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Domain Decomposition Preconditioners for Higher-Order Domain Decomposition Preconditioners for Higher-Order Discontinuous Galerkin Discretizations by Laslo Tibor Diosady S.M., Massachusetts Institute of Technology (2007) B.A.Sc., University of Toronto (2005) Submitted to the Department of Aeronautics and Astronautics ACHIES in partial fulfillment of the requirements for the degree of A CHE S Doctor of Philosophy Y at the APR 2 MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2011 @ Massachusetts Institute of Technology 2011. All rights reserved. Author.................................. o A a s-ona-es Department of Aeronautics and Astron201s C(1-1 A -r-,\ Sept 2A2011 C ertified by .............................. David Darmofal Professor of Aeronautics and Astronautics /,* ThefisSupervisor C ertified by .............................. ........... Alan Edelman Pr es or of Mathematics i~ t=----is Committee C ertified by .............................. .......... Jaime Peraire Professorbf Aeronautics and Astronautics Thesis Committee A ccepted by .............................. Eytan H. Modiano Professor of Aeronautics and Astronautics Chair, Committee on Graduate Students 2 Domain Decomposition Preconditioners for Higher-Order Discontinuous Galerkin Discretizations by Laslo Tibor Diosady Submitted to the Department of Aeronautics and Astronautics on Sept 23, 2011, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Aerodynamic flows involve features with a wide range of spatial and temporal scales which need to be resolved in order to accurately predict desired engineering quantities. While computational fluid dynamics (CFD) has advanced considerably in the past 30 years, the desire to perform more complex, higher-fidelity simulations remains. Present day CFD simu- lations are limited by the lack of an efficient high-fidelity solver able to take advantage of the massively parallel architectures of modern day supercomputers. A higher-order hybridizable discontinuous Galerkin (HDG) discretization combined with an implicit solution method is proposed as a means to attain engineering accuracy at lower computational cost. Domain decomposition methods are studied for the parallel solution of the linear system arising at each iteration of the implicit scheme. A minimum overlapping additive Schwarz (ASM) preconditioner and a Balancing Do- main Decomposition by Constraints (BDDC) preconditioner are developed for the HDG discretization. An algebraic coarse space for the ASM preconditioner is developed based on the solution of local harmonic problems. The BDDC preconditioner is proven to converge at a rate independent of the number of subdomains and only weakly dependent on the solution order or the number of elements per subdomain for a second-order elliptic problem. The BDDC preconditioner is extended to the solution of convection-dominated problems using a Robin-Robin interface condition. An inexact BDDC preconditioner is developed based on incomplete factorizations and a p-multigrid type coarse grid correction. It is shown that the incomplete factorization of the singular linear systems corresponding to local Neumann problems results in a non- singular preconditioner. The inexact BDDC preconditioner converges in a similar number of iterations as the exact BDDC method, with significantly reduced CPU time. The domain decomposition preconditioners are extended to solve the Euler and Navier- Stokes systems of equations. An analysis is performed to determine the effect of boundary conditions on the convergence of domain decomposition methods. Optimized Robin-Robin interface conditions are derived for the BDDC preconditioner which significantly improve the performance relative to the standard Robin-Robin interface conditions. Both ASM and BDDC preconditioners are applied to solve several fundamental aerodynamic flows. Numerical results demonstrate that for high-Reynolds number flows, solved on anisotropic meshes, a coarse space is necessary in order to obtain good performance on more than 100 processors. Thesis Supervisor: David L. Darmofal Title: Professor of Aeronautics and Astronautics 4 Acknowledgments I would like to thank all of those who helped make this work possible. First, I would like to thank my advisor Prof. David Darmofal, for giving me the opportunity to work with him. I am grateful for his guidance and encouragement throughout my studies at MIT. Second, I would like to thank my committee members Profs. Alan Edelman and Jaime Peraire for their criticism and direction throughout my PhD work. I would also like to thank my thesis readers Prof. Qiqi Wang and Dr. Venkat Venkatakrishnan for providing comments and suggestions on improving this thesis. This work would not have been possible without the contributions of the ProjectX