JSTOR - Stabilized Wavelet Approximations of the Stokes Problem

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Stabilized Wavelet Approximations of the Stokes Problem Claudio Canuto; Roland Masson Mathematics of Computation, Vol. 70, No. 236. (Oct., 2001), pp. 1397-1416. Stable URL: http://links.jstor.org/sici?sici=0025-5718%28200110%2970%3A236%3C1397%3ASWAOTS%3E2.0.CO%3B2-6 Mathematics of Computation is currently published by American Mathematical Society.

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hI1THEilI.ATICS or COhlPL~T.ATION Volumc 70, hTumbcr 236. Pagcs 1397-1-116 S 0025-5718(00)01263-1 Article electronicall? p u b l i r l ~ e d011 J u l y 21. 2000

STABILIZED WAVELET APPROXINIATIONS

OF THE STOKES PROBLEM

CL.AUDIO CANLTO AND ROLAND hIASSON

XBSTR-ACT.TVe propose a n e ~ vconsistent. residual-based stabilization of the Stolces problem. Tlie stabilizing term involves a pseudo-cliffercntial operator. defined via a ~vavcletexpansion of the test pressures. This yields control on the full L2-norm of the resulting approximate pressure independently of any discretization parameter. Tlie methocl is particularly ~vellsuited for being applied ~vitliinan aclapti~ediscretization strategy. T5k detail the realization of the stabilizing term tlirougli biortliogonal spline wa\-elcts, and we provide soiile numerical results.

Wavelet bases are being increasingly used in the numerical solution of partial differential and integral ecllsations (see, e.g.. [Da2, Co] ancl the references therein). There are many aspects in a discretization procedure for such equations that can benefit from the features of these bases. TVavelets share xi~ithother multilevel methods the capability of easily preconditioning the discrete realizations of symlnetric positive definite operators. More typical of wavelets is their ortliogonality to certain classes of smooth functions (e.g.. polynomials), a feature whicli can be exploited in the cornpression of dense matrices ancl-in a more general context-in the design of adaptive discretization strategies. The finite-dimensional space. xi~hicliis used in a Galerltin-type approximation, is aclaptivell- constrlscted by including in it precisell- those lvavelet basis functions that have the potential of representing the rnost significant structures of the solution. From this point of view, wavelet projection rnethocls can be viewed as meshless snethods, ~ v i t ha highly flexible meclianisrn for aclding/removing degrees of freedom.

lT7aveletswere originally introcl~~cecl

in unbounded domains, with a shift invariant property (see [IIe]). Currently, wavelet bases are available ancl easily computable on fairll- general dornains in an arbitrary dimension. A popular st,rategl-of construction consists of decomposing the domain into the union of srnooth irnages of a tensor product reference domain. The wavelets are themselves images of tensor product ~vaveletson sucli a reference dolnain; t,his alloxi~sthe by now well-developed wavelet Received by the editor June 3 , 1999 and, in revised form, October 18, 1999,

2000 hIathematics Subject Classlficatlon. Primary 65N30, 65N12, 42C15.

K e y words and phrases. Stokes problem. inf-sup condition, stabilization, ~vaveletbases.

Tliis ~vorlcwas partially supported by tlie European Coniiilission within tlie T M R project

(Training and hlobility for Researchers) TVa\-elets and hIultiscale hletliods in Numerical Anal?-sis and Simulation, No. ERB FhIRX CT98 0184, and by tlie Italian f~nlcls~ I U R S40% T .Analisi Nnmerica.

13!)8

I!I.XlrDIO C.ASUTO A N D ROL-ASD hl.I\SSOS

technolog>- on tohe lmit interval to be efficiently exploited (see [DS. CTU. Ck12.

