A NNALI DELLA
S CUOLA N ORMALE S UPERIORE DI P ISA Classe di Scienze
D ANIELE B OFFI
F RANCO B REZZI
L UCIA G ASTALDI
On the convergence of eigenvalues for mixed formulations
Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4
esérie, tome 25, n
o1-2 (1997), p. 131-154
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131-
On the Convergence of Eigenvalues for Mixed Formulations
DANIELE BOFFI - FRANCO BREZZI - LUCIA GASTALDI
Vol. XXV (1997), pp. 131-154
Abstract. Eigenvalue problems for mixed formulation show peculiar features that make them substantially different from the corresponding mixed direct problems.
In this paper we analyze, in an abstract framework, necessary and sufficient con-
ditions for their convergence.
1. - Introduction
In a
general
way, we say that a variationalproblem
is written in mixedform
if it fits thefollowing
abstractsetting.
We assume that(1)
4) and E are Hilbert spaces,(2) (p)
andb(~/r~, ~)
are bilinear forms on (D x (D and (D x Erespectively,
and,
tosimplify
thepresentation,
we also assume that(4) a (~, .)
issymmetric
andpositive
semidefinite.Setting
:=(a (~P, ~P ) ) 1 ~2 (which
ingeneral
willonly
be a seminorm on1»
this
immediately gives
Properties (1)
to(4)
will be assumed to holdthroughout
all the paper.For any
given pair ( f, g)
in 1>’ x E’ we consider now theproblem
It is known
that,
in order to haveexistence, uniqueness
and continuous de-pendence
from the data forproblem (6)
it is necessary and sufficient that the bilinear formsa ( ~ , .)
and&(-, -) satisfy
thefollowing
conditionswhere the kernel K is defined as:
EXAMPLE 1. Stokes
problem.
We take (D =(HJ(Q))2, 8
=L2 (S2)~~, a(1/I, ~p)
=b(*, ~) = -(div *, ~) where,
asusual, (.,.)
is the innerproduct
inL2 (S2)
or in(L2 (S2))2.
It is easy to see that(7)
and(8)
are satisfied. Moreover if wetake g =
0 the solution of(6)
is related to the solution of the Stokesproblem
by
therelations 1/1
= uand X
= p.Approximations
based- on thisapproach
areclassical and are
usually
calledapproximations
in theprimitive
variables.EXAMPLE 2. Dirichlet
problem
withLagrange multipliers.
We take 4S =E =
a(1fr, cp)
=(V1/I, Vcp)
andb(~, ~) _ (~, (duality
betweenand
~).
It is easy to see that(7)
and(8)
are satisfied.Moreover,
for everyf
EH-1 (Q) and g
e theunique
solution of(6)
is relatedto the solution of
by
therelations 1fr
= uApproximations
based on thisapproach
where first introduced
by
B abuska[2].
EXAMPLE 3.
Mixed formulation of
second order linearelliptic problems.
We take4$ =
H(div; Q), E
=L2(S2), cp)
=(~, ~) = (div *, ~).
It is easy to see that(7)
and(8)
are satisfied. Moreover if we takef
= 0 the solution of(6)
is related to the solution of theproblem
by
therelations x =
uand 1/1
= V u.Approximations
based on thisapproach
where first introduced
by
Raviart-Thomas[24].
EXAMPLE 4. Biharmonic
problem.
We take (D =Hl(Q), 8
=Ho’(0), a (1/1, cp)
=(1/1, cp)
and&(~, ~) = -(V1/l, V~).
It is easy to see that(7)
issatisfied,
but(8)
is not.
However,
if SZ is smoothenough
and we takef =
0 then(6)
has aunique solution,
related to the solution ofby
the relations X = uand 1/1
_ - 0 u .Approximations
based on thisapproach
where first introduced
by
Glowinski[16]
andanalyzed by
Ciarlet-Raviart[13]
and Mercier
[20].
For many other
examples
of mixed formulations ofboundary
valueproblems
related to various
applications
in fluidmechanics and in continuous mechanicswe
refer,
forinstance,
to[9].