team past and present (Julie Andren, Garrett Barter, Krzysztof Fidkowski, Bob Haimes, Josh Krakos, Eric Liu, JM Modisette, Todd Oliver, Mike Park, Huafei Sun, Masa Yano). Partic- ularly, I would like to acknowledge JM whose been a good friend and office mate my entire time at MIT. I would like to thank my wife, Laura, my parents, Klara and Levente, and my brother, Andrew, for their encouragement throughout my time at MIT. Without their love and sup- port this work would never have been completed. Finally, I would like to acknowledge the financial support I have received throughout my graduate studied. This work was supported by funding from MIT through the Zakhartchenko Fellowship and from The Boeing Company under technical monitor Dr. Mori Mani. This research was also supported in part by the National Science Foundation through TeraGrid resources provided under grant number TG-DMS110018. Contents 1 Introduction 1.1 Motivation .......... ................. 1.2 Background .......... ....... .......... 1.2.1 Higher-order Discontinu ous Galerkin Methods .... 1.2.2 Scalable Implicit CFD Thr................ 1.2.3 Domain Decomposition Theory ..... ....... 1.2.4 Large Scale CFD simul ations ...... ....... 1.2.5 Balancing Domain Dec )mposition by Constraints . 1.3 Thesis Overview ....... .. ............. .. 1.3.1 Objectives ....... ................. 1.3.2 Thesis Contributions . ................. 2 Discretization and Governing Equations 2.1 HDG Discretization ............ ........... 2.2 Compressible Navier-Stokes Equations ............. 2.3 Boundary Conditions ............ ........... 2.4 Solution M ethod ...... ................... 3 Domain Decomposition and the Additive Schwarz Method 3.1 Domain Decomposition .............. ....... 3.2 Additive Schwarz method .................... 3.3 Numerical Results ................. ....... 4 Balancing Domain Decomposition by Constraints 4.1 A Schur Complement Problem .. ............... 4.2 The BDDC Preconditioner ................... 4.3 The BDDC Preconditioner Revisited .............. 4.4 Convergence Analysis ............... ....... 4.5 Robin-Robin Interface Condition ................ 4.6 Numerical Results ........................ 5 Inexact BDDC 5.1 A Note on the BDDC Implementation ...... ....... 5.2 BDDC using Incomplete Factorizations ............. 5.2.1 Inexact Harmonic Extensions .............. 5.2.2 Inexact Partially Assembled Solve ........... 5.2.3 One or Two matrix method ........................ 81 5.3 Inexact BDDC with p-multigrid correction .................... 82 5.3.1 Inexact Harmonic Extensions ............. .......... 82 5.3.2 Inexact Partially Assembled Solve .................... 83 5.4 Numerical Results ..... ............................ 84 5.4.1 Inexact BDDC using Incomplete Factorizations ............. 85 5.4.2 Inexact BDDC with p-multigrid correction ................ 89 6 Euler and Navier-Stokes Systems 101 6.1 Linearized Euler Equations ........... ................. 101 6.2 1D Analysis ......... ............................ 103 6.3 2D Analysis . * .... ... .. ... .. .. 107 6.4 Optimized Robin Interface Condition ....................... 111 6.5 Numerical Results ....... ............. ............. 118 7 Application to Aerodynamics Flows 127 7.1 HDG Results . .............. .............. ....... 127 7.2 D G Results ............... ...................... 133 8 Conclusions 143 8.1 Summary and Conclusions .............. ............. .. 143 8.2 Recommendations and Future Work ........ ............. .. 145 Bibliography 147 A 2D Euler Analysis: Infinite Domain 157 B BDDC for DG Discretizations 163 B.1 DG Discretization .... .............. ............. .. 164 B.2 The DG discretization from a new perspective .................. 167 B .3 BD D C . ... ................. 172 B .4 A nalysis .......... .............. ............. .. 174 B.5 Numerical Results ........................... ...... 178 List of Figures 3-1 Sample grid and non-overlapping partition ......... ........... 38 3-2 Overlapping partition ... ............................ 39 3-3 Additive Schwarz method applied to sample Poisson problem ......... 40 3-4 Coarse basis functions for additive Schwarz method ............... 42 3-5 Additive Schwarz method with coarse space applied to sample Poisson problem 43 3-6 Plot of solution for 2d scalar model problems ............ ...... 44 3-7 Plot of meshes used 2d scalar model problems ............. ..... 45 3-8 GMRES convergence history for advection-diffusion boundary layer problem with p = 10-6, p = 2 and n = 128 on isotropic structured mesh ........ 48 4-1 Coarse basis functions for BDDC . .............. .......... 59 4-2 Partially Assembled solve applied to sample Poisson problem ......... 59 4-3 Effect of harmonic extensions for sample Poisson problem ........... 59 4-4
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