Homogenization, Multiple scales problems in Biomathematics, Mechanics, Physics and Numerics, CIMPA AIMS, Cape Town 2007

354 Pages·2015·19.14 MB·English

by  A ABDULLE

Multiple scales problems in
Biomathematics, Mechanics,
Physics and Numerics
CIMPA School, Cape Town 2007
Editors:
Assyr Abdulle, Jacek Banasiak, Alain Damlamian and Mamadou Sango
GAKUTO International Series
Math. Sci. Appl., Vol.** (2009)
GAKKOTOSHO
Multiple scales problems in Biomathematics,
TOKYOJAPAN
Mechanics, Physics and Numerics, pp. i-v
Preface
This volume contains a collection of lectures presented at the 2007 CIMPA-UNESCO-
South Africa School “Multiple scales problems in Biomathematics, Mechanics, Physics and
Numerics” held at the African Institute of Mathematical Sciences in Muizenberg, South
Africa on 6th-18th of August 2007. The School primarily focused on the presentation of the
state of the art in homogenization theory, multiscale methods for homogenization problems,
asymptotic methods as well as modeling issues in a variety of applications in physics and
biology involving multiple scales.
This School was organized under the auspices of the CIMPA (Centre International de
Math´ematiques Pures et Appliqu´ees), a non-profit international organization based in Nice,
France, whose purpose, as a Category 2 Institute of UNESCO, is to promote international
cooperation in higher education and research in mathematics for the benefit of developing
countries (see http://www.cimpa-icpam.org).
TheSchoolconsistedof8coursesdeliveredby10invitedlecturerscomingfromCameroon,
France, Scotland and South Africa and it attracted 43 participants from countries as diverse
as Burkina Faso, Cameroon, the Democratic Republic of Congo, Morocco, Nigeria, Scotland,
Slovakia, South Africa, Tunisia and Zimbabwe. The lectures targeted postgraduate students
and young researchers and thus contained a blend of educational material with survey of
cutting edge research.
In parallel with the School, two mini-workshops were organized: one on functional ana-
lytic methods in applied sciences and the other on numerical methods for problems arising
from multiple scales models. The talks at the mini-workshops were presented by regular
participants of the School and invited speakers.
This volume is based on the courses given by the invited lecturers. It also contains three
invited lectures delivered during the workshop. The first part of the book gives a thorough
survey of recent developments in homogenization theory including the periodic unfolding
method and the sigma-convergence theory. This provides an introduction to these topics as
well as an account of its recent developments. New results in the homogenization of linear
and nonlinear elliptic eigenvalue problems in domains with fine grained boundaries are also
presented. On the computational side, new numerical methods, the so-called heterogeneous
multiscale method, is discussed for homogenization problems. Several numerical examples
and a detailed convergence theory of various numerical methods based on finite elements
are presented. Here again, the lecture allows for a rather complete presentation of the
developments of this new method, successful in several applications over the past few years.
ii CIMPA School Cape Town 2007
The second part of the book discusses the asymptotic analysis of singularly perturbed
problems, applications of the asymptotic analysis to biology as well as numerical methods
for singularly perturbed problems. Here the reader can see how modeling based on the
recognition of multiple time scales in a complex model allows to construct a systematic way
ofaggregatingvariablesleadingtoasignificantdecreaseinthedimensionofthemodels. This,
in turn, greatly facilitates its robust analysis without compromising accuracy of the results.
Thelecturesofthissecondpartcoveraggregationmethodsindiscretetimepopulationmodels
and describe a modified classical Chapman-Enskog asymptotic procedure which can be used
for aggregation of variables in continuous time population models and kinetic models. They
also provide a survey of numerical methods designed to treat singularly perturbed problems.
A. Damlamian was the main organizer of the School from the CIMPA side, which funded
participationofAfricanstudentsfromoutsideSouthAfrica,includingtwoscholarshipsspecif-
ically targeted for young women mathematicians. The local organizing committee consisted
of M. Sango and J. Banasiak. However, the School would not have been possible without
the significant help and contributions of many other people and institutions. Special thanks
must go to Prof F. Hahne and the staff of AIMS for providing an excellent infrastructure and
support throughout the School. Thus, the organizers were able to focus purely on academic
matters - a rare feat as far as organization of conferences is concerned. AIMS also provided
financial support for some of the participants. The organizers are also extremely grateful for
financial support received from the National Research Foundation of South Africa, which
funded all South African participants as well as supported two lecturers of the School. Or-
ganizers received generous support from the French Embassy in South Africa, the Hanno
Rund Fund of the School of Mathematical Sciences of the University of KwaZulu-Natal and
the Commission for Development and Exchange of the International Mathematical Union.
