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...