Let us now consider the
problem
of discretization. Assume that we aregiven
two families of finite dimensionalsubspaces 4Sh
andEh
of (D and E,respectively.
We consider the discretizedproblem:
It is known that discrete
analogues
of(7)
and(8)
are sufficient to ensuresolvability
of the discreteproblem together
withoptimal
error bounds. Moreprecisely
if the spaces4$h and Eh satisfy
thefollowing
conditionsthere exists a >
0, independent
ofh,
such thatwhere the discrete kernel
Kh
is defined asand
there exists
f3
>0, independent
ofh,
such thatthen we have
unique solvability
of(13)
and thefollowing
error estimateAs we shall see, conditions
(14)
and(15)
are also necessary forhaving (16),
in a suitable sense.
We turn now to the
eigenvalue problems.
As we can see from the ex-amples above,
theeigenvalue problem
which isnaturally
associated with thecorresponding boundary
valueproblem
in strong form(namely (9), (10), (11)
or
(12))
does notcorrespond
totaking (À1fr, ÀX)
asright-hand
side of(6).
Instead, according
with the different cases, the naturaleigenvalue problem
isobtained
by taking 0)
or(0, 2013~x)
asright-hand
side of(6).
One expects,as for instance in
[21],
that(14)
and(15), together
with suitable compactnessproperties,
are sufficient to ensuregood
convergence of theeigenvalues.
How-ever, when the
problem
is set in mixed variationalform, compactness
is more delicate to deal with. In aprevious
paper[5]
we showedthat,
for theparticular
case of
Example 3,
even if theoperator mapping g
into u in(11)
isclearly compact, assumptions (14)
and(15)
are not sufficient toavoid,
forinstance,
the presence ofspurious eigenvalues
in the discretespectrum.
Here we address amore
general problem,
in abstractform,
and we look for sufficient(and,
pos-sibly, necessary)
conditions in order to havegood approximation properties
forthe
eigenvalue problems having
either(À1fr,O)
or(0,
at theright-hand
side. As we shall see, in each of the two cases,
(14)
and(15) might
be neithernecessary nor sufficient for that.
Our
approach
turns out to be more similar to the one of[14]
rather than the one of[8]
or[1]. Important
references for thestudy
ofeigenvalue problems
in mixed form are
[21], [3], [23].
As far as thesuf,ficient
conditions areconcerned,
we haveonly
littleimprovements
over theprevious
papers. Forinstance,
our bilinear forma(., -)
is notsupposed
to bepositive
definite as inthe
previous
literature.Moreover, previous
related papers dealmostly
with casesin which the two
components
of the solution of the directproblem
are bothconvergent,
while weaccept
discretizations that canproduce singular global
matrices. On the other
hand, having
assumed symmetry ofa(., .),
we do not have to consideradjoint problems
as in[14].
However, inpractical
cases,the actual
gain
isnegligible.
Themajor
interest of the paper, in ouropinion,
consists in
showing
that our sufficient conditions are,mostly,
also necessary,thus
providing
a severe test forassessing
whether agiven
discretization is.suitable for
computing eigenvalues
or not. Thisjustifies,
in ouropinion,
theapparently
excessivegenerality
of our abstractapproach. Indeed,
as we shall see, convergence of discreteeigenvalues
does not evenimply,
for mixedformulations,
the
nonsingularity
of thecorresponding global
matrices.Finally
wepoint
out that in this paper we do not look for apriori
esti-mates for
eigenvalues
andeigenvectors,
butonly
deal with convergence. This is somehow inagreement
with the fact that necessary conditions are amajor
issuehere.
However,
in most cases, apriori
error estimates can bereadily
deducedchecking
the laststep
in theproofs
of sufficient conditions and/orapplying
thegeneral
instrumentsof,
say,[7], [21], [3].
An outline of the paper is the
following.
In Section 2 we state theproblem
and relate the convergence of the
spectrum
with the uniform convergence of the resolvent operators. Moreover wepoint
out the role of the discrete conditions(DEK)
and(DIS)
is order to have existence and boundedness of the different components of the solution of(13).
