DOCSLIB.ORG

Domain Decomposition Methods for Problems in H(Curl)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Domain Decomposition Methods for Problems in H(Curl) Domain Decomposition Methods for Problems in H (curl) by Juan Gabriel Calvo A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics New York University September 2015 Professor Olof B. Widlund ©Juan Gabriel Calvo All rights reserved, 2015 Dedication To my family. iv Acknowledgements First, my deepest gratitude goes to my advisor Olof Widlund. I thank him profoundly for his direction, guidance, support and advice during four years. I would also like to thank the rest of my committee: Professors Berger, Good- man, O'Neil and Stadler. In addition, I also thank Dr. Clark Dohrmann of the SANDIA-Albuquerque laboratories for his help and comments throughout my re- search. Finally I thank my Alma Mater, Universidad de Costa Rica, for the support during my academic education at NYU. v Abstract Two domain decomposition methods for solving vector field problems posed in H(curl) and discretized with N´ed´elecfinite elements are considered. These finite elements are conforming in H(curl). A two-level overlapping Schwarz algorithm in two dimensions is analyzed, where the subdomains are only assumed to be uniform in the sense of Peter Jones. The coarse space is based on energy minimization and its dimension equals the number of interior subdomain edges. Local direct solvers are based on the overlapping subdomains. The bound for the condition number depends only on a few geometric parameters of the decomposition. This bound is independent of jumps in the coefficients across the interface between the subdomains for most of the different cases considered. A bound is also obtained for the condition number of a balancing domain de- composition by constraints (BDDC) algorithm in two dimensions, with Jones sub- domains. For the primal variable space, a continuity constraint for the tangential average over each interior subdomain edge is imposed. For the averaging operator, a new technique named deluxe scaling is used. The optimal bound is independent of jumps in the coefficients across the interface between the subdomains. Furthermore, a new coarse function for problems in three dimensions is intro- duced, with only one degree of freedom per subdomain edge. In all the cases, it is established that the algorithms are scalable. Numerical results that verify the results are provided, including some with subdomains with fractal edges and others obtained by a mesh partitioner. vi Contents Dedication . iv Acknowledgements . v Abstract . vi List of Figures . x List of Tables . xii 1 Introduction 1 1.1 An overview . 1 1.2 Iterative methods for solving linear systems . 4 1.3 Organization of the dissertation . 7 2 Finite Element spaces 9 2.1 Sobolev spaces . 9 2.2 Triangulations . 11 2.3 Nodal finite elements . 12 2.4 N´edel´ecfinite elements . 13 2.4.1 N´ed´elecelements in two dimensions . 14 2.4.2 N´ed´elecelements in three dimensions . 15 2.5 An inverse inequality . 17 vii 2.6 A discrete Helmholtz decomposition . 18 3 A model problem 20 3.1 Introduction . 20 3.2 Weak form and discretization . 21 3.3 Notation . 22 3.4 Some implementation details . 25 3.5 Some previous work on vector-valued problems . 27 4 John and Jones domains 29 4.1 Introduction . 29 4.2 Definitions and properties . 30 5 Domain Decomposition methods 35 5.1 Abstract Schwarz theory . 35 5.1.1 Schwarz methods . 35 5.1.2 Convergence theory . 39 5.2 Overlapping Schwarz methods . 41 5.3 BDDC methods . 43 6 An overlapping Schwarz algorithm for N´ed´elecvector fields in 2D 50 6.1 Introduction . 50 6.2 Technical tools . 51 6.2.1 Coarse functions . 52 6.2.2 Cutoff functions . 