Last but not least, the School would not have been successful without the enthusiasm of the
students who duly attended and actively participated in all lectures and activities for the
full two weeks.
December 2008
Assyr Abdulle, Jacek Banasiak, Alain Damlamian & Mamadou Sango
The web page of the school is : http://maths.za.net/index.php?cf=5.
Multiple scales problems in Biomathematics, Mechanics, Physics and Numerics iii
List of participants
School Lecturers Invited Workshop Lecturers
A. Abdulle (UK) M.K. Banda (SA)
P. Auger (France) J.P. Lubuma (SA)
J. Banasiak (SA) K.C. Patidar (SA)
A. Damlamian (France)
P. Donato (France)
F. Ebobisse Bille (SA)
G. Nguetseng (Cameroon)
E. Perrier (France)
B.D. Reddy (SA)
M. Sango(SA)
School Participants
J. Absalom (Zimbabwe) V. Melicher (Slovakia)
V. Aizebeokhai (Nigeria) F. Minani (SA)
A. Al Ahouel (Tunisia) J. Mtimunye (SA)
A. Barka (Morocco) B. Nana Nbendjo (Cameroon)
J. Busa (Slovakia) L. Nkague Nkamba (Cameroon)
A. Chama (SA) K. Okosun (Nigeria)
A. Chirigo (Zimbabwe) S. C. Oukoumie Noutchie (SA)
A. Goswami (SA) J. Y. Semegni (SA)
S. Faleye (SA) L. Signing (Cameroon)
W. Lamb (UK) J. M. Tchoukouegno Ngnotchouye (SA)
M. A. Luruli (SA) A. Traore (Ivory Coast)
J. Malka Koubemba (DRC) A. Udomene (Nigeria)
A. Masekela (SA) J. Urombo (Zimbabwe)
B. Matadi Maba (SA) T.T. Yusuf (Nigeria)
K. Matlawa (SA) J.d.D.ZabsonreJeandeDieu(BurkinaFaso)
iv CIMPA School Cape Town 2007
Multiple scales problems in Biomathematics, Mechanics, Physics and Numerics v
CONTENTS
Part I. Homogenization, elasticity and multiscale methods
1) The Periodic Unfolding Method in Homogenization (D. Cioranescu, Alain Damla-
mian and G. Griso) page 1
2) The periodic unfolding method in perforated domains and applications to
Robin problems (D. Cioranescu, P. Donato and R. Zaki) page 37
3) Homogenization of linear and nonlinear spectral problems for higher-order
elliptic problems in varying domains (M. Sango) page 69
4) Σ-Convergence of Parabolic Differential Operators (G. Nguetseng) page 95
5) The Finite Element Heterogeneous Multiscale Method: a computational
strategy for multiscale PDEs (A. Abdulle) page 135
6) Mathematical Aspects of Elastoplasticity (F. Ebobisse Bille and B. D. Reddy)
page 185
Part II. Asymptotic analysis, numerical methods and applications
1) Asymptotic analysis of singularly perturbed dynamical systems of kinetic type
(J. Banasiak) page 221
2) Aggregation methods of time discrete models: review and application to host-
parasitoid interactions (P. Auger, C. Lett and T. Nguyen-Huu) page 257
3) Numerical schemes that preserve properties of the solutions of the Burgers
equation for small viscosity (R. Anguelov, J.K. Djoko and J.M.-S. Lubuma) page 279
4) Implicit-Explicit (IMEX) Schemes and Relaxation Systems (M.K. Banda)
page 303
5) Numerical Methods for Multi-Parameter Singular Perturbation Problems
(K.C. Patidar) page 329
PART I
Homogenization, elasticity
and multiscale methods

GAKUTO International Series
Math.Sci. Appl. Vol. ** (2009)
Multiple scales problems in Biomathematics,
Mechanics, Physics and Numerics, pp. 1–35
The Periodic Unfolding Method in Homogenization
D. Cioranescu, Alain Damlamian & G. Griso
Abstract: We give a detailed presentation of the Periodic Unfolding Method and how it
applies to periodic homogenization problems. All the proofs are included as well as some
examples.
1. Introduction.
The notion of two-scale convergence was introduced in 1989 by G. Nguetseng [27] and
further developed by G. Allaire [1] with applications to periodic homogenization. It was
generalized to some multi-scale problems by A. I. Ene and J. Saint Jean Paulin [17], G.