Section 3 and 4 are devoted to the
analysis
of theeigenvalue problems
associated to
(6)
when theright-hand
side is of the type(~, 0)
or(0, -ÀX), respectively.
In both cases we state sufficient and necessary conditions for thegood approximation
of thespectrum.
At the end of each section we will show how the knowngood approximations
of theproblems
in theexamples
abovesatisfy
our sufficient conditions for convergence ofeigenvalues
andeigenvectors,
and more
generally
we discuss thevalidity
of otherpossible approximations
inlight
of our conditions.2. - Statement of the
problems
Let H be a Hilbert space and T : H - H a
selfadjoint
compact operator.To
simplify
thepresentation
we assume that T isnonnegative.
We are interested in the
eigenvalues k
E R definedby
In the above
assumptions
it is well-known that there exists a sequence andan associated orthonormal basis
luil
such thatWe also set, for i E N,
Ei = span(ui).
The
following mapping
will be useful. Let m : N 2013~ N be theapplication
which to every N associates the dimension of the space
generated by
theeigenspaces
of the first N distincteigenvalues;
that isClearly, ~,m ( 1 ) , ... , (N
EN)
will now be the first N distincteigenvalues
of
(17).
Assume that we are
given,
forevery h
>0,
aselfadjoint nonnegative
operator
Th :
H - H with finite range. We denoteby À?
e R theeigenvalues
of the
problem
Let
Hh
be the finite-dimensional range ofTh
anddim Hh
=:N (h ) ;
thenTh
admitsN (h )
realeigenvalues
denotedÀ?
such thatThe associated discrete
eigenfuntions u 7, i
=1,..., N (h), give
rise to anorthonormal basis of
Hh
withrespect
to the scalarproduct
of H. LetWe assume that
It is a classical result in
spectrum perturbation theory
that(22) implies
thefollowing
convergence property foreigenvalues
andeigenvectors:
F),
for E and F linearsubspaces
of H, represents the gap between E and F and is definedby
Viceversa,
it is not difficult to prove that(23) is a
sufficient condition for(22).
We are interested in
having (23)
foreigenvalue problems
in mixed form.Let us therefore go back to the abstract framework
already
used in theintroduction,
with theassumptions
therein. Inparticular
assume, for the moment, that(7)
and(8)
are satisfied and that(13)
has a solution for every( f, g)
in1>’ x E’. Problems
(6)
and(13)
definethen,
in a natural way, two operatorsS( f, g) = (1fr, X) (solution
of(6))
andSh ( f, g) = (1/!h, Xh ) (solution
of(13)).
It is well-known
(see [9])
that(DIS)
and(DEK) (cfr. equations (15)
and
(14)) imply
that the discreteoperator Sh
is bounded from(D’ h
xEg
to~ x E,
uniformly
in h(see (16)). Moreover,
the converse holds true, as it isproved
in thefollowing
Lemma 1. Before it we introduce thefollowing
notation: for every h > 0 we define
LEMMA 1.
If
there exists a constant C suchthat for
allf
E (D’ and g E E’for
all h >0,
then(DIS)
and(DEK)
areverified.
PROOF. Let
1frh belong
toKh,
then(1frh, 0, f, 0)
satisfies(13) with f , CPh)
:=for all CPh E
4Sh.
Hence theinequality (26) gives (DEK)
with a -1/(C2Ma), Ma being
thecontinuity
constant of a(see (3)).
If
Xh E Sh,
then(0~,/,0)
satisfies(13), with (1, CPh)
forall CPh E
4Sh.
Hence theinequality (26) yields (DIS)
withfl
=1 / C.
0REMARK 1. In the statement of Lemma 1 we
implicitly
assumed that theoperator Sh
was defined for everyf
and g. However, as it can beclearly
seenin the
proof,
this was notreally
necessary. Indeed it is sufficient to assumethat there exists a constant C > 0 such that for
every h
> 0 and for everyquadruplet (1frh,
xh,f, g)
E4Sh
xEh
x 1>’ x E’satisfying (13),
one hasThis should not
surprise,
as(13)
isalways
a linearsystem
with a square matrix.D Consider now theeigenvalue problem.
