaPh.Dthesis.
I owe my deepest gratitude to my supervisor Martin Vohralík, director of the Serena
team at INRIA Paris. It has been an honor, as well as a pleasure, to work with you.
Thank you, Martin, for the continuous and excellent guidance, patience, ideas, and
immenseknowledgethatyougavemethoughoutmyPh.D.
I would like to express my warmest gratitude to my co-supervisor, Caroline Japhet,
especially for sharing her specialist knowledge of the domain decomposition method
and for the continuous support and optimism she gave me. This thesis would have
beensomuchmoredifficulttocompletewithout you,Caroline.
I would alsolike to thank Michel Kern, my co-supervisor for the useful discussions we
had, and the help he provided during the CEMRACS summerschool, which led to the
fourthchapterofthis thesis.
I am very grateful to ANDRA, the French national radioactive waste management
agency,whofundedthis thesis overthree years.
Three years ago, I would never have imagined that I would be capable of writing my
thesis in English. That I was able to do so is largely due to the efforts of Richard
James,andIwouldlike offerhim myspecialthanks.
I cannot fail to mention here, the valuable contribution Elyes Ahmed made to the
fourthchapter, atthe CEMRACSsummerschool. Thanks Elyes!
I would like to acknowledge all the members of Pomdapi team, which later became
the Serena team, who provided such a friendly atmosphere at work: Jean E. Roberts,
Jérôme Jaffré, François Clément, Pierre Weis, Géraldine Pichot, Vincent Martin, Iain
Smears, Fatma Cheikh, Nabil Birgle, Mohamed Hedi Riahi, Émilie Joannopoulos,
Clément Franchini. Special thanks goes to Zuki Tang with whom I shared an office at
Inria Rocquencourt, and another special thanks goes to Jad Dabaghi, Patrik Daniel,
and Khadija Talali with whom I shared an office (as well as a lot of food!) at Inria
Paris. Thanks forthe helpofourteam assistants,who supportedmeduring mythesis,
Nathalie Bonte,CindyCrossouard, VirginieCollette,andKevin Bonny.
vi Acknowledgements
Myfamily’ssupporthasplayedamajorrole...
I would like to thank my mother-in-law, Mariam, for her encouragement and her
prayers. Please, accept my thanks! I would also thank my father-in-law Houssein,
Ayat, Mouhamad, and Karim for their kindness. Your kind wishes mean a great deal
tome.
I’d like to thank my grandparents "Jedo" Mahmoud and "Teta" Najwa for their love,
and my grandmother’s non-stop questioning, for the last three years, about when I
willfinish. Iwonderwhat sheis goingtoaskabout,nowihave infactfinished.
I express my special thanks to my two lovely brothers Ali and Moustapha and
Moustapha’s wife Zaineb and their lovely little daughter Sarah. Thank you for your
constantsupportandencouragement.
Mylovelypatienthusband,dearMayssam,Ioweyouthanksforalltheyearsyouhave
been by my side and always believed in me. I am very thankful that I have you in my
life and appreciate all you have done for me. Without your love, presence, support,
comfort, understanding, guidance, prayers, and encouragement,this thesis would not
have beenpossible. Mygratitude anddeepestappreciation gotoyou,Mayssam.
My beloved parents, Mum Zeinab Chahine and Dad Mehdi Ali Hassan, I am forever
indebted to you, for all the continuous love, support, sacrifices, prayers, and encour-
agement that you have given me throughout my life, to make me the person I am
today. Iam eternallygratefulyou,MumandDad.
Mum, thank you for coming to Paris twice to support me, and once for my defense.
Thank you also for the delicious strawberry jam and Za’atar you made for me. The
best moments in my life are when you are beside me. Because I have you, Mum, I
have everything.
Dad, you told me, "Never be afraid because you are my daughter." Dad you knew
that I could achieve this thesis, and you already prepared my gift for this day. Now I
have finished, I want to thank you for being my rock, and for asking me to send you
my programming code to find the bug when I had one. I remember how we laughed
becausewebothknewthatyou,neverhaving written anycode,wouldn’teverfindit!
Mum, Dad, Mayssam, and I followed and shared the same dream. This thesis
was really done by all four of us together. Mum, Mayssam, and I dedicate this thesis
toyouDad,the mostspecialperson inourlife...