BCUj). Flnid dj-ualnics is a challellging ficlcl of applictxtion for wavelet-based adaptive discret,ization rnetl~ods,since tjpically a flow exhibits well-localizcd structures and/or coherent vortcx patterns. For illcolnpressihlc iiows, the Stokas problem is a siniplified rnoclel ~vhicllneglects convection and focusses on the viscous effects and thc divergence-frcx constraiilt. For these reasons. it has received considerable attentioli by m-avelet a.ddicts. beginnillg with the pioneering nrorii of Lemari6 [L:. in ~vhichdivergence-fi.t:e wavelet basts mere constn~ctcclin an lunbollnded doinail1 (see also [UI). Hon~t~ver, for arbitrary k)ounded c1orna.ius and boundary conditions, divergence-free basis finictions are tlificult to build; lieilce, one restorts t,o the discreti~at~ion of t,lie continuit?- equation: thus coping with the well-ltno~vn inf-sup condit~on(see IBF]) which ties together discrete velocity and pressure spaces. Several xvaxelet discretization lnet~hodshave been recently proposed whicll f~~lfill that, condition (see [DKUl, 91a21). This task is relativelj- easy in t,lie ca.se of nonadaptive al>proxinlations, ~vllenthe discrete velocity aud pressure spaces are uniform, i.e.. 2111 the ~i~avelet basis f~nlnctioas1111 to a certain maximal level arc included. The situation becomes considerably lnore intriguing in t l i ~ adaptive case, when the need for fillfillirlg the iaf-sup condition contrast,^ against thc desire to choose, the discrete velocity and pressnrc spaces as iiideporrdent,ly as possible, guided oiily by the local structure of the flow. Instead of satisfging the inf-sup couditiolz, one (:all circurnue~n,t it. This alternative approacli, first- proposed by EIughes, Franca and Balestra [EIFB] and n o ~ i ~ popular in the finite-clement co~nni~lnitp (see, e.g.. iI3Fj). can be rcalized by appending a suitable consistent (i.e., vanislzing for t,llt, cxact solution) stabilization tern1 l o the continility equation. It prevents the onsci of spurious oscillat,ions in discrete pressure, making the approximate problelil \~rc:llposed. The same effect call be achieved by adding and then st,nt,ically condensing auxiliarl- velocity functions, the "1)llbbles" (see [BFHR]). The typical stabilization term acts at the elenlental levcl, and. through tbtl choice of local tuning paramei,ers, it provides stabilization bj- coiltrolling a mesh-dependent wc:ighted nor111 of tilo pressure gra.dient,. Among the features of wavelets (as well as ot,ller ~riultilevelbases, see. e.g., [BPI), we recall the possibilitj- of easily rel,rt~sentingnornls and inner protlilcts in Sobolev spac(:s of fractional ; ~ n deven negativt, order. This call be exploited to design new stabilized discretizations of operator equations. Sucli forlnulations are opt,imal froin the point of view of thc functional se1.tiag. Soine results on the use of \\-;welets in this clircct-ion already exist. In [Bj: a ~nultilevelleast-square stabilization of the Stokes problem is considered. ~vllereasin [BCT] a multilc~-(~l SUPG-type st abilization of the convection-diffusion ecluat,ion >-ielcls control on soille norill of fi.act,ional order 112 f o t~he solutioii. Iu the present paper, we exploit mt~veletst o clcsigll a lleii~,consist,t:nt, residualbased stabilization tol.lil for the Stoktls problem. wl-iich replaces thc classical term int,rotluced in IHFB] for finite elemellts. Our tcrln yields direct colitrol of the f ~ dL2l norm of the pressurt., independently of any discretizatiol~parameter. Furthermore, basic,alll- no information on the discrete velocity space is needed. Consequently. for t,he discretization of the problem in an the rrlcthocl is particl~larly~ i ~ esuited ll adaptive fra~nem;orkas described a1)ovt. Technically speaking. our tc\rrn exploit,^ the expa~isioliof the discrete pressures ill a ~vaveletbasis (associated \iritl-~a possibly aoncolifornling deconiposition of the dolllaill into macroelelnents), 8.11~1the existelice of a local dual basis. -4 local right-inverse of the tlivc.~.genceoperai,ol.is easily built

STABILIZED T'vAVELET APPROSIT\IATIONS O F T H E S T O K E S P R O B L E h I

13'3'3

on the space spanned by the latter basis. This operator is used to define the test functions for the residual of the momentum equation, thus yielding the stabilization device. Tlie content of the paper is as follows. First, we introduce our stabilization tern1 in an abstract setting, and we prove that it implies a uniforln inf-sup condition for both the continuous and the discrete velocity-pressure pairs of spaces. This yields optimal a priori ancl a posteriori error estimates. ^\Jest,we detail a particular construction of the stabilizing wavelets, based on the Cohen-Daubechies-Feauveau [CDF] biortliogonal spline wavelets on the real line. Finally. we describe the results of some numerical tests which demonstrate the feasibility of the proposed method. Tlie following notation will be used througliout the paper. If, for i = 1,2. 1% are nonnegative f~lnctionsdefined on sets A, xi~hich may depend on certain parameters, then lVl(al) 5 1V2(a2)means the existence of a constant c independent of these parameters such that ( a l ) 5 clY2(a2): Val E Al , \a2 t AD IIoreover, i\il(al) 1Y2(a2)means lYl(al) 5 1V2(a2)and 1V2(a2)5 lYl(al).

