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Metric Analysis Solvers April 2018 Eingereicht von Dipl.-Ing. Christoph Hofer, BSc. Angefertigt am Doktoratskolleg “Computational Mathematics” Betreuer und Erstbeurteiler O. Univ.-Prof. Dipl.-Ing. Dr. Ulrich Langer Fast Multipatch Isogeo- Zweitbeurteiler Prof. Dr. Giancarlo Sangalli metric Analysis Solvers April 2018 Dissertation zur Erlangung des akademischen Grades Doktor der technischen Wissenschaften im Doktoratsstudium Technische Wissenschaften JOHANNES KEPLER UNIVERSITAT¨ LINZ Altenbergerstraße 69 4040 Linz, Osterreich¨ www.jku.at DVR 0093696 2 Abstract This thesis is devoted to the generalization of the Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) method to linear algebraic systems arising from the Isogemetric Analysis (IgA) of linear elliptic boundary value problems, like stationary diffusion or heat conduction problems. This IgA version of the FETI-DP method is called Dual-Primal Isogeometric Tearing and Interconnect (IETI-DP) method. The FETI-DP method is well established as parallel solver for large-scale systems of finite element equations, especially, in the case of heterogeneous coefficients having jumps across subdomain interfaces. These methods belong to the class of non-overlapping domain decomposition methods. In practise, a complicated domain can often not be represented by a single patch, instead a collection of patches is used to represent the computational domain, called multi-patch domains. Regarding the solver, it is a natural idea to use this already available decomposition into patches directly for the construction of a robust and parallel solver. We investigate the cases where the IgA spaces are continuous or even discontinuous across the patch interfaces, but smooth within the patches. In the latter case, a stable formulation is obtained by means of discontinuous Galerkin (dG) techniques. Such formulations are important for various reasons, e.g, if the IgA spaces are not matching across patch interfaces (different mesh-sizes, different spline degrees) or if the patches are not matching (gap and overlapping regions). Using ideas from dG-FETI-DP methods, we extend IETI-DP methods in such a way that they can efficiently solve multi-patch dG-IgA schemes. This thesis also provides a theoretical foundation of IETI-DP methods. We prove the quasi-optimal depen- dence of the convergence behaviour on the mesh-size for both version. Moreover, the numerical experiments indicate robustness of these methods with respect to jumps in the coefficient and a weak dependence on the spline degree. All algorithms are implemented in the C++ library G+Smo. Finally, this thesis investigates space-time methods for linear parabolic initial-boundary value problems, like instationary diffusion or heat conduction problems. The focus is again on efficient solution techniques. The aim is the development of solvers which are on the one hand robust with respect to certain parameters and on the other hand parallelizeable in space and time. We develop special block smoothers that lead to ro- bust and efficient time-parallel multigrid solvers. The parallelization in space is again achieved by means of IETI-DP methods. i ii Zusammenfassung Diese Arbeit ist der Verallgemeinerung der so genannten “Dual-Primal Finite Element Tearing and Interconnecting” (FETI-DP) Methode auf lineare Gleichungssysteme ge- widmet, die aus der isogeometrischen Analyse (IgA) von linearen elliptischen Rand- wertproblemen, wie etwa stationäre Diffusions- oder Wärmeleitproblemen, entstehen. Wir nennen diese Verallgemeinerung “Dual-Primal Isogeometric Tearing and Inter- connecting” (IETI-DP) Methode. FETI-DP ist eine weit verbreitete parallele Methode zum Lösen von großen linearen Gleichungssystemen, die aus der Diskretisierung mit- tels Finiten Elementen entstehen. Insbesondere lässt sich diese Methode parallelisieren und ist besonders geeignet um Probleme mit springenden Koeffizienten zu lösen. Komplexe Geometrien werden als Vereinigung von “einfachen” Gebieten, genannt “Pat- ches”, repräsentiert und als “Multi-patch” Geometrien bezeichnet. Diese bereits beste- hende Zerlegung des Rechengebiets in Teilgebiete (Patches) bietet einen natürlichen Zugang zur Konstruktion von effizienten und parallelisierbaren Lösern. Wir betrachten die Verwendung von IgA Räumen, welche im Inneren der Patches glatte Ansatzfunk- tionen besitzen, aber nur stetig oder sogar unstetig entlang der Patch-Schnittstellen sind. Im Falle von unstetigen IgA Räumen verwenden wir sogenannte unstetige Galer- kin (dG) Techniken um eine stabile Formulierung zu erhalten. Diese ermöglichen es uns verschiedene Spline-Grade oder Gitterfeinheiten auf benachbarten Patches, sowie klei- ne Löcher bzw. Überlappungen zwischen benachbarten Patches, zu betrachten. Basierend auf der dG-FETI-DP Methode erweitern wir die IETI-DP Methode dahinge- hend, sodass sie auch für Multi-patch dG-IgA Schemen anwendbar ist. In dieser Arbeit beweisen wir die quasi-optimale Abhängigkeit des Konvergenzverhalten von der Gitter- feinheit, sowohl für die stetige als auch die unstetige Version der IETI-DP Methode. In numerischen Experimenten beobachten