Table Of ContentToFausto
Luca Formaggia . Fausto Saleri . AlessandroVeneziani
Solving Numerical PDEs:
Problems, Applications,
Exercises
LucaFormaggia FaustoSaleri†
MOX–DipartimentodiMatematica MOX–DipartimentodiMatematica
“F.Brioschi” “F.Brioschi”
PolitecnicodiMilano,Italy PolitecnicodiMilano,Italy
AlessandroVeneziani
DepartmentofMathematics
andComputerScience
EmoryUniversity, Atlanta,Georgia,USA
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Password:978-88-470-2411-3
Translatedby:SimonG.Chiossi,DipartimentodiMatematica,PolitecnicodiTorino,Italy,
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ThetranslationisbasedontheoriginalItalianedition:
L.Formaggia,F.Saleri,A.Veneziani:Applicazioniedesercizidimodellisticanumericaperproblemi
differenziali,©Springer-VerlagItalia,Milano2005
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Preface
ThisbookresultsfromtheexperienceoftheauthorsinthecoursesonNumer-
ical Methods in Engineering, Partial Differential Equations and Numerical
Partial Differential Equations given to undergraduate and graduate students
at the Politecnico di Milano (Italy), EPFL Lausanne (Switzerland), the Uni-
versity of Bergamo (Italy) and Emory University (Atlanta, GA, USA). In
these courses we introduce students to methods for the numerical approx-
imation of Partial Differential Equations (PDEs). One of the difficulties in
this field is to identify the right trade-off between theoretical concepts and
theirpracticaluse.Inthiscollectionofexamplesandexercisesweaddressthis
difficultybyprovidingexerciseswithan“academic” flavor,stemmingdirectly
from basic concepts of Numerical Analysis (and often inspired by exams of
the courses mentioned above), as well as exercises, marked by the symbol *,
whichaddressapracticalproblemthatthestudentisencouragedtoformalize
in terms of PDE, analyze and solve numerically. These problems are derived
from the knowledge gained by the authors in projects developed in colla-
boration with scientists of different fields (geology, mechanical engineering,
biology, etc.) and industries.
The book consists of four parts. The first part recalls basic concepts and
introduces the notation which will be used throughout the text, which is
however quite standard. Therefore, it may be skipped by readers already ac-
quainted with the general subject and more interested in applications and
exercises. More precisely, in Chapter 1 we give a brief overview of elemen-
tary functional analysis and fundamentals of linear algebra, while Chapter 2
addresses fundamentals of composite interpolation, which is the basis of the
finite element method, and of numerical differentiation, basis for the finite
difference method. The second part covers steady elliptic problems, with a
dominance of diffusion effects in Chapter 3, the treatment of advection and
reactiontermsbeing giveninChapter4. Finiteelementsisthediscretization
methodofreferenceforthosechapters,butsomeexercisesaredevelopedwith
finite differences. The third part covers time-dependent problems, parabolic
equations Chapter 5, linear hyperbolic equations Chapter 6, as well as ba-
vi Preface
sic problems in incompressible fluid dynamics (Navier-Stokes equations) in
Chapter 7. Time discretization is carried out mainly with finite differences,
but we address space-time finite elements as well.
The fourth part contains complementary material collected into two ap-
pendices. Appendix A recalls some implementation details of sparse matri-
ces,whichisofinterestforreadersmoreinterestedinprogrammingnumerical
methodsforPartialDIfferentialEquations.Finally,AppendixBgivesashort
biography of mathematicians and scientists often cited in the text. It is not
intended to be an exhaustive collection, but rather to give an historical per-
spective and an insight on people who gave fundamental contributions to
numerical analysis and applied mathematics.
The approach we have followed is to allow different ways of using this
book,bysplittingthesolutionofeachexerciseintothreeparts,Mathematical
Analysis of the problem, Numerical Approximation, Analysis of the results.
Exercises marked by * have a preliminary section dedicated to the Model
formulation. The rationale behind this subdivision is to make easier for the
reader to select the specific aspects on which (s)he is interested.
Many exercises are taken from the companion book Numerical Modeling
for Partial Differential Equations, by A. Quarteroni (Springer, 2010), where
theyareproposedwithoutsolution.Therefore,thecitedmonographprovides
the main background of this book. However, the summary of the main the-
oretical results provided in the introductory chapter should make this text
self-contained.
We have assumed different levels for the reader also for the software
tools adopted in the book. For a basic reader interested in a quick check
of the theoretical results, we provide MATLAB programs, most of which
are described in the text and downloadable from the Springer website:
http://extras.springer.com. In particular, the code fem1D, written by the au-
thors, is equipped with a simple graphical user interface for 1D (elliptic,
parabolic, hyperbolic) problems, so that the user does not need any specific
programming knowledge for obtaining the numerical results.
