Numerical Approximations to Multiscale Solutions in Partial Differential Equations
Many problems of fundamental and practical importance have multiple scale solutions. The direct numerical solution of multiple scale problems is difficult to obtain even with modern supercomputers. The major difficulty of direct solutions is the scale of
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Springer-Verlag Berlin Heidelberg GmbH
James F. Blowey Alan W. Craig Tony Shardlow Editors
Frontiers in Numerical Analysis Durham 2002
Springer
James Blowey Alan Craig University ofDurham Oepartment of Mathematical Sciences South Road ORI )LE Ourham. United Kingdom e-mail: [email protected] [email protected]
Tony Shardlow University of Manchester Oepartment of Mathematics Oxford Road MI3 9PL Manchester. United Kingdom e-mail: shardlow®maths.man.ac.uk Libfll)' of Coogress CataIpgiDg-m-Publicarlcrt Data LMS-EPSRC Numerica! Analysis SIlIIllIlfI' Sebool (IOth: 2002; Ullivasity ofDurham) Frootiers in lIlIlneric.a.l analysis; Dwbam 20021 James F. Blowey, Alao W. Crai&. Tony Shardlow, editon. p. an. - (UDivlnital) IncIudes bibliographical rdlRllC5 UId index. ISBN 978-3-S40-44319-3 ISBN 978-3-642-SS692-O (eBook) 00110. 1007/978-3-642-55692-0 1. Nlllll5ic:allIIalysis-Ca!grmses. L B~, James F. n. Craig, Alan W. m ShardJow, Tony.lV. Title. QA297.LS92oo2 ~19.4-dc21
2003050594
ISBN 9"78-3-540-44319-3 Mathematics Subject Classification (2000); 35-XX, 65-XX Thil work il lubject to COpyrighL Ali right5 an' relerved, whether the whole or put of the nu.terial il concerne 0 might
:=
10, fv dx
have a very fine structure).
Example 2.3. Composite materials in mixed form, i.e., the same problem of the previous example, but now with: a = -aV''ljJ in 0; V' . a = f in 0; 'ljJ = 0 on 80 V := :E x O.
(4.14)
We notice now that the quantity which appears at the left-hand side of (4.14) does not depend on x or on y anymore, but depends only on the matrix M. In order to simplify the notation, we assume from now on that when taking infimums and supremums we implicitly discard the value O. Hence we can set . Jt(M):= yElR mf xElR" sup S
yTMx
II X II X II YII y '
(4.15)
and we remember that, if we have a sequence of matrices Mk, then Jt(Mk) clearly depends on k. If we want a condition that is actually independent of k we must then require that there exists a Jt > 0, independent of k, such that inf sup
yElR S xElR"
yTMx
Ilxllx Ilylly
2:
Jt
>0
(4.16)
for all matrices M in the sequence M k . Condition (4.16) is usually called the inf-sup condition. From the above discussion, going somewhat backward, it is not difficult to see that, for a given Jt > 0, it is equivalent to requiring that inf sup
yElR S xElR"
or that
yTMx
Ilxllx Ilylly
2:
Jt,
(4.17)
Stability of Saddle-Points in Finite Dimensions \j
Y E JR" with y f=- 0,
supr
xElR
yTMx
-11-112 JLllylly, X X
39
(4.18)
or finally that \j
y E JR"
\ {O},
:3
x E JRr
\ {O} such that yT Mx
2 JLllxllx Ilylly. (4.19)
In order to analyze in a deeper way the dependence of the constant JL on M and on the norms II '11x and 11·lly, it will be convenient to limit the possible choice of the norms. Hence, we assume that for each k we are given an r( k) x r( k) symmetric positive definite matrix 3k with entries ~ij, and an s(k) x s(k) symmetric positive definite matrix Hk with entries 'flij. Again, for the sake of simplicity, we shall drop the index k and just