-

2. AN ABSTRACT

FORhI OF THE STXBILIZXTION hIETHOD

>

Given a bounded clornain R c Rd (d 2) wit11 Lipscliitz boundary dR,1%-ew ant to approximate the Stoltes problem submitted to homogeneous boundary conditions:

+

Let us introduce the function spaces X = ( H i ( R ) ) dfor velocities and A l -. L 2 ( R ) / R (so that L2(R) = l l I e s p a n ( 1 ) ) for pressures. Existence and uniqueness of a solution P) E 2 x !lI folloxi~classically from the assumption E 9' = ( H - l ( R ) ) d . Tliroughout t,he paper, both the (L'(R))"-inner product and the !lI-inner product will be denoted by (., .); the symbol (., .) will indicate the duality pairing betxi~een 2' ancl 2. Let us equip 2 by the norm . c ' > ? = (V!?: and A l by the norm

(c:

1

llqlln.r = (q3 In order to define the stabilizing operator and the approximation spaces, let us assume that I11 is split as follo1i~s: where the complementarl- spaces A I o ancl M are closed subspaces of JI and satisfy the following conditions: i) ,b4 admits a Riesz basis \ZIP = {I$,: X t CP}, i.e.. ,tU = clos spanap with AEL7'

for all q = (qx)xEc7,E t2(Cp): ii) ,llois a(possiblS- emptl-) finite-dimensional subspace of d l : if d l o # @.t hen a of r;' is associated to it, such that the uniforln finite-dimensional subspace znf-sup condition

zo

390

>0

:

inf qEl1In

sup

zE.eaI ~ I q2 l \ I

-

CLAI-DIO CATT;TO AND ROLAND 3,IASSON

1,100

As a coi1sequenc.e of i) allti the Riosz represellt;al-,iotheorenl, tllc.rc3 cxist,s a dual {(Ti : X t CP} c ]If;: thiis: bioitllogonal set $"

and an\- q t -44 call be rcpresenteti as

(-7,

Next,, nTc.associate a fill~ct,ion E 2 aud a coefficient c~ > 0 to each X t LP. 'I'l~ese f ~ ~ n c t ~ i oarltl n s coefficiel~tsare chosen in such a way tllitt tht: operat;or S (ihrmallp) defined as

i)

.f is 1~oun3cdflorn ,l/l t o

ii) there exists a, constant (2.10)

-

thus there ~ x i s t s coilstdllt

C,

> U bi~chthat

9,> O silch t,l\at

-(Cq.

2 ~w~l~ll~,I,

vq t M

;

iii) t,hc orthog;malit?; relation 4

(2.11)

%qo t -?/I(,.Vq E

(Cqi,: Sq) = 0.

";u

11oltls.

LT7(> shall see holow ho\v ssllclr conditiolls call h r fillfilled. Finally. let us select, (b3- some aclaptive proccdurc., that me will not dotail lserc) a t 12") and iItir = ,\lo S J U !\r) . 1 ~set _JU1\1, = span{(,'f : finite sllbset hP C CP;1 ~ US + + Flirl.klernsore, lei, us select a fillit,(;-dimcnsiona1 suk~spaccXhlC X: colitailsilig t hc suhspacc. defin(~lin (2.6). Note that need not contain ally of the velocities + + UX 1~1iicherster illto tlre clefinition of t,he stahiliziilg opcrator S. 1% consider the follon-ing consist,e~~tljstabilizt~ciGalerltiil d i s c ~ ~ t i z a t i oofn prob

lem (2.1)-(2.3): Find 17 E ancl p E ~\l.\~' such that,

xAIv

f.\,.

is the residual of the lnoll~eiltu~li eqilatiori, and pressures are split as q = q0 $-q . t~ i; x M equals j / ('T. qj//$x,, = I / ? - ? 2

B!(zZ, ,r), ( Z q)]

1 1 (2.q) ![-2x,71 r)'!-TxA31

2 x 211 o r -glI7x I\Il\p; and

+ 6qi/;I.