A reader interested in more advanced problems but still not on imple-
mentation details would appreciate the exercises developed with the code
Freefem++, authored by O.Pironneau, F.Hecht and A.Le Hyaric. The main
advantages of using this code is the proximity of the syntax to the weak for-
mulation of the problems and the relative easiness of installation of the code
with different operating systems.
Finally, for the sake of readers with an interest in more advanced numer-
ical programming, a few (simple) exercises have been completed with a 3D
extensionoftheproblemproposed.Inthiscase,thecodeweuseistheFinite
Element library LifeV (see www.lifev.org). This library is a joint project of
different research groups (EPFL, Politecnico di Milano, Emory and INRIA
Rocquencourt, France) working in different fields of Scientific Computing.
The purpose of these examples is to encourage the reader to explore the
fascinating (yet difficult) world of professional implementation of numerical
Preface vii
methods for PDEs. Asa matteroffact, wetrulybelieve that a deepcompre-
hension of the Numerical Modeling of Differential Problems requires a good
knowledge of all the aspects, ranging from the most theoretical to the most
practical ones, including implementation techniques.
Notice that for the sake of space, we give in the text only the synopsis of
theprograms,orsomesnapshotsofthecodesaswellasprogrammingdetails
whenever necessary for the understanding of the algorithm. The reader is
encouraged to have a look at the complete code at the companion web sites
mentioned above.
We thank all the colleagues and students who contributed to improve this
text. In particular, we thank Alfio Quarteroni for many fruitful suggestions,
the team of developers of the code LifeV, in particular Alessio Fumagalli at
MOX, Politecnico di Milano, Tiziano Passerini at the Department of Mathe-
maticsandComputerScienceatEmoryUniversity.NicolaParoliniofMOXis
gratefully acknowledged for providing an interesting exercise for the chapter
on Navier-Stokes, and Joshua Keller of Emory University for the simulations
andtheimagesofthecover.WethankFrancescaBonadeiatSpringerforthe
editorial work and the patience during the preparation of the text. Alessan-
dro Veneziani wishes to thank the Emory’s team, Lucia Mirabella, Marina
Piccinelli, Tiziano Passerini, Luca Gerardo Giorda, Mauro Perego, Alexis
Aposporidis, Marta D’Elia, Umberto Villa and Luca Bertagna for reading
and supporting the improvement of the manuscript. Luca Bertagna has also
kindly provided a 3D Navier-Stokes computation carried out with LifeV and
shown in one of the exercises. The help of Michele Benzi (and his impressive
knowledgeofthehistoryofNumericalAnalysis)isgreatlyacknowledged.Er-
rors or imprecision are obviously under our own responsibility and we thank
in advance all the colleagues and students who will give us their feedback.
When we wrote the Italian edition of this book in 2005, Fausto was a
fundamental component of the Authors’ team in conveying to students our
passion for Mathematics and its applications in any field of science (and
life). We wrote this English edition, which is a revision and an extension
of the Italian one, hoping that this book contributes in keeping his and our
enthusiasm alive. To his memory, friendship and passion we dedicate this
work.
Milan, Atlanta, August 2011 Luca Formaggia
Alessandro Veneziani
Contents
Preface ....................................................... v
Part I Basic Material
1 Some fundamental tools .................................. 3
1.1 Hilbert spaces.......................................... 4
1.2 Distributions........................................... 5
1.3 The spaces Lp and Hs .................................. 6
1.4 Sequences in lp......................................... 10
1.5 Important inequalities................................... 11
1.6 Brief overview of matrix algebra.......................... 13
2 Fundamentals of finite elements and finite differences..... 17
2.1 The one dimensional case: approximation by piecewise
polynomials............................................ 17
2.2 Interpolation in higher dimension using finite elements ...... 22
2.2.1 Geometric preliminary definitions .................. 22
2.2.2 The finite element ................................ 29
2.2.3 Parametric Finite Elements........................ 35
2.2.4 Function approximation by finite elements........... 44
2.3 The method of finite differences .......................... 46
2.3.1 Difference quotients in dimension one ............... 46
Part II Stationary Problems
3 Galerkin-finite element method for elliptic problems...... 65
3.1 Approximation of 1D elliptic problems .................... 67
3.1.1 Finite differences in 1D ........................... 89
3.2 Elliptic problems in 2D.................................. 100
3.3 Domain decomposition methods for 1D elliptic problems..... 131
3.3.1 Overlapping methods ............................. 131
3.3.2 Non-overlapping methods ......................... 141
x Contents
4 Advection-diffusion-reaction (ADR) problems ............ 147
4.1 Preliminary problems ................................... 149
4.2 Advection dominated problems........................... 157
4.3 Reaction dominated problems............................ 190
Part III Time Dependent Problems
5 Equations of parabolic type............................... 205
5.1 Finite difference time discretization ....................... 206
5.2 Finite element time discretization......................... 263
6 Equations of hyperbolic type ............................. 277
6.1 Scalar advection-reaction problems ....................... 277
6.2 Systems of linear hyperbolic equations of order one ......... 310
7 Navier-Stokes equations for incompressible fluids ......... 333
7.1 Steady problems ....................................... 334
7.2 Unsteady problems ..................................... 366
Part IV Appendices
A The treatment of sparse matrices ......................... 395
A.1 Storing techniques for sparse matrices..................... 395
A.1.1 The COO format................................. 399
A.1.2 The skyline format ............................... 401
A.1.3 The CSR format ................................. 404
A.1.4 The CSC format ................................. 405
A.1.5 The MSR format................................. 406
A.2 Imposing essential boundary conditions ................... 409
A.2.1 Elimination of essential degrees of freedom .......... 409
A.2.2 Penalization technique ............................ 410
A.2.3 “Diagonalization” technique........................ 411
A.2.4 Essential conditions in a vectorial problem........... 413
B Who’s who ............................................... 417
References.................................................... 425
Subject Index ................................................ 431
1
Some fundamental tools
Mathematical modeling of real-life problems in engineering, physics or life
sciencesoftengivesrisetopartialdifferentialproblemsthatcannotbesolved
analyticallybutneedanumericalschemetoobtainasuitableapproximation.