For the sake ofsimplicity,
let usassume for the moment that there exist two Hilbert spaces
HD
and7/s
such that we canidentify
and such that
hold with dense and continuous
embedding,
in acompatible
way.The restrictions of
S and Sh
to xT~s
define now twooperators
fromx
T~s
into itself.As a consequence of
(16)
and Lemma1,
it is immediate to prove thefollowing proposition.
PROPOSITION 1. Assume that
(DIS)
and(DEK)
hold. ThenSh
converges uni-formly
to S in xHE) if and only if
S(from HD
xHE
intoitself)
is compact.This
proposition
concludes our convergenceanalysis
for theeigenvalue problems
associated to(6)
and(13).
However in theapplications
one findsmore often
eigenvalue problems
associated to(6)
and(13)
when one of the two components of the datum is zero. Let us set theseeigenvalue problems
in theirappropriate
abstract frameworkintroducing
thefollowing
operators:and their
adjoints
We shall say that
(6)
is aproblem of the
type(f )
if theright-hand
sidein
(6)
satisfies g = 0.Similarly,
we shall say that(6)
is aproblem of the
typex
if theright-hand
side in(6)
satisfiesf
= 0.Correspondingly,
we shallstudy
theapproximation
of theeigenvalues
of thefollowing operators:
Whenever the associated discrete
problem?
aresolvable,
we can introduce thediscrete counterparts of
TD
and7s
as: ,, -,
In the
remaining
part of this section we aregoing
to relate thesolvability
andboundedness - of the discrete
operators
with either(DIS)
or(DEK).
PROPOSITION 2.
If (DEK) (see (14))
holds and g =0,
~thenproblem ( 13)
hasat least one solution
(1frh, Xh ).
Moreover1frh
isuniquely
determinedby f
and(where
a is the constantappearing
in(14)).
PROOF. Let
1frh
be theunique
solution ofa(1frh, = (f,
for all ~Oh inKh. Clearly 1frh exists,
isunique
and satisfies(34).
Now look for Xh inEh
such thatXh)
=( f ,
-a(1frh,
for all E As theright-hand
side is in the
polar
set ofKh,
thesystem
iscompatible
and hence has at leastone solution. 1:1
PROPOSITION 3. Assume that there exists a constant C > 0 such
that for
everyh > 0
and for
everyquadruplet
Xh,f, 0)
E(Dh
X8h
x (D’ x S’satisfying (13)
one has
then the operator
T; is defined in
all ~’ and(DEK)
holds with a =1 / (C2 Ma ), Ma
being
thecontinuity
constanto, f a (see (3)).
PROOF. With the same
proof
as in Lemma 1 we see that(DEK)
holds true.The
solvability
of(13)
is now a consequence ofProposition
2. 0PROPOSITION 4. Assume
that the following
weak discreteinf-sup
conditionholds:
for
every h >0,
there exists a constantflh
> 0 such thatThen for
every g E S~ andf
=0 problem ( 13)
has at least one solution(1frh, Xh )
and
Xh is uniquely
determinedby
g.PROOF. The
assumption (36) implies that,
with obviousnotation, Bh
issurjective.
Hence for g E S/ there exists at least one1fr g
E4$h
such thatg. Then find
1frk
eKh
such that =OCh.
Finally,
take Xh such that = for alle
4$h.
Such a Xhexists, by
the same argument used in theproof
ofProposition
2.Finally
observe that solves(13)
with(0, g)
asright-hand
side.To see the
uniqueness,
assume that(i
=1, 2)
are two solutions.Clearly a(1frt -
= 0 for all eOCh. Taking = 1frt - 1frl
oneobtains that
1frt -1frl)
= 0 and hence as a issymmetric
andpositive semidefinite,
= 0 for all e(use (5)).
Nowxl)
= 0 for all in and(DISh ) implies xt
=xl.