wir des Weiteren Robustheit bezüglich Sprün- ge in den Koeffizienten und eine schwache Abhängigkeit von dem Spline-Grad. Die vorgestellten Algorithmen sind in der C++ Bibliothek G+Smo implementiert. Zuletzt untersuchen wir sogenannte Raum-Zeit Methoden für lineare parabolische Anfangs-Randwertprobleme, wie etwa instationäre Diffusions- oder Wärmeleitungs- probleme. Wie in den anderen Kapitel liegt der Fokus erneut auf effizienten Lösungs- strategien. Ziel ist es Löser zu entwickeln, die einerseits robust gegenüber verschieden- sten Parametern sind und andererseits sich bezüglich Ort und Zeit parallelisieren las- sen. Dazu entwickeln wir Glätter, die zu robusten zeit-parallelen Multigrid Methoden führen. Die zusätzliche Parallelisierung im Ort wird mittels IETI-DP erreicht. iii iv Acknowledgments First of all, I would like to express my deepest gratitude to my supervisor Prof. Ulrich Langer, for his supervision, guidance and support during my Ph.D. studies. I am very grateful for his scientific and non-scientific advices, and for the time he spent reviewing our reports, papers and this thesis. Moreover, I would like to thank him for giving me the latitude to follow also my own interest in the frame of the project, and for encouraging and enabling me to visit many international and national conferences. At the same time I want to thank Prof. Giancarlo Sangalli for co-refereeing this thesis. The results in this thesis were achieved in the NFN project “Geometry + Simulation”, sub-project 3 (NFN S117-03), and in the Doctoral Program “Computational Math- ematics” (W1214), project DK4, which were funded by the Austrian Science Fund (FWF). Both programs provided an excellent scientific environment. This support is gratefully acknowledged. I want to thank the former speaker of the DK, Prof. Peter Paule, and the current speaker, Veronika Pillwein, for their valuable support during my studies. I want to thank the Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences (OAW), at Linz for the permission to use the distributed memory cluster RADON1. I am very grateful for all the fruitful discussions and helpful hints from my colleges of the Institute of Computational Mathematics, Institute of Applied Geometry, the RICAM groups “Computational Methods for PDEs” and “Geometry in Simulations”. Especially, I want to thank Angelos Mantzaflaris for providing assistance for using and contributing to G+Smo and Ioannis Toulopoulos for his extraordinary support and advice during my studies. Finally, I want to express my sincere thanks to my parents, who supported me during all my studies, and to my girlfriend Katharina, for her faith in me, her understanding during difficult periods and her love. Christoph Hofer Linz, April 2018 v vi Contents 1 Introduction 1 2 Preliminaries 11 2.1 Model Problem . 11 2.1.1 Function Spaces . 11 2.1.2 Variational Formulation . 14 2.2 Isogeometric Analysis . 15 2.2.1 Univariate B-Splines . 16 2.2.2 Tensor-Product B-Splines . 18 2.2.3 B-Spline Geometries and Geometrical Mapping . 19 2.2.4 Multi-Patch Geometries . 20 2.2.5 Isogeometric Discretization . 20 2.2.6 Approximation Properties . 21 2.2.7 Important Discrete Inequalities . 22 2.3 Galerkin Discretizatons . 23 2.3.1 Continuous Galerkin Discretization . 23 2.3.2 Discontinuous Galerkin Discretization . 25 3 Continuous Galerkin IETI-DP Methods 29 3.1 Derivation of the Method . 29 3.1.1 Schur Complement System . 30 3.1.2 Local Spaces and Jump Operator . 31 3.1.3 Saddle-Point Formulation . 33 3.1.4 Intermediate Space and Primal Constraints . 35 3.1.5 IETI - DP . 36 3.1.6 Preconditioning . 37 3.1.7 BDDC - Preconditioner . 38 3.2 Implementation of the IETI-DP method . 40 3.2.1 Choosing a Basis for WΠ ...................... 41 −1 3.2.2 Application of Se ......................... 42 3.2.3 Application of Ie and IeT ...................... 43 3.2.4 Application of the Preconditioner . 43 −1 3.2.5 Summary for the Application of F and MsD ........... 44 vii viii CONTENTS 3.3 Analyzing the Condition Number . 45 3.3.1 General Results . 45 3.3.2 Condition Number Estimate . 47 3.4 Numerical Examples . 51 3.4.1 Homogeneous Diffusion Coefficients . 51 3.4.2 Jumping Diffusion Coefficients . 52 3.4.3 Weak Scaling . 52 3.4.4 Dependence on the Degree p .................... 54 4 Discontinuous Galerkin IETI-DP Methods 57 4.1 Derivation of the Method . 58 4.1.1 Basic Setup and Local Space Description . 58 4.1.2 Schur Complement and Discrete Harmonic Extensions . 61 4.1.3 Global Space Description . 63 4.1.4 Intermediate Space and Primal Constraints . 66 4.1.5 IETI-DP and preconditioning . 68 4.2 Analysis of the Condition Number . 69 4.2.1 Auxiliary Results . 70 4.2.2 Condition Number Bound . 79 4.3 Numerical Examples . 82 4.3.1 Homogeneous Diffusion Coefficient . 84 4.3.2 Inhomogeneous Diffusion Coefficient . 84 4.3.3 Dependence on hk=hl ........................ 86 4.3.4 Weak Scaling . 86 4.3.5 Dependence on the Degree p .................... 88 4.3.6 Performance of cG-IETI-DP and dG-IETI-DP Methods . 90 4.4 Segmentation Crimes . 90 4.4.1 Motivation and Setup . 91 4.4.2 Discrete dG-Scheme . 94 4.4.3 Numerical Examples .
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