52 6.2.3 Estimates for auxiliary functions . 55 6.3 Main result . 60 viii 6.3.1 The coarse space component . 60 6.3.2 Local subspaces . 63 6.4 Numerical results . 66 7 A BDDC deluxe algorithm for N´ed´elecvector fields in 2D 80 7.1 Introduction . 80 7.2 Primal constraints . 81 7.3 Deluxe averaging . 82 7.4 Technical tools . 83 7.4.1 Convergence analysis . 83 7.4.2 Discrete curl extensions . 84 7.4.3 Estimates for auxiliary functions . 85 7.4.4 A stability estimate . 87 7.5 Condition number for the BDDC deluxe algorithm . 94 7.6 Numerical results . 96 8 An overlapping Schwarz algorithm for N´ed´elecvector fields in 3D 104 8.1 Introduction . 104 8.2 A coarse space . 105 8.3 Numerical Results . 106 Bibliography 110 ix List of Figures 2.1 Nodal basis function for nodal elements. 13 2.2 N´ed´elecbasis function for edge elements. 15 6.1 Type 1, 2, 3 and METIS subdomains . 67 6.2 Domain decomposition with Type 2 and 3 subdomains . 68 6.3 Domain decomposition with L-shaped subdomains . 71 6.4 Condition number: dependence on H/δ for the two-level overlapping Schwarz method in 2D . 72 6.5 Domain decomposition with short subdomain edges. 72 6.6 Domain decomposition with METIS subdomains in 2D . 74 6.7 Condition number: dependence on the entries of B . 76 6.8 Condition number: dependence on the number of subdomains for B not a multiple of the identity . 77 6.9 Coefficient distribution for problems with discontinuous coefficients inside the subdomains . 79 6.10 Coefficient distribution for problems with discontinuous coefficients inside the subdomains and across the interface . 79 7.1 Domain decomposition with Type 1 and 4 subdomains . 97 x 7.2 Condition number: dependence on H=h for the BDDC deluxe algo- rithm in 2D with Type 1 and 2 subdomains . 99 7.3 Condition number: dependence on H=h for the BDDC deluxe algo- rithm in 2D with Type 4 subdomains . 99 8.1 Domain decomposition with METIS subdomains in 3D . 107 xi List of Tables 3.1 Assembling time for the stiffness matrix in 3D . 27 6.1 Condition numbers and iteration counts: scalability for the two-level overlapping Schwarz method in 2D . 69 6.2 Condition numbers and iteration counts: dependence on H=h for the two-level overlapping Schwarz method in 2D . 69 6.3 Condition numbers and iteration counts: dependence on H/δ for the two-level overlapping Schwarz method in 2D and Type 1, 2 and 3 subdomains . 70 6.4 Condition numbers and iteration counts: dependence on H/δ for the two-level overlapping Schwarz method in 2D and L-shapped subdomains . 70 6.5 Condition numbers and iteration counts: subdomains with small edges for the two-level overlapping Schwarz method in 2D . 71 6.6 Condition numbers and iteration counts: discontinuous and random coefficient distribution for the two-level overlapping Schwarz method in2D................................... 73 6.7 Condition numbers and iteration counts: experiments with METIS subdomains for the two-level overlapping Schwarz method in 2D . 74 xii 6.8 Condition numbers and iteration counts: comparison for additive, hybrid and multiplicative Schwarz operators in 2D . 75 6.9 Condition numbers and iteration counts: discontinuous coefficients inside subdomains for the two-level overlapping Schwarz method in 2D.................................... 78 6.10 Condition numbers and iteration counts: discontinuous coefficients inside subdomains and across the interface for the two-level over- lapping Schwarz method in 2D . 78 7.1 Condition numbers and iteration counts: scalability for the BDDC deluxe method in 2D . 98 7.2 Condition numbers and iteration counts: dependence on H=h for the BDDC deluxe method in 2D and Type 1 and 2 subdomains . 100 7.3 Condition numbers and iteration counts: dependence on H=h for the BDDC deluxe method in 2D and Type 4 subdomains . 