0PROPOSITION 5. Assume that there exists a constant C > 0 such
that for
everyh > 0
and for
everyquadruplet (1/!h,
Xh,0, g)
E xEh
x ~~ x E’satisfying (13)
one has
then the operator
T h
isdefined
in all S’ and the weak discreteinf-sup
condition(DISh )
holds. Ingeneral, (37)
does notimply (DIS).
PROOF. Remark first that the
assumption (37) implies that,
with obviousnotation, Bh
isinjective,
thereforeBh
will besurjective
and thisimplies (DISh ).
In order to see that
(DIS)
cannot be deduced ingeneral,
consider the casewhen a w
0, 4$h
=Eh
and b is h times the scalarproduct
in ElPROPOSITION 6. Assume that there exists a constant C > 0 such
that for
every h > 0and for
everyquadruplet (~h,
Xh,0, g)
E(Dh
XSh
x (D’ x S’satisfying (13)
one has
then both
T~
andC~
oSh
oCs
aredefined
on S’ and(DIS)
holdswith f3
=I/ C.
PROOF. Remark first
that,
fromProposition 4, problem (13)
has at least one solution for every g EE’,
but now the estimate(38)
ensures that such solution isunique.
HenceC~
oSh
oCs
is also well-defined in ~’. Letnow ~h
be anelement of
Eh,
andlet g
E E’ be suchthat
= 1 and~g, ~h ~
=Taking 1fr;
=C
oSh
oCE9
we havePROPOSITION 7.
If there
exists C > 0 such thatfor
every h >0,
then(DIS)
holdswith f3
=1/C.
PROOF. The same
proof
as in Lemma 1. 0We see from
Propositions
3 and7,
that forproblems
of the type~o~
theestimate
(35)
on1frh implies (DEK)
and the estimate(40)
on Xhimplies (DIS).
Analogue properties
do notentirely
hold forproblems
of thetype (0)
3. - Problems of the
type ( ) 0
In this
section, together
with(1)-(4),
we assume that(EK)
and(IS)
areverified. We also assume that we are
given
a Hilbert spaceHp (that
we shallidentify
with its dual spaceH.)
such thatwith continuous and dense
embeddings.
We consider theeigenvalue problem
which in the formalism of the
previous
section can be writtenWe assume that the operator
Tp
iscompact
fromH(D
to (D.Suppose
now that we aregiven
two finite dimensionalsubspaces 4$h
andEh
of (D and E,respectively.
Then theapproximation
of(42)
readsthat is
We are now
looking
for necessary and sufficient conditions that ensure the uniform convergence of7~
toTo
in1» which,
as we have seen,implies
the convergence of
eigenvalues
andeigenvectors (see (23)).
To start
with,
we look for sufficient conditions.We introduce some notation. Let and
E)
be thesubspaces
of 4$ andE,
respectively, containing
all thesolutions *
e 4$ andrespectively,
ofproblem (6) when g
=0;
thatis,
with the formalism of theprevious section,
Notice that the
following
inclusion holds true:The spaces and
E)
will be endowed with the natural norm: thatis,
forinstance,
DEFINITION 1. We say that the weak
approximability
ofE§
is verifiedif there exists
cvl (h), tending
to zero as h goes to zero, such that for everyNotice
that,
inspite
of its appearance,(48)
is indeed anapproximability property. Actually
as EKh,
we haveX)
- X - for everyX I
EEh,
whichhas, usually,
to be used toverify (48).
DEFINITION 2. We say that the strong
approximability of
is verified if there existsúJ2(h), tending
to zero as h goes to zero, such that for every1fr
E thereexists *’
EKh
such thatTHEOREM 1. Let us assume that
(DEK)
isverified (see ( 14)).
Assume moreoverthe weak
approximability no
and thestrong approximability (Do .
Then thesequence
T;
convergesuniformly
toTD
in(D),
that is there existsúJ3(h),
tending
to zero as h goes to zero, such thatPROOF. Let
f
EHo
and let(1fr, x ) E (DI
xE)
be solution of(6): (1fr, X )
=S ( f, 0) .
As we assumed(DEK) Proposition
2 ensures thatT~
is well definedon 4)’. Recall
that 1fr
:=7(/).