Prooj. 'Taking (c',q ) = ((2, r )

E

lj,

flit:

norm in 2 x 31 t,s scaled as

)7 x M and using (2.10) x i d (2.11). one gcts

< &a' + $b2 with all

Or1 tile other hand. by (2.9) and the Holder inerlua!it>- job app~,opriatechoict: of 17 > 0. we ~ : I T T C (

A

)

< ~

+

~ A ~ I : i ~5 i~ i ,~ ~~ Z ~ i~~ r~ ~~ i~~ iS lr ~~ ~ i 1 ~

l'lirxrefore, we obtain

. r). (

I

r )

(1 - 6

c2

2 3,

)1

+ - Ll*

,

~;)/IT/c( u

This gi.res the desired rcs~lltif ;\Icl = @. From now oil. lot us asslline that A10 # In. + Recallilig !2.6), lvt 'Go t .yo c X be s~~clrr t h a t (77. 7% r.0) do 11 6I!,? Ij7.o i ! 2 q ~ Iw ith //'Zoll = Y llroJiAI.-; > 0 to be delilied. Then,

>

(

r

(

-

)

=

-(V~2.O&)+(V.1~~r,~)i(O.ifg.1~o)

2 2 -~ll~~llA~I r~. ol ln~~ ~ , ~ ~ l l + ~ ~ 3i or oI ll~l o. ,~~~1 .

Usillg again applopriatc IXolder iacqualities for thc first and secolid t el llss. we gc.t

CLAUD10 CAYUTO AND ROLAND hIASSON

1102

Suinniing up. \ire obtain

Now. 1%-echoose

-, =

3,130 The conclusion follo~vs.t aking into account (2.4). 4

Proposition 2.1 allo~vsus to establish classical abstract a priori and a posteriori error estimates between the solutions of problen~s(2.1)-(2.3)and (2.12)-(2.13). Since proofs are by now standard, we only give the results, in which the dependence upon the stabilization parameter S has been made explicit.

Proposition 2.2. Under the cond~tzons(2 5)-(2.6) on the pressure spaces. (2 9)(2.11) o n the stabzlzzzng operator 2 and (2.16) o n the stabzlzzatzon parameter 6.the followzng a przorz estzmate holds:

( c- u'l12 + 611211~-I ~ ) / n j

in_f

V'EX\L

1 + +1 b -~ 1 ' 2 ~ G V ~ ~ , z i )

( G-~

-

Proposition 2.3. linder the aboue condztzons, the followzng a posterzorz estzmate holds:

(2.19)

+

sup

inf

6 ~ ' / ' (V . u'. Q

-

q)

+

11&11.t{

Q E M qt-%f,$P

/ ( r & . Z(Q - q)) 1 .

1f7eend this section by indicating a natural way of choosing the auxiliary func+ tions u);. X E CP,in such a wa>-that the crucial condition (2.10) is easily fulfilled.

Proposition 2.4. For any X E C P , let

@:

E 2 be such that

Furthermore; set

i n the definition (2.8) of the operator satisfied.

2. T h e n , conditions

(2.10) and (2.11) are

Proof. For any q = CXtLP qXv: E il4, we have -(Vq.Sq)

=

(q. c .Sq) =

1q,(q.V.)I ; XELP

=

1q:. XECP

and (2.10) follows from (2.5). On the other hand. (2.11) is a consequence of the inclusion GP c Ail:.

Remark 2.5. The choice (2.21) for the coefficients of S? is the simplest one from the theoretical point of view. A control of the L2(R)-norm of the pressure is assured as -cvell if we relax the condition into ex -- 1.VX E C P . This enhances flexibility and efficiency in our stabilization scheme, allowing for a local (in scale and position) tuning of the coefficients.