100 7.4 Condition numbers and iteration counts: experiments with METIS subdomains for the BDDC deluxe method in 2D . 101 7.5 Condition numbers and iteration counts: discontinuous and random coefficient distribution for the BDDC deluxe method in 2D . 102 7.6 Condition numbers and iteration counts: discontinuous coefficients inside subdomains for the BDDC deluxe method in 2D . 103 7.7 Condition numbers and iteration counts: discontinuous coefficients inside subdomains and across the interface for the BDDC deluxe method in 2D . 103 xiii 8.1 Condition numbers and iteration counts: scalability for the two-level overlapping Schwarz method in 3D . 107 8.2 Condition numbers and iteration counts: dependence on H=h for the two-level overlapping Schwarz method in 3D . 108 8.3 Condition numbers and iteration counts: dependence on H/δ for the two-level overlapping Schwarz method in 3D . 108 8.4 Condition numbers and iteration counts: discontinuous and random coefficient distribution for the two-level overlapping Schwarz method in3D...................................109 xiv Chapter 1 Introduction 1.1 An overview We are interested in solving linear systems that arise from the discretizations of Partial Differential Equations (PDEs). There are different means to obtain an approximate solution for a PDE numerically, such as finite elements and finite differences, which end up with large and ill-conditioned linear systems of algebraic equations. Solving these linear systems is often a hard problem: direct methods require much time and memory, and many iterative methods will converge slowly, because of the large condition numbers of the matrices; see Section 1.2. The main goal in this dissertation will be to construct preconditioners for such linear systems. Domain decomposition refers to the process of subdividing the solution of a large linear system into smaller problems whose solutions can be used to produce a good and scalable preconditioner for the system of equations that results from discretizing the PDE on the entire domain. They are typically used as precondi- 1 tioners for Krylov space iterative methods, such as the Conjugate Gradient (CG) method or the Generalized Minimal Residual (GMRES) method.
Load more

Recommended publications
  • Newton-Krylov-BDDC Solvers for Nonlinear Cardiac Mechanics
    Newton-Krylov-BDDC solvers for nonlinear cardiac mechanics Item Type Article Authors Pavarino, L.F.; Scacchi, S.; Zampini, Stefano Citation Newton-Krylov-BDDC solvers for nonlinear cardiac mechanics 2015 Computer Methods in Applied Mechanics and Engineering Eprint version Post-print DOI 10.1016/j.cma.2015.07.009 Publisher Elsevier BV Journal Computer Methods in Applied Mechanics and Engineering Rights NOTICE: this is the author’s version of a work that was accepted for publication in Computer Methods in Applied Mechanics and Engineering. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Computer Methods in Applied Mechanics and Engineering, 18 July 2015. DOI:10.1016/ j.cma.2015.07.009 Download date 07/10/2021 05:34:35 Link to Item http://hdl.handle.net/10754/561071 Accepted Manuscript Newton-Krylov-BDDC solvers for nonlinear cardiac mechanics L.F. Pavarino, S. Scacchi, S. Zampini PII: S0045-7825(15)00221-2 DOI: http://dx.doi.org/10.1016/j.cma.2015.07.009 Reference: CMA 10661 To appear in: Comput. Methods Appl. Mech. Engrg. Received date: 13 December 2014 Revised date: 3 June 2015 Accepted date: 8 July 2015 Please cite this article as: L.F. Pavarino, S. Scacchi, S. Zampini, Newton-Krylov-BDDC solvers for nonlinear cardiac mechanics, Comput. Methods Appl. Mech. Engrg. (2015), http://dx.doi.org/10.1016/j.cma.2015.07.009 This is a PDF file of an unedited manuscript that has been accepted for publication.