Let1frh
:=T ( f )
and letx 1
be such that is a solution of(13) (such x 1 might
not beunique).
In order toprove the uniform convergence of to
To,
we have to estimate the differenceI I ’~’ - ’fh I I ~ .
We do itby bounding
the where1fr1
isgiven by (49),
and thenby using
thetriangular inequality.
We haveThe result then follows
immediately
from the strongapproximability
of~o
and the
weak approximability
of Inparticular
we can takeW3 (h)
=(1
++ D
In the
following
theorem we shall see that theassumptions
of Theorem 1are
also,
in a sense, necessary for the uniform convergence of7~
toTcp
in,C(I~~,, ~).
THEOREM 2. Assume that the sequence
T; is
bounded in£(4)’, 4»,
and con-verges
uniformly to Tcp in 4» (see (50)). Then,
theellipticity in the
kernelproperty
(DEK)
holds true. Moreover, both the strongapproximability of ~o
andthe weak
approximability
aresatisfied.
PROOF. The
(DEK) property
can be obtainedapplying Proposition
3. Let1fr
be an element of4$f.
Thenby
definition of~
there isf
EHcp
suchthat 1fr
=Tcpf. Define 1fr1
:= Uniform convergenceimplies
the strongapproximability
of~o .
In a similar way, let X be an element of
sf!.
Thenby
definition ofSf!,
X =
CsoSoCcpf
for somef
EHcp.
Theremight
be more than one suchf.
Wechooser
suchCsoSoCcpf
=X }
Let_ _
- o
:=
Correspondingly
let1frh
:=T;1
and let Xh be such that(1frh, X h )
isa solution of
(13)
with the sameright-hand
side(such
Xhmight
not beunique).
Then we obtain
which
gives (48)
withúJ1(h) =
that is theweak approximability
D
EXAMPLE 1. We go back to the
Example
1 of the Introduction(Stokes problem).
= V =(Ho (S2))2
and E =Q
= It is easy to seethat if S2
is,
forinstance,
a convexpolygon, ~o
is and~o
is thesubspace
of(H2 (S2) f1 Ho (S2))2
made of freedivergence
functions(see [19]).
Inparticular
we can check^’ i ( u ( I 2
o 0
(with
standardnotation,
here and in thefollowing,
we denoteby II. Ilk
the normin
for kEN). Let Vh
andQh
be finite dimensionalsubspaces
of Vand
Q respectively.
The weakapproximability
willsurely
hold ifwhich is satisfied
by
all choices of finite element spaces that one mayseriously
think to use in
practice.
The
strong approximability
of(DH,
which now readsis more
delicate,
asu 1
has to be chosen inKh.
If thepair (Vh, Qh)
satisfies theinf-sup
condition(DIS)
then the propertytrivially
holds.Remark, however,
thatthe
typical
way toproving
theinf-sup
condition is toshow, following [15],
that:for
every u in V there inVh
suchv c II!!II v (C independent of u qh)
=0 Vqh
EQh, which
is more difficult thanproving (52)
directly.
Moreover there are choices of elements that fail tosatisfy
theinf-sup
condition,
for which(52)
holds true. Forinstance,
we may think to the so-calledQ 1 - Po element,
wherewhere,
with standard finite elementnotation,
for kinteger >
0Pk (D)
denotesthe space of
polynomials
ofdegree k
on a domainD,
andQk(D)
the space ofpolynomials
ofdegree k separately
in each variable.Hence,
hereQ (K)
is the set of bilinear
polynomials
on K, andPo (K)
is the set of the constantfunctions on K. We may assume, for
simplicity,
that Q is a square and that thedecomposition Th
is madeby
2N x 2Nequal subsquares.
It is known that thischoice of elements does not
satisfy
theinf-sup
condition: the operatorBh
hasa non trivial kernel
(the
checkerboardmode),
andby discarding
it we still haveat best
(DISh )
withflh - h (see [22], [18], [6]). Nevertheless,
for u E4) H
C K,we can construct
ul
as follows:let M
be the vector inVh
which is bilinear in each square of the N x N(coarser) grid
and agrees with u at the vertices of the coarsergrid.
