Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations

774 Pages·2008·7.94 MB·English

by  Tarek Poonithara Abraham Mathew (auth.)

Lecture Notes
in Computational Science 61
and Engineering
Editors
TimothyJ.Barth
MichaelGriebel
DavidE.Keyes
RistoM.Nieminen
DirkRoose
TamarSchlick
Tarek P. A. Mathew
Domain Decomposition
Methods for the Numerical
Solution of Partial
Differential Equations
With40Figuresand1Table
ABC
TarekPoonitharaAbrahamMathew
[email protected]
ISBN978-3-540-77205-7 e-ISBN978-3-540-77209-5
LectureNotesinComputationalScienceandEngineeringISSN1439-7358
LibraryofCongressControlNumber:2008921994
MathematicsSubjectClassification(2000):65F10,65F15,65N22,65N30,65N55,
65M15,65M55,65K10
(cid:1)c 2008Springer-VerlagBerlinHeidelberg
Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis
concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,
reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication
orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,
1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare
liableforprosecutionundertheGermanCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply,
evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws
andregulationsandthereforefreeforgeneraluse.
Coverdesign:WMXDesignGmbH,Heidelberg
Printedonacid-freepaper
9 8 7 6 5 4 3 2 1
spinger.com
In loving dedication to my (late) dear mother,
and to my dear father and brother
Preface
These notes serve as an introduction to a subject of study in computational
mathematics referred to as domain decomposition methods. It concerns divide
and conquermethodsforthenumericalsolutionandapproximationofpartial
differential equations, primarily of elliptic or parabolic type. The methods in
this family include iterative algorithms for the solution of partial differential
equations, techniques for the discretization of partial differential equations
on non-matching grids, and techniques for the heterogeneous approximation
of partial differential equations of heterogeneous character. The divide and
conquer methodology used is based on a decomposition of the domain of the
partial differential equation into smaller subdomains, and by design is suited
forimplementationonparallelcomputerarchitectures.However,evenonserial
computers, these methods can provide flexibility in the treatment of complex
geometry and heterogeneities in a partial differential equation.
Interest in this family of computational methods for partial differential
equations was spawned following the development of various high perfor-
mance multiprocessor computer architectures in the early eighties. On such
parallel computer architectures, the execution time of these algorithms, as
well as the memory requirements per processor, scale reasonably well with
the size of the problem and the number of processors. From a computational
viewpoint, the divide and conquer methodology based on a decomposition
of the domain of the partial differential equation, yields algorithms having
coarse granularity, i.e., a significant portion of the computations can be im-
plemented concurrently on different processors, while the remaining portion
requires communication between the processors. As a consequence, these al-
gorithms are well suited for implementation on MIMD (multiple instruction,
multiple data) architectures. Currently, such parallel computer architectures
canalternativelybesimulatedusingacluster ofworkstationsnetworkedwith
highspeedconnectionsusingcommunicationprotocolssuchasMPI(Message
Passing Interface) [GR15] or PVM (Parallel Virtual Machines) [GE2].
VIII Preface
The mathematical roots of this subject trace back to the seminal work of
H.A.Schwarz[SC5]inthenineteenthcentury.Schwarzproposedaniterative
method, now referred to as the Schwarz alternating method, for constructing
harmonic functions on regions of irregular shape which can be expressed as
theunion of subregionsof regular shape (such asrectangles and spheres). His
motivation was primarily theoretical, to establish the existence of harmonic
functions on irregular regions, and his method was not used in computations
until recently [SO, MO2, BA2, MI, MA37, DR11, LI6, LI7, BR18].
A general development of domain decomposition methodology for par-
tial differential equations occurred only subsequent to the development of
parallel computer architectures, though divide and conquer methods such as
Kron’s method for electrical circuits [KR] and the substructuring method
[PR4] in structural engineering, pre-date domain decomposition methodol-
ogy. Usage of the term “domain decomposition” seems to have originated
around the mid-eighties [GL2] when interest in these methods gained mo-
mentum. The first international symposium on this subject was held in Paris
in 1987, and since then there have been yearly international conferences on
this subject, attracting interdisciplinary interest from communities of engi-
neers, applied scientists and computational mathematicians from around the
globe.