    [Show full text]
  • Preconditioning the Coarse Problem of BDDC Methods—Three-Level, Algebraic Multigrid, and Vertex-Based Preconditioners
    Electronic Transactions on Numerical Analysis. Volume 51, pp. 432–450, 2019. ETNA Kent State University and Copyright c 2019, Kent State University. Johann Radon Institute (RICAM) ISSN 1068–9613. DOI: 10.1553/etna_vol51s432 PRECONDITIONING THE COARSE PROBLEM OF BDDC METHODS— THREE-LEVEL, ALGEBRAIC MULTIGRID, AND VERTEX-BASED PRECONDITIONERS∗ AXEL KLAWONNyz, MARTIN LANSERyz, OLIVER RHEINBACHx, AND JANINE WEBERy Abstract. A comparison of three Balancing Domain Decomposition by Constraints (BDDC) methods with an approximate coarse space solver using the same software building blocks is attempted for the first time. The comparison is made for a BDDC method with an algebraic multigrid preconditioner for the coarse problem, a three-level BDDC method, and a BDDC method with a vertex-based coarse preconditioner. It is new that all methods are presented and discussed in a common framework. Condition number bounds are provided for all approaches. All methods are implemented in a common highly parallel scalable BDDC software package based on PETSc to allow for a simple and meaningful comparison. Numerical results showing the parallel scalability are presented for the equations of linear elasticity. For the first time, this includes parallel scalability tests for a vertex-based approximate BDDC method. Key words. approximate BDDC, three-level BDDC, multilevel BDDC, vertex-based BDDC AMS subject classifications. 68W10, 65N22, 65N55, 65F08, 65F10, 65Y05 1. Introduction. During the last decade, approximate variants of the BDDC (Balanc- ing Domain Decomposition by Constraints) and FETI-DP (Finite Element Tearing and Interconnecting-Dual-Primal) methods have become popular for the solution of various linear and nonlinear partial differential equations [1,8,9, 12, 14, 15, 17, 19, 21, 24, 25].
    [Show full text]
  • Multigrid Solvers for Immersed Finite Element Methods and Immersed Isogeometric Analysis
    Computational Mechanics https://doi.org/10.1007/s00466-019-01796-y ORIGINAL PAPER Multigrid solvers for immersed finite element methods and immersed isogeometric analysis F. de Prenter1,3 · C. V. Verhoosel1 · E. H. van Brummelen1 · J. A. Evans2 · C. Messe2,4 · J. Benzaken2,5 · K. Maute2 Received: 26 March 2019 / Accepted: 10 November 2019 © The Author(s) 2019 Abstract Ill-conditioning of the system matrix is a well-known complication in immersed finite element methods and trimmed isogeo- metric analysis. Elements with small intersections with the physical domain yield problematic eigenvalues in the system matrix, which generally degrades efficiency and robustness of iterative solvers. In this contribution we investigate the spectral prop- erties of immersed finite element systems treated by Schwarz-type methods, to establish the suitability of these as smoothers in a multigrid method. Based on this investigation we develop a geometric multigrid preconditioner for immersed finite element methods, which provides mesh-independent and cut-element-independent convergence rates. This preconditioning technique is applicable to higher-order discretizations, and enables solving large-scale immersed systems at a computational cost that scales linearly with the number of degrees of freedom. The performance of the preconditioner is demonstrated for conventional Lagrange basis functions and for isogeometric discretizations with both uniform B-splines and locally refined approximations based on truncated hierarchical B-splines. Keywords Immersed finite element method · Fictitious domain method · Iterative solver · Preconditioner · Multigrid 1 Introduction [14–21], scan based analysis [22–27] and topology optimiza- tion, e.g., [28–34]. Immersed methods are useful tools to avoid laborious and An essential aspect of finite element methods and iso- computationally expensive procedures for the generation of geometric analysis is the computation of the solution to a body-fitted finite element discretizations or analysis-suitable system of equations.