STABILIZED llrl\. E L E T .SPPROXIhI4TIOSS O F T H E S T O K E S P R O B L E h I

1403

In this section. we shall construct a Riesz basis 9 p for the pressure space. as well as its companion biorthogonal basis G p 3 for which it is easy to define auxiliary velocities fulfilling condition (2.20). At first. we consider the reference domain fi =]0. l[d.where simple tensor product arguments can be used. starting from biorthogonal bases on the interval (0.1) s u i t a b l ~nlodified a t the endpoints. Next. we shoxv how our construction can be transported on a domain n. ~vllichis the smooth image of the reference domain. Finally. we deal -cvith Inore general domains, using domain decomposition ideas. 3.1. Biorthogonal wavelets on the interval 10.1[. As usual this construction is obtained b>-restriction to ] O . 1 [ and adaptation a t the edges 0 and 1 of biorthogonal wavelets on the line. The present construction onl>-differs from existing ones (see [CDV], [AJP] for the basic ideas and also [DKU2], [LIal], [GT] for various specific topics) in our choice of H i ( 0 . 1 ) boundary conditions for the dual wavelets, while the primal wavelets (defining the pressure) have optimal order of approximation. This specific choice is needed in order t o build the stabilizing ivarelets $'%y tensor products. The construction on the line (see [CDF]) starts from a pair of compactly sup(xvith integer ported scaling functions (d,); of supports [-nxo, m l ] and [-??Lo, edges) sat,isfying the two scale relations

for finite rnaslts h and

h. and the biort11ogonalit~-relations (d, d ( x - k)) = 6k for all k E Z.

The primal and dual multiresolution analysis (LIRA) spaces

are spanned by the biorthogonal compactly supported bases

It is sho-cvn in [CDF] that the primal and dual wavelet spaces

lz17, (R) := V , + l ( ~ n ) V , ( R ) ~E . 3(x):= V,+,(x) n &(R)-L

are spanned by the biorthogonal compactly supported wavelet bases

-cx7herev := figmo(2. - nx) and := &ij,o(2. - nx) are thc mother xvavelets obtained from the wavelet masks g, = (-l)mhl-n, and ij, = ( - l ) m l z l ~ m .The

multiscale biorthogonal basis are defined (at least formally) by Q :=

U

U Q j and 9 := 6ou U j

EN

$j:

3 EX

it is convenient to denote them by Q := {qx,X E C), G := {GA3X E C), where X stands for (j,k) with 1X1 := j. In [CDF] such cornpactl>-biorthogonal generators ( d , $) are built with arbitrary smoothness. We denote by r and 7. the supremum of their smoothness measured in HS. In addition, such generators can be built with arbitraq- order of approxilnations n and fi in the sense that the integer translates of d and span the polynomials of and Then it is sho~vnthat Q (and s~-mmetricallyG) is stable in H S ( R ) for the range min(?. 6 ) < s < min(r, 72) in tlle sense that f llHb(C) (EXEC 22X1Sl(f~ j ~ ) ( ~ ) ~ / ~ for all f E HS(R).

4

N

-

Hgpotheszs 3.1. IT-e shall assume in the folloiving that the pressure generators are chosen so that 72 2, fi 2 and r > 0. 7. > 1.

>

>

Starting frorn a pair of biorthogonal gznerators on tlle line (a, G ). all the constructions of new LIRA spaces & and & on the interval share the basic ideas introduced in [CDV' and [AJP] to retain: (i) the .'interiorn scaling functions on the line whose supports are in [do2-3. 1 -

d12-3] for 4 and [do2-'. 1 - d12-31 for V,. (where d = (do.d l ) , d = (do.dl)

are pairs of nonnegatis-e integer parameters);

truncated linear

(ii) at the edges 0 and 1, only the rr (for 55) and 6 (for colnbinations of scaling functions that correspond to the reproduction on ] O . l [

of the rnonolnials of degrees a = 0.. . . . n - 1 (for &) and a = 0... . fi 1

(for V, ).

6)

.

-

Then, the optirnal orders of approximation n and 17, and the nestedness are preserved. This s t r a t e g ~enables us. in addition, to take into account hon~ogeneousboundary conditions at the edges 0 or 1 for the primal or dual LIRA. It suffices to retain in the previous definition o n l ~the rnonolnials satisfying the sarne boundary conditions at the edges E = 0 . 1 . For our purposes. the dual LIRA will satisfy holnogeneous Dirichlet boundary conditions at both edges while the primal LIRA does not satisfS an> boundary condition. Recalling that Supp d = [-mo. m l ] . the primal LIRA 55 is defined as follox~rs. with CL = (CLo. CLl) = (-1. -1): @j,ic,,O,ll,

k = mo t d o , . . . . 2 j - m l

-

dl

,

0 0 , k ) Oj,kl[O,lj

a:')

:=

a =CLo+l; ...; nal -l+dl = k = - T n o - l ( ( - l ) a ~ a ,0 0 . - k ) h3.2.i-k I [ O , l l '

{d,~~~dl

a = C L ~ t. l. . . ,n - 1

55 := span (@I0));i span (a;"')

; i span

( a(1) j ).