Early literature on domain decomposition methods focused primarily on
iterative procedures for the solution of partial differential equations. As the
methodology evolved, however, techniques were also developed for coupling
discretizations on subregions with non-matching grids, and for constructing
heterogeneous approximations of complicated systems of partial differential
equationshavingheterogeneouscharacter.Thelatterapproximationsarebuilt
by solving local equations of different character. From a mathematical view-
point, these diverse categories of numerical methods for partial differential
equations may be derived within several frameworks. Each decomposition of
a domain typically suggests a reformulation of the original partial differen-
tial equation as an equivalent coupled system of partial differential equations
posed on the subdomains with boundary conditions chosen to match solu-
tions on adjacent subdomains. Such equivalent systems are referred to in
these notes as hybrid formulations, and provide a framework for develop-
ingnoveldomaindecompositionmethods.Divideandconqueralgorithmscan
be obtained by numerical approximation of hybrid formulations. Four hybrid
formulations are considered in these notes, suited for equations primarily of
elliptic type:
• The Schwarz formulation.
• The Steklov-Poincar´e (substructuring or Schur complement) formulation.
• The Lagrange multiplier formulation.
• The Least squares-control formulation.
Alternative hybrid formulations are also possible, see [CA7, AC5].
Preface IX
The applicability and stability of each hybrid formulation depends on
the underlying partial differential equation and subdomain decomposition.
For instance, the Schwarz formulation requires an overlapping decomposi-
tion, while the Steklov-Poincar´e and Lagrange multiplier formulations are
based on a non-overlapping decomposition. The least squares-control method
can be formulated given overlapping or non-overlapping decompositions.
Within each framework, novel iterative methods, discretizations schemes on
non-matching grids, and heterogeneous approximations of the original par-
tial differential equation, can be developed based on the associated hybrid
formulations.
In writing these notes, the author has attempted to provide an accessible
introduction to the important methodologies in this subject, emphasizing a
matrix formulation of algorithms. However, as the literature on domain de-
composition methods is vast, various topics have either been omitted or only
touched upon. The methods described here apply primarily to equations of
ellipticorparabolictype,andapplicationstohyperbolicequations[QU2],and
spectral or p-version elements have been omitted [BA4, PA16, SE2, TO10].
Applications to the equations of elasticity and to Maxwell’s equations have
also been omitted, see [TO10]. Parallel implementation is covered in greater
depth in [GR12, GR10, FA18, FA9, GR16, GR17, HO4, SM5, BR39]. For
additional domain decomposition theory, see [XU3, DR10, XU10, TO10]. A
broader discussion on heterogeneous domain decomposition can be found in
[QU6], and on FETI-DP and BDDC methods in [TO10, MA18, MA19]. For
additional bibliography on domain decomposition, see http://www.ddm.org.
Readers are assumed to be familiar with the basic properties of ellip-
tic and parabolic partial differential equations [JO, SM7, EV] and tradi-
tional methods for their discretization [RI, ST14, CI2, SO2, JO2, BR28, BR].
Familiarity is also assumed with basic numerical analysis [IS, ST10], com-
putational linear algebra [GO4, SA2, AX, GR2, ME8], and elements of op-
timization theory [CI4, DE7, LU3, GI2]. Selected background topics are re-
viewed in various sections of these notes. Chap. 1 provides an overview of
domain decomposition methodology in a context involving two subdomain
decompositions.Fourdifferenthybridformulationsareillustratedforamodel
coercive 2nd order elliptic equation. Chapters 2, 3 and 4 describe the ma-
trix implementation of multisubdomain domain decomposition iterative al-
gorithms for traditional discretizations of self adjoint and coercive elliptic
problems. These chapters should ideally be read prior to the other chapters.
Readers unfamiliar with constrained minimization problems and their saddle
point formulation, may find it useful to review background in Chap. 10 or
in [CI4], as saddle point methodology is employed in Chaps. 1.4 and 1.5 and
in Chaps. 4 and 6. With a few exceptions, the remaining chapters may be
read independently.
X Preface
The author expresses his deep gratitude to the anonymous referees who
made numerous suggestions for revision and improvement of the manuscript.