    [Show full text]
  • Using Algebraic Multigrid in Inexact BDDC Domain Decomposition Methods
    Using Algebraic Multigrid in Inexact BDDC Domain Decomposition Methods Axel Klawonn, Martin Lanser, and Oliver Rheinbach 1 Introduction Traditionally, domain decomposition methods use sparse direct solvers as building blocks, i.e., to solve local subdomain problems and/or the coarse problem. Often, the sparse direct solvers can be replaced by spectrally equivalent preconditioners without loss of convergence speed. In FETI-DP and BDDC domain decomposition methods, such approaches have first been introduced in [9, 8, 4], and have since then successfully been used in large parallel codes [6, 1]. 2 An Inexact BDDC Method 2.1 A BDDC Preconditioner for the Assembled System Let us briefly describe the BDDC preconditioner which can directly be applied to a linear system Au = b (1) arising from a finite element discretization of a partial differential equation on a d computational domain W ⊂ R ; d = 2;3. The variant discussed here was first intro- duced in [9]. Let Wi; i = 1;:::;N; be a nonoverlapping domain decomposition of SN W such that W = i=1 W i. Each subdomain Wi is discretized using finite elements, Axel Klawonn, Martin Lanser Mathematisches Institut, Universitat¨ zu Koln,¨ Weyertal 86-90, 50931 Koln,¨ Germany, e-mail: faxel.klawonn,[email protected] Oliver Rheinbach Institut fur¨ Numerische Mathematik und Optimierung, Fakultat¨ fur¨ Mathematik und Infor- matik, Technische Universitat¨ Bergakademie Freiberg, Akademiestr. 6, 09596 Freiberg, e-mail: [email protected] 1 2 Axel Klawonn, Martin Lanser, and Oliver Rheinbach the corresponding local finite element spaces are denoted by Wi; i = 1;:::;N, and the product space is defined by W = W1 ×:::×WN.
    [Show full text]
  • Nonlinear Bddc Methods with Approximate Solvers∗
    Electronic Transactions on Numerical Analysis. Volume 49, pp. 244–273, 2018. ETNA Kent State University and Copyright c 2018, Kent State University. Johann Radon Institute (RICAM) ISSN 1068–9613. DOI: 10.1553/etna_vol49s244 NONLINEAR BDDC METHODS WITH APPROXIMATE SOLVERS∗ AXEL KLAWONNyz, MARTIN LANSERyz, AND OLIVER RHEINBACHx Abstract. New nonlinear BDDC (Balancing Domain Decomposition by Constraints) domain decomposition methods using inexact solvers for the subdomains and the coarse problem are proposed. In nonlinear domain decomposition methods, the nonlinear problem is decomposed before linearization to improve concurrency and robustness. For linear problems, the new methods are equivalent to known inexact BDDC methods. The new approaches are therefore discussed in the context of other known inexact BDDC methods for linear problems. Relations are pointed out, and the advantages of the approaches chosen here are highlighted. For the new approaches, using an algebraic multigrid method as a building block, parallel scalability is shown for more than half a million (524 288) MPI ranks on the JUQUEEN IBM BG/Q supercomputer (JSC Jülich, Germany) and on up to 193 600 cores of the Theta Xeon Phi supercomputer (ALCF, Argonne National Laboratory, USA), which is based on the recent Intel Knights Landing (KNL) many-core architecture. One of our nonlinear inexact BDDC domain decomposition methods is also applied to three-dimensional plasticity problems. Comparisons to standard Newton-Krylov-BDDC methods are provided. Key words. nonlinear BDDC, nonlinear domain decomposition, nonlinear elimination, Newton’s method, nonlinear problems, parallel computing, inexact BDDC, nonlinear elasticity, plasticity AMS subject classifications. 68W10, 68U20, 65N55, 65F08, 65Y05 1. Introduction. Nonlinear BDDC (Balancing Domain Decomposition by Constraints) domain decomposition methods were introduced in [24] for the parallel solution of nonlinear problems and can be viewed as generalizations of the well-known family of BDDC methods for linear elliptic problems [11, 15, 35, 37].
    [Show full text]
  • 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.