Dealing with numericalmodeling requires first of all anunderstanding ofthe
underlying differential problem. The type of differential problem, as well as
issues of well-posedness and regularity of the solution may indeed drive the
selection of the appropriate simulation tool. A second requirement is the
analysis of numerical schemes, in particular their stability and convergence
characteristics. Last, but not least, numerical schemes must be implemented
in a computer language, and often aspects which look easy “on paper” arise
complex implementation issues, particularly when computational efficiency
is at stake.
Theseconsiderationshavedriventheselectionoftheexercisesinthisbook
and their solution, with the precise intent of giving to the reader practical
examples of mathematical and numerical modeling, as well as computer im-
plementation.
We will consider differential problems of the following general form: find
u:Ω →R such that
L(u)=f on Ω, (1.1)
where L is a differential operator (often linear in u) andΩ is an open subset
of Rd, with d either 1 or 2. We will also give some examples of problems in
three dimensions.
Inmostapplicationsofnumericalmethodswithpracticalinterestthecom-
putationaldomainisbounded.Therefore,whennototherwisestatedwecon-
sider Ω bounded. Problem (1.1) will be supplemented by suitable boundary
conditions and – for time-dependent problems – by initial conditions.
Accordingtothetypeofformulationchosenfor(1.1)itwillbenecessaryto
introduce certain function spaces, besides discussing the regularity required
FormaggiaL.,SaleriF.,VenezianiA.:SolvingNumericalPDEs:Problems,Applications,
Exercises. DOI10.1007/978-88-470-2412-0_1, ©Springer-VerlagItalia2012
4 1 Somefundamentaltools
ScalarProduct
HilbertSpace
(·,·)
u = (u,u)
Norm
Banachspace
d(u,v)= u−v
Distance
Completemetricspace
d(·,·)
∀{vn}∈V,n=1,2,...:
m(cid:2)lnim→∞d(vn,vm)=0⇒∃v∈V :nl→im∞vn=v
Fig. 1.1 Therelationshipamongacompletemetricspace(bottombox),aBanachspace
(middle)andaHilbertspace(top)
fortheproblemdataandthedomainΩ itself,andgiveanappropriatemean-
ing to the operations of differentiation and integration.
In this introductory chapter we recall just some basic results that are in-
strumentaltotheunderstandingofthesolutionoftheproposedexercises.The
interested reader may find more complete treatment of mathematical mod-
eling and analysis of partial differential equations in [RR04, Sal08, Eva10],
while advanced text in functional analysis are [Bre11, AF03, Leo09]. For the
numerical aspects the main reference is [Qua09]. For the linear algebra part
we refer to [GV96, Str03].
1.1 Hilbert spaces
Let V be a real vector space equipped with a metric defined by a distance
d:V ×V →R between elements u,v ∈V. A sequence of elements v of V
n
with n=1,2,..., is called Cauchy sequence if for any ε>0 there is a q such
that for m,n ≥ q we have d(v ,v ) ≤ ε . The space V is called complete if
m n
any Cauchy sequence converges to an element v ∈V.
A real vector space V where a norm ||·|| can be defined is equipped by a
distance d(u,v) ≡ ||u−v||. And, if the space is complete is called a Banach
space.Finally,aBanachspaceV eq(cid:2)uippedwithascalarproduct,denotedby
(·,·) has a norm, namely (cid:7)v(cid:7) = (v,v) for any v ∈ V. Hence it features
V
a metric for which V is complete. In this case, V is called Hilbert space, see
Fig. 1.1.
Given two Hilbert spaces V1 and V2 one may consider an operator
L:V1 →V2,