Let now M be the vector inVh
with thefollowing properties.
It vanishes at the vertices of the coarser
grid;
itstangential component
vanisheson the
midpoints
of theedges
of the coarsergrid;
its normal component at themidpoints
of eachedge e
of the coarsergrid
is chosen in such a way that~(M2013M2013M)’~ = 0,
andfinally
the values at the center of each element K of the coarsergrid
are chosen tosatisfy
= 0 for qh =sign(x - xc) and qh = sign(y - Yc) (where (xc, Yc)
is the center ofK).
It is not difficultto check that
ul - u -E- u
satisfies(52)
withco2(h) = O(h).
For a similarconstruction see
[9],
pages 241-242. We have here a firstexample
in which theeigenvalues
areapproximated correctly
eventhough
theglobal
matrix associatedto
(13)
issingular.
EXAMPLE 2. Dirichlet
problem
withLagrange multipliers.
Here (D = V =and E = M =
N’~(9~).
It is well-known(see
e.g.[17]),
thatif Q
is,
forinstance,
a convexpolygon,
then~o - H 2(Q)
n and= H 1 ~2 ( a S2 ) .
Let now be aregular
sequence ofdecompositions
of Q(see
e.g.[12]),
be aregular
sequence ofdecompositions
of8Q, k1
andk2
beintegers
with 1 andk2 >
0. SetIt is trivial to check that
(48)
and(49)
hold for every choice of{~~}, kl
and
k2.
Inparticular (DEK),
which is now a sort of Poincareinequality, only requires
thatMh
C M contains at least aTih
such that1 ) ~ 0.
Note that to have
(DIS)
one must ask rather strictcompatibility
conditionson
kl
andk2,
see[2]. Therefore,
for ageneral choice, solvability
of
(13) might
fail.Nevertheless,
as we have seen, convergence of theeigenvalues
is assured under weaker
assumptions.
4. - Problems of the
type 0
g
In this
section, together
with( 1 )-(4),
we assumethat,
for everygiven
g E E’and
f
=0, problem (6)
has aunique
solution(~, X )
and that there exists a constant C(independent
ofg)
such thatIt is easy to see that this
implies (IS)
but not(EK) (see Example
4 of theIntroduction).
Moreover we assume that we aregiven
a Hilbert spaceH8 (that
we shall
identify
with its dual spaceH~ )
such thatwith continuous and dense
embeddings.
Forsimplicity,
we assume that forevery ~
ES,
we have(with
constantequal
to1 ).
We consider the
eigenvalue problem
which in the formalism of Section 2 can be written
As we shall see,
problems
of the type(0)
9 are moreclosely
related to theabstract
theory
of[14]
thanproblems
of theprevious
type().
From now on we assume that the
operator Ts
iscompact
fromHs
into E.We introduce two finite dimensional
subspaces 4$h
andEh
of (D andE, respectively.
Then theapproximation
of(57)
readsthat is
We are now
looking
for necessary and sufficient conditions that ensure the uniform convergence ofT h
toTE
in whichimplies
the convergence ofeigenvalues
andeigenvectors (see (23)).
To start
with,
we look for sufficient conditions.We introduce some notation. and
Ej
be thesubspaces
of (D andS
respectively, containing
all thesolutions *
e 4$and X E E, respectively,
ofproblem (6)
whenf
=0;
thatis,
with the formalism of Section2,
It will also be useful to define the space
4$),
as theimage
ofC~
o S oCs (from
E’ to0).
As
before,
the spaces and~/
will be endowed with their naturalnorms
(see
for instance(47)).
DEFINITION 3. We say that the weak
approximability
with respect toa(., .)
is verified if there existsw4(h), tending
to zero as h goes to zero, such that for every x EE)
and for every wh EKh
Notice that
(62)
is indeed anapproximation
property, as wealready pointed
out for its
counterpart (48).
DEFINITION 4. We say that the strong
approximability of Eo H
is verified if there exists cv5(h), tending
to zero as h goes to zero, such that for every X EEo
there existsXI
EEh
such thatNotice that
(62)
and(63)
are(much)
weaker forms ofassumption
H7of
[14].