    [Show full text]
  • Software Concepts and Algorithms for an Efficient and Scalable Parallel Finite Element Method
    Fakultät Mathematik und Naturwissenschaften, Institut für Wissenschaftliches Rechnen SOFTWARE CONCEPTS AND ALGORITHMS FOR AN EFFICIENT AND SCALABLE PARALLEL FINITE ELEMENT METHOD Der Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden zur Erlangung des akademischen Grades Dr. rer. nat. eingereichte Dissertation von Thomas Witkowski geboren am 24. Mai 1982 in Loslau/Polen. Die Dissertation wurde in der Zeit von 07/2007 bis 12/2012 am Institut für Wissenschaftliches Rechnen angefertigt. Tag der Einreichung: ... Tag der Verteidigung: ... Gutachter: Prof. Dr. rer. nat. habil. A. Voigt Technische Universität Dresden ... ... Contents 1 Introduction 5 1.1 Overview ..................................... 6 1.2 Technicalnotes .................................. 6 2 Adaptivemeshesforfiniteelementmethod 9 2.1 Data structures of adaptive meshes . 9 2.2 Error estimation and adaptive strategies . 10 2.3 Meshstructurecodes............................... 11 3 Scalable parallelization 15 3.1 Formaldefinitions ................................ 18 3.2 Distributedmeshes................................ 20 3.2.1 Mesh structure codes for parallel mesh adaptivity . 23 3.2.2 Mesh partitioning and mesh distribution . 25 3.2.3 Parallel DOF mapping . 26 3.2.4 Efficiency and parallel scaling . 29 3.2.5 Limitations of coarse element based partitioning . 32 3.3 Linearsolvermethods .............................. 33 3.4 FETI-DP ..................................... 35 3.4.1 Implementationissues . 39 3.4.2 Numerical results . 42 3.5 Extensions of the standard FETI-DP . 48 3.5.1 InexactFETI-DP............................. 48 3.5.2 MultilevelFETI-DP ........................... 50 3.6 ANavier-Stokessolver .............................. 57 3.6.1 Implementationissues . 59 3.6.2 Numerical results . 60 3.6.3 Diffuse domain approach . 61 3.7 Softwareconcepts................................. 62 4 Multi-mesh method for Lagrange finite elements 69 4.1 Virtualmeshassembling............................. 70 4.1.1 CouplingtermsinsystemsofPDEs .
    [Show full text]
  • Fakultät Für Mathematik Und Informatik
    Fakultät für Mathematik und Informatik Preprint 2017-04 Axel Klawonn, Martin Kühn, and Oliver Rheinbach Adaptive FETI-DP and BDDC Methods with a Generalized Transformation of Basis for Heterogeneous Problems ISSN 1433-9307 Axel Klawonn, Martin Kühn, Oliver Rheinbach Adaptive FETI-DP and BDDC Methods with a Generalized Transformation of Basis for Heterogeneous Problems TU Bergakademie Freiberg Fakultät für Mathematik und Informatik Prüferstraße 9 09599 FREIBERG http://tu-freiberg.de/fakult1 ISSN 1433 – 9307 Herausgeber: Dekan der Fakultät für Mathematik und Informatik Herstellung: Medienzentrum der TU Bergakademie Freiberg ADAPTIVE FETI-DP AND BDDC METHODS WITH A GENERALIZED TRANSFORMATION OF BASIS FOR HETEROGENEOUS PROBLEMS AXEL KLAWONN∗, MARTIN KÜHN∗, AND OLIVER RHEINBACH† July 26, 2017 Abstract. In FETI-DP (Finite Element Tearing and Interconnecting) and BDDC (Balancing Domain Decomposition by Constraints) domain decomposition methods, the transformation-of-basis approach is used to improve the convergence by combining local assembly with a change of basis. Suitable basis vectors can be constructed by recently introduced adaptive coarse space approaches. The resulting FETI-DP and BDDC methods fulfill a condition number bound independent of hetero- geneities in the problem. The adaptive method with a transformation of basis presented here builds on a recently introduced adaptive FETI-DP approach for elliptic problems in three dimensions and uses a coarse space constructed from solving small, local eigenvalue problems on closed faces and on a small number of edges. In contrast to our earlier work on adaptive FETI-DP, the coarse space correction is not implemented using balancing (or deflation), which requires the use of an exact coarse space solver, but using local transformations.