ST4BILIZED T\.A\

E L E T APPROXILIATIOSS O F T H E STOICES PROBLELI

1405

C'L

Tlle dual LIRA is defined similarly with d = (do.dl) and = (0,O). These definitions of V, and hold ~vllenes-erJ J Q for a coarse les-el J O such that there is no overlapping of the subscripts in tlie definitions of and 5;'". In order to obtain biorthogonal LIRA spaces. the parameters d and d are chosen to match tlle dimensions of separately at both edges. i.e., for E = 0 , l and

>

(3.4)

m- + d,

-

+ CL,

= m,

+ d,

-

-

?l. t CLE.

Then, tlle LIRA spaces admit local biorthogonal Riesz bases @, and &, if and only if txvo (one at each edge) fixed (independent of J ) matrices are nonsingular. which will always be assumed in the following. This property 1s shown in [DKC2j for spline biorthogonal wavelets. Tlle biorthogonal bases cP, and 5, mill include a fixed number of lnodlfied scaling functions at the edges xvhile all the other basis functions are scaling functions on the real line d, i, restricted to the inters-al. The construction of local blorthogonal Riesz bases (Q,. G,)of the was-elet spaces n I>'-. 11; .= %+1 n IjL leads t o a fixed number of locally supT I > := ported modified wavelets at the edges while all the other xvavelets are restrictions of was-elets on the real line (3.2). In addition, tlle modified was-elets at the edges are st111 dyadic dilate fnnctlons. For example, tlle primal was-elet basls writes

xvitll u,+1 C\ = fit), C\(2.)and e),+l,21,~-3-1(1 .) = &v, 2 7 - 3 - 1 ( 1 - 2.). In order to prove the boundedness of the stabilizing operator defined in the next subsection. we need to extend hIeyer's "vaguelette lernma" stated on the line t o the interval 10. l[. Let us first recall tlie lemma on the line (see, e.g.. [Alel). -

Lemma 3.2. Let 7 be a compactly supported functzon such that some a > 0 and SR 7 = 0. T h e n for all f E L2(X)

rj E

Ha(R) for

o r equiualently

Lemma 3.3. Let rj be a compactly supported functzon such that r j E H u ( R ) for some a > 0 and rj = 0 LIIoreouer. let i l i O ) . a = 0 , . . U O . be compactly supported fc~nctzons of Ha(jO.m[): szmzlurly. let i 7 F ) . 3 = 0. . . . 111. be compactly supported functzons of H u ( ]- x.1[) For j > 10.let us set K , = (0. . . .23 1) and. for any h - E K,,

sE

-

1408

CL.SUDI0 C A S U T O .SND R O L A N D hIASSON

Then for all f E L2()0.1 [ )

Proof. Let

f the extension by 0 of f

to

X.Applying Lenxna 3 . 2 one has

lib01.

);7 on the line so On the other hand. one can always extend the functions that these extensions are compactly supported functions of H n ( X ) with vanishing mean values. Applying again Lelnnla 3 . 2 to these extensions yields (3.5). Finally. . (3.6) is equivalent b> transposition of the operator f + ( f , q3 h)3230~ E K ,property to (3.5).

For exanlple, applying Lelnllla 3 . 3 to the primal and dual wavelet bases Q := Uj2jo Q j and G := $j0 U Uj,30 Gj. ~ v ededuce that, they are biortllogonal Riesz bases. Using the cornbination of inverse and direct estimates in 11; and interpolation theor>-will yield more precisel>-the stability of 9 in H&(]O. 1[)for the range 0 5 s < ~min(.r:7 2) and s>-mmetricallyfor (see. e.g., [Co], [Dal]). Qjo U

3.2. Pressure and stabilizing wavelets on the reference domain. Let us consider the biorthogonal d1RA in ~ ~ ( f i )

The corresponding I\-avelet spaces 11; tensor product waxrelet bases

(6)(and similar1~-

where E := {0,1)" {(O.. . . , 0 ) ) and Ojm = QJ if lye decornpose V,, (fi) as

and correspondingly

-

-

with

E,

= 0,

-

yo(R) = span{qo) 9Kfjo

ifij0 I1 and A