DEFINITION 5. An operator
rIh
from (1)(or
from asubspace
ofit)
into4$h
is called a Fortin operator withrespect
to the bilinear form b and thesubspace
Eh
C E if itverifies,
forall w
in itsdomain,
The
following assumptions
for a Fortinoperator
will be useful.- There exists a constant
Cn, independent
of h such that:- There exists
cv6 (h), tending
to zero as h goes to zero, such that for every w it holdsNotice that
(66)
isstrongly
related toassumption
H5 of[14]. However, being
interested in convergence, we have to assume that
w6(h)
goes to0,
while[14]
only
assumes it to be bounded andputs
it in theright-hand
side of apriori
estimates. On the other
hand,
as we shall see,(66)
isactually
necessary forhaving
convergence ofeigenvalues.
This was notpointed
out in[14]
for the verygood
reasonthat, first,
their interest was in apriori
bounds(and
not onnecessity) and, second, they
weredealing
with directproblems (and
not witheigenvalues).
In
particular (66)
is not necessary forhaving pointwise
convergence ofT h
toTs
where(DEK)
and(DIS)
are sufficient. Notice that(as
it is alsopointed
out in
Proposition
1 of[14]) (IS)
and(DIS) imply (65), but,
as we shall see later in this paper,(EK), (DEK), (IS)
and(DIS) (all together) imply pointwise
convergence but not
(66).
THEOREM 3. Let us assume that there exists a Fortin operator
(see (64))
~~ ,
~~h satisfying (65)
and(66).
Assume moreover that thestrong
ap-proximability of Ej is verified (see (63))
as well as the weakapproximability
ofwith respect to a
(see (62)).
Then the sequenceTh
converges toTs uniformly from H~
into E, that is there existstending to
zero as h goes to zero, suchthat
PROOF. We remark first
that,
as it iswell-known, (7)
and(64)-(65) imply (DIS) (see [15]
or[9]).
Thanks toProposition 4, Th
is then well defined.Let g E
Hs
and let(1fr, x ) e V%
xS~
be the solution of(6)
withf
= 0.Recall that X - Let Xh :=
T~g
and let1frh
be such thatXh )
is asolution of
(13) (such 1frh might
be notunique).
In order to prove the uniform convergence ofT h
toT s
we have to find apriori
estimates for the errorLet
g
be such that(g,
X -xhlls and Ilglls’
= 1.hence
Ilglls’
= 1(see (47)).
Then we haveLet us estimate
separately
the two terms in theright-hand
side:Using (65)
we obtain thefollowing
estimate forPutting together (68), (69)
and(70)
andusing (63)
we obtainTo conclude the
proof
it remains toestimate 111/1 - 1frh Ila.
Thanks to thetriangular inequality
and to(66)
we boundusing
also(62)
and
(64).
Notice thatbelongs
toKh.
which,
due to(66), gives
and
(67)
holds withREMARK 2. In Theorem 3 we have
proved
the uniform convergence ofT h
to
Tg
in~(77s, E).
However in Section 2 we have seen that the convergence of thespectrum
isequivalent
to the uniform convergence ofT h
toTg
in~(77s).
Indeed the latter holds under the weaker
assumption
that there exists a Fortin operatorsatisfying only (66)
as we shall see in thefollowing
theorem. DTHEOREM 4. Let us assume that there exists a Fortin operator
(see (64))
I>~,
-~satisfying (66).
Assume moreover that both thestrong approximability (see (63))
and the weakapproximability of E
withrespect
to a(see (62))
are
verified.
Then the sequenceT h
convergesuniformly to Tg
inHg.
PROOF. We observe that
(7)
and(64) imply
the weak discreteinf-sup
con-dition
(DISh).
Thanks toProposition 4, Th
is then well defined.Let g
EHg
and let(1fr, x) e 4$%
x be the solution of(6)
withf
= 0.Recall that X =
Tgg.
Let Xh : := and let1frh
be such that(1frh, Xh )
is asolution of