Allaire and M. Briane in [2], J. L. Lions, D. Lukkassen L. Persson and P. Wall [25],
In 1990, T. Arbogast, J. Douglas and U. Hornung [5] defined a “dilation” operation to
study homogenization for a periodic medium with double porosity. This technique was
used again by A. Bourgeat, S. Luckhaus and A. Mikelic [7], G. Allaire and C. Conca [3],
G. Allaire, C. Conca and M. Vanninathan [4], M. Lenczner [20-21], M. Lenczner et all
[22-24], D. Lukkassen, G. Nguetseng and P. Wall [26], by J. Casado D´ıaz [8] , J. Casado
D´ıaz et al. [9-11].
It turns out that the dilation technique reduces two-scale convergence to weak conver-
gence in an appropriate space. Combining this approach with ideas from Finite Element
approximations, wegive here a verygeneral and quitesimple method, the “periodic unfold-
ing method”, to study classical or multi-scale periodic homogenization. It a fixed-domain
method (the dimension of the fixed domain depends on the number of scales) that applies
as well to problems with holes and truss-like structures or in linearized elasticity. We pre-
2 D. Cioranescu, Alain Damlamian & G. Griso
sented this method in [12]. A preliminary version of the proofs can be found in the survey
of A. Damlamian [15]. Here the complete proofs of the results announced in [12] are given
as well as more recent developments.
The periodic unfolding method is essentially based on two ingredients. The first one is
the unfolding operator T defined in Section 2, where its properties are investigated. Let
ε
Ω be a bounded open set and Y a reference cell in IRn. By definition, the operator T
ε
associatestoanyfunctionv inLp(Ω), afunctionT v inLp(Ω ×Y), whereΩ isthesmallest
ε ε ε
finite union of cells εY containing Ω. An immediate (and very interesting) property of T
ε
is that it enables to transform any integral over Ω in an integral over Ω ×Y. Indeed, one
ε
has (Proposition 2.6)
(cid:90) (cid:90)
1
w(x) dx ∼ T (w)(x,y) dx dy, ∀w ∈ L1(Ω). (1.1)
ε
|Y|
Ω Ω×Y
If {v } is a bounded sequence in Lp(Ω), weakly converging to v in Lp(Ω), the sequence
ε
{T (v )} weakly converges to v in Lp(Ω×Y) (Proposition 2.10). This allows to show that
ε ε (cid:98)
the two-scale convergence in the Lp(Ω)-sense of a sequence of functions {v }, is equivalent
ε
to the weak convergence of the sequence of unfolded functions {T (v )} in Lp(Ω × Y)
ε ε
(Proposition 2.12).
In Section 2 are also introduce an local average operator Mε and an averaging operator
Y
U , the later being in some sense, the inverse of the unfolding operator T .
ε ε
The second ingredient of the periodic unfolding method consists of separating the char-
acteristic scales by decomposing every function ϕ belonging to W1,p(Ω) in two parts. In
Section 3 this is achieved by using the local average. In Section 4, the original proof of
this scale-splitting, inspired by the Finite Element Method, is given. The confrontation of
the two method of Sections 3 and 4, is interesting in itself (Theorem 3.5 and Proposition
4.7). In both approaches, ϕ is written as ϕ = ϕ1 +εϕ2 where ϕ1 is a macroscopic part,
ε ε ε
designed not to capture the oscillations of order ε (if there are any ), while the microscopic
part ϕ2 is designed to do so. The main result states that from any bounded sequence {w }
ε ε
in W1,p(Ω), weakly convergent to some w, one can always subtract a subsequence (still
denoted {w }) such that w = w1 +εw2 with
ε ε ε ε
(i) w1 (cid:42) w weakly in W1,p(Ω),
ε
(ii) T (w ) (cid:42) w weakly in Lp(Ω;W1,p(Y)),
ε ε
(1.2)
(iii) T (w2) (cid:42) w weakly in Lp(Ω;W1,p(Y)),
ε ε (cid:98)
(iv) T (∇w ) (cid:42) ∇ w+∇ w weakly in Lp(Ω×Y),
ε ε x y (cid:98)
where w belongs to Lp(Ω;W1,p(Y)). Convergence (1.2)(iii) shows that if the proper scaling
(cid:98) per
is used, oscillatory behaviour can be turned into weak (or even strong) convergence, at the
price of an increase in the dimension of the problem.