Stabilized Wavelet Approximations of the Stokes Problem

Stabilized Wavelet Approximations of the Stokes Problem Claudio Canuto

y

Roland Masson

z

May 19, 1999

Abstract. We propose a new consistent, residual-based stabilization of the Stokes problem. The stabilizing term involves a pseudo-dierential operator, de ned via a wavelet expansion of the test pressures. This yields control on the full L2 -norm of the resulting approximate pressure, independently of any discretization parameter. The method is particularly well-suited for being applied within an adaptive discretization strategy. We detail the realization of the stabilizing term through biorthogonal spline wavelets, and we provide some numerical result. Key words. Wavelet bases, Stokes problem, inf-sup condition, stabilization. AMS subject classi cations. 42C15, 65N55, 65M70.

1 Introduction Wavelet bases are being increasingly used in the numerical solution of partial dierential and integral equations (see, e.g., [Da2, Co] and the references therein). There are many aspects in a discretization procedure for such equations that can bene t from the features of these bases. Wavelets share with other multilevel methods the capability of easily preconditioning the discrete realizations of simmetric positive de nite operators. More typical of wavelets is their orthogonality to certain classes of smooth functions (e.g., polynomials), a feature which can be exploited in the compression of dense matrices and { in a more general context {in the design of adaptive discretization strategies. The nite dimensional space, which is used in a Galerkin-type approximation, is adaptively constructed by including in it precisely those wavelet basis functions that have the potential of representing the most signi cant structures of the solution. From this point of view, wavelet projection methods can be viewed as meshless methods, with a highly exible mechanism for adding/removing degrees of freedom. This work was partially supported by the European Commission within the TMR project (Training and Mobility for Researchers) Wavelets and Multiscale Methods in Numerical Analysis and Simulation, No. ERB FMRX CT98 0184, and by the Italian funds Murst 40% Analisi Numerica. y Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy, e{mail: [email protected]. z D epartement Informatique et Mathematiques Appliquees, Institut Francais du Petrole, BP 311, 92852 Rueil Malmaison Cedex, France, e{mail: [email protected].

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Stabilized Wavelet Approximations of the Stokes Problem

2

Wavelets were originally introduced in unbounded domains, with a shift invatiant property (see [Me]). Nowadays, wavelet bases are available and easily computable on fairly general domains in an arbitrary dimension. A popular strategy of construction consists of decomposing the domain into the union of smooth images of a tensor product reference domain. The wavelets are themselves images of tensor product wavelets on such reference domain; this allows the by-now well-developed wavelet technology on the unit interval to be eciently exploited (see [DS, CTU, CM2, BCU]). Fluid dynamics is a challenging eld of application for wavelet-based adaptive discretization methods, since typically a ow exhibits well-localized structures and/or coherent vortex patterns. For incompressible ows, the Stokes problem is a simpli ed model which neglects convection and focusses on the viscous eects and the divergence-free constraint. For these reasons, it has received considerable attention by wavelet addicts, beginning with the pioneering work of Lemarie [L], in which divergence-free wavelet bases where constructed in an unbounded domain (see also [U]). However, for arbitrary bounded domains and boundary conditions, divergence-free basis functions are dicult to build; hence, one restorts to the discretization of the continuity equation, thus copying with the well-known inf-sup condition (see [BF]) which ties together the discrete velocity and pressure spaces. Several wavelet discretization methods have been recently proposed, which ful ll that condition (see [DKU1, Ma2]). This task is relatively easy in the case of nonadaptive approximations, when the discrete velocity and pressure spaces are uniform, i.e., all the wavelet basis functions up to a certain maximal level are included. The situation become considerably more intrigued in the adaptive case, when the need of ful lling the inf-sup condition contrasts the desire of choosing the discrete velocity and pressure spaces as much independently as possible, only guided by the local structure of the ow. Instead of satisfying the inf-sup condition, one can circumvent it. This alternative approach, rst proposed by Hughes, Franca and Balestra [HFB] and now popular in the nite element community (see, e.g., [BF]), can be realized by appending a suitable consistent (i.e., vanishing for the exact solution) stabilization term to the continuity equation. It prevents the onset of spurious oscillations in the discrete pressure, making the approximate problem well-posed. The same eect can be achieved by adding and then statically condensing auxiliary velocity functions (see [BFHR]). The typical stabilization term acts at the elemental level, and, through the choice of local tuning parameters, it provides stabilization by controlling a mesh-dependent, weighted norm of the pressure gradient. Among the features of wavelets (as well as other multilevel bases), we recall the possibility of easily representing norms and inner products in Sobolev spaces of fractional and even negative order. This can be exploited to design new stabilized discretizations of operator equations. Such formulations are optimal from the point of view of the functional setting. Some results in this direction already exist. In [B], a multilevel least-square stabilization of the Stokes problem is considered, whereas in [BCT] a multilevel SUPG-type stabilization of the convection-diusion equation yields control on some norm of fractional order 1/2 for the solution. In the present paper, we exploit wavelets to design a new, consistent, residual-based stabilization term for the Stokes problem, which replaces the classical term introduced in [HFB] for nite elements. Our term yields the direct control of the full L2 -norm


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of the pressure, independently of any discretization parameter. Furthermore, basically no