Deep gratitude is also expressed to Prof. Olof Widlund who introduced the
authortothissubjectovertwentyyearsago,toProf.TonyChan,forhiskind
encouragement to embark on writing a book extending our survey paper on
this subject [CH11], and to Prof. Xiao-Chuan Cai, Prof. Marcus Sarkis and
Prof.JunpingWangfortheirresearchcollaborationsandnumerousinsightful
discussions over the years. The author deeply thanks Prof. Timothy Barth
for his kind permission to use the figure on the cover of this book, and for
use of Fig. 5.1. To former colleagues at the University of Wyoming, and to
professors Myron Allen, Gastao Braga, Benito Chen, Duilio Conceic¸˜ao, Max
Dryja,FredericoFurtado,JuanGalvis,EtereldesGonc¸alves,RaytchoLazarov,
Mary Elizabeth Ong, Peter Polyakov, Giovanni Russo, Christian Schaerer,
Shagi-Di Shih, Daniel Szyld, Panayot Vassilevski and Henrique Versieux, the
authorexpresseshisdeepgratitude.Deepappreciationisalsoexpressedtothe
editors of the LNCSE series, Dr. Martin Peters, Ms. Thanh-Ha LeThi, and
Mr. Frank Holzwarth for their patience and kind help during the completion
of this manuscript. Finally, deep appreciation is expressed to Mr. Elumalai
Balamurugan for his kind assistance with reformatting the text. The author
welcomescommentsandsuggestionsfromreaders,andhopestopostupdates
at www.poonithara.org/publications/dd.
January 2008 Tarek P. A. Mathew
Contents
1 Decomposition Frameworks................................ 1
1.1 Hybrid Formulations..................................... 2
1.2 Schwarz Framework...................................... 9
1.3 Steklov-Poincar´e Framework .............................. 16
1.4 Lagrange Multiplier Framework ........................... 27
1.5 Least Squares-Control Framework ......................... 36
2 Schwarz Iterative Algorithms .............................. 47
2.1 Background............................................. 48
2.2 Projection Formulation of Schwarz Algorithms .............. 56
2.3 Matrix Form of Schwarz Subspace Algorithms .............. 66
2.4 Implementational Issues .................................. 72
2.5 Theoretical Results ...................................... 77
3 Schur Complement and Iterative
Substructuring Algorithms ................................107
3.1 Background.............................................108
3.2 Schur Complement System ...............................110
3.3 FFT Based Direct Solvers ................................125
3.4 Two Subdomain Preconditioners ..........................140
3.5 Preconditioners in Two Dimensions ........................155
3.6 Preconditioners in Three Dimensions.......................162
3.7 Neumann-Neumann and Balancing Preconditioners ..........175
3.8 Implementational Issues ..................................185
3.9 Theoretical Results ......................................192
4 Lagrange Multiplier Based Substructuring:
FETI Method .............................................231
4.1 Constrained Minimization Formulation.....................232
4.2 Lagrange Multiplier Formulation ..........................239
4.3 Projected Gradient Algorithm.............................241
4.4 FETI-DP and BDDC Methods ............................250
XII Contents
5 Computational Issues and Parallelization ..................263
5.1 Algorithms for Automated Partitioning of Domains ..........264
5.2 Parallelizability of Domain Decomposition Solvers ...........280
6 Least Squares-Control Theory: Iterative Algorithms ......295
6.1 Two Overlapping Subdomains.............................296
6.2 Two Non-Overlapping Subdomains ........................303
6.3 Extensions to Multiple Subdomains........................310
7 Multilevel and Local Grid Refinement Methods ...........313
7.1 Multilevel Iterative Algorithms............................314
7.2 Iterative Algorithms for Locally Refined Grids ..............321
8 Non-Self Adjoint Elliptic Equations: Iterative Methods ....333
8.1 Background.............................................334
8.2 Diffusion Dominated Case ................................340
8.3 Advection Dominated Case ...............................348
8.4 Time Stepping Applications ..............................364
8.5 Theoretical Results ......................................366
9 Parabolic Equations .......................................377
9.1 Background.............................................378
9.2 Iterative Algorithms .....................................381
9.3 Non-Iterative Algorithms .................................384
9.4 Parareal-Multiple Shooting Method........................401
9.5 Theoretical Results .....................................408
10 Saddle Point Problems ....................................417
10.1 Properties of Saddle Point Systems ........................418
10.2 Algorithms Based on Duality .............................426
10.3 Penalty and Regularization Methods.......................434
10.4 Projection Methods......................................437
10.5 Krylov Space and Block Matrix Methods ...................445
10.6 Applications to the Stokes and Navier-Stokes Equations ......456
10.7 Applications to Mixed Formulations of Elliptic Equations.....474
10.8 Applications to Optimal Control Problems..................489
11 Non-Matching Grid Discretizations ........................515
11.1 Multi-Subdomain Hybrid Formulations.....................516
11.2 Mortar Element Discretization: Saddle Point Approach.......523
11.3 Mortar Element Discretization: Nonconforming Approach.....551
11.4 Schwarz Discretizations on Overlapping Grids...............555
11.5 Alternative Nonmatching Grid Discretization Methods .......559
11.6 Applications to Parabolic Equations .......................564