    [Show full text]
  • 65 Numerical Analysis
    e Q (e t o 5 M SectionsSet 1Q (Section 65)MR September 2012 65 NUMERICAL ANALYSIS MR2918625 65-06 FRecent advances in scientific computing and matrix analysis. Proceedings of the International Workshop held at the University of Macau, Macau, December 28{30, 2009. Edited by Xiao-Qing Jin, Hai-Wei Sun and Seak-Weng Vong. International Press, Somerville, MA; Higher Education Press, Beijing, 2011. xii+126 pp. ISBN 978-1-57146-202-2 Contents: Zheng-jian Bai and Xiao-qing Jin [Xiao Qing Jin1], A note on the Ulm-like method for inverse eigenvalue problems (1{7) MR2908437; Che-man Cheng [Che-Man Cheng], Kin-sio Fong [Kin-Sio Fong] and Io-kei Lok [Io-Kei Lok], Another proof for commutators with maximal Frobenius norm (9{14) MR2908438; Wai-ki Ching [Wai-Ki Ching] and Dong-mei Zhu [Dong Mei Zhu1], On high-dimensional Markov chain models for categorical data sequences with applications (15{34) MR2908439; Yan-nei Law [Yan Nei Law], Hwee-kuan Lee [Hwee Kuan Lee], Chao-qiang Liu [Chaoqiang Liu] and Andy M. Yip, An additive variational model for image segmentation (35{48) MR2908440; Hai-yong Liao [Haiyong Liao] and Michael K. Ng, Total variation image restoration with automatic selection of regularization parameters (49{59) MR2908441; Franklin T. Luk and San-zheng Qiao [San Zheng Qiao], Matrices and the LLL algorithm (61{69) MR2908442; Mila Nikolova, Michael K. Ng and Chi-pan Tam [Chi-Pan Tam], A fast nonconvex nonsmooth minimization method for image restoration and reconstruction (71{83) MR2908443; Gang Wu [Gang Wu1], Eigenvalues of certain augmented complex stochastic matrices with applications to PageRank (85{92) MR2908444; Yan Xuan and Fu-rong Lin, Clenshaw-Curtis-rational quadrature rule for Wiener-Hopf equations of the second kind (93{110) MR2908445; Man-chung Yeung [Man-Chung Yeung], On the solution of singular systems by Krylov subspace methods (111{116) MR2908446; Qi- fang Yu, San-zheng Qiao [San Zheng Qiao] and Yi-min Wei, A comparative study of the LLL algorithm (117{126) MR2908447.
    [Show full text]
  • BDDC Domain Decomposition Algorithms
    BDDC Domain Decomposition Algorithms Olof B. Widlund Courant Institute, New York University 75th Anniversary of Mathematics of Computation November 3, 2018 Olof B. Widlund BDDC Domain Decomposition Algorithms In this talk, I will introduce the basics of BDDC, give examples of successful applications and talk about some recent work. My last few PhD students were all involved in the development of the BDDC family. I have also worked with Dohrmann and with Beir~aoda Veiga, Pavarino, Scacchi, Zampini, Oh, and Calvo. Recently, the focus has been on small coarse problems for BDDC, adaptive choices of the coarse problems, and isogeometric analysis problems for elasticity including the almost incompressible case.. Problems considered BDDC stands for Balancing Domain Decomposition by Constraints and this family of algorithms was introduced by Clark Dohrmann in 2003 following the introduction of the FETI{DP (Dual Primal Finite Element Tearing and Interconnecting) algorithms by Charbel Farhat et al in 2000. Dohrmann remains a main provider of ideas and analysis. Olof B. Widlund BDDC Domain Decomposition Algorithms My last few PhD students were all involved in the development of the BDDC family. I have also worked with Dohrmann and with Beir~aoda Veiga, Pavarino, Scacchi, Zampini, Oh, and Calvo. Recently, the focus has been on small coarse problems for BDDC, adaptive choices of the coarse problems, and isogeometric analysis problems for elasticity including the almost incompressible case.. Problems considered BDDC stands for Balancing Domain Decomposition by Constraints and this family of algorithms was introduced by Clark Dohrmann in 2003 following the introduction of the FETI{DP (Dual Primal Finite Element Tearing and Interconnecting) algorithms by Charbel Farhat et al in 2000.
    [Show full text]
  • Parallel Adaptive FETI-DP Using Lightweight Asynchronous Dynamic Load Balancing
    Received: Added at production Revised: Added at production Accepted: Added at production DOI: xxx/xxxx RESEARCH ARTICLE Parallel adaptive FETI-DP using lightweight asynchronous dynamic load balancing Axel Klawonn1,2 | Martin J. Kühn3 | Oliver Rheinbach4 1Department of Mathematics and Computer Science, University of Cologne, Germany Summary 2Center for Data and Simulation Science, University of Cologne, Germany A parallel FETI-DP domain decomposition method using an adaptive coarse space 3Parallel Algorithms Team, CERFACS is presented. The implementation builds on a recently introduced adaptive FETI-DP (Centre Européen de Recherche et de approach for elliptic problems in three dimensions and uses small, local eigenvalue Formation Avancée en Calcul Scientifique), France problems for faces and, additionally, for a small number of edges. The condition 4Institut für Numerische Mathematik und number of the preconditioned operator then satisfies a bound which is independent Optimierung, Technische Universität of coefficient heterogeneities in the problem. The computational cost of the local Bergakademie Freiberg, Germany eigenvalue problems is not negligible, and also a significant load imbalance can be Correspondence introduced. As a remedy, certain eigenvalue problems are discarded by a theory- Martin J. Kühn, Parallel Algorithms Team, CERFACS (Centre Européen de Recherche guided heuristic strategy, based on the diagonal entries of the stiffness matrices. et de Formation Avancée en Calcul Additionally, a lightweight pairwise dynamic load balancing strategy is implemented Scientifique). Email: for the eigenvalue problems. The load balancing is supervised by an orchestrating [email protected] rank using asynchronous point-to-point communication. The resulting method shows Present Address good weak and strong scalability up to thousands of cores while fast convergence is 42 Avenue Gaspard Coriolis, 31057 Toulouse Cedex 01, France obtained even for heterogeneous problems.
    [Show full text]
  • Substructuring Preconditioners for H-P Mortar FEM Silvia Bertoluzza, Micol Pennacchio, Christophe Prud’Homme, Abdoulaye Samake
    Substructuring Preconditioners for h-p Mortar FEM Silvia Bertoluzza, Micol Pennacchio, Christophe Prud’Homme, Abdoulaye Samake To cite this version: Silvia Bertoluzza, Micol Pennacchio, Christophe Prud’Homme, Abdoulaye Samake. Substructuring Preconditioners for h-p Mortar FEM. ESAIM: Mathematical Modelling and Numerical Analysis, EDP Sciences, 2016, 50 (4), pp.1057-1082. 10.1051/m2an/2015065. hal-01066606 HAL Id: hal-01066606 https://hal.archives-ouvertes.fr/hal-01066606 Submitted on 22 Sep 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. SUBSTRUCTURING PRECONDITIONERS FOR h-p MORTAR FEM. SILVIA BERTOLUZZA, MICOL PENNACCHIO, CHRISTOPHE PRUD’HOMME, AND ABDOULAYE SAMAKE Abstract. We build and analyze a substructuring preconditioner for the mortar method in the h-p finite element framework. Particular attention is given to the construction of the coarse component of the preconditioner in this framework, in which continuity at the cross points is not required. Two variants are proposed: the first one is an improved version of a coarse preconditioner already presented in [12]. The second is new and is built by using a Discontinuous Galerkin interior penalty method as coarse problem. A bound of the condition number is proven for both variants and their efficiency and scalability is illustrated by numerical experiments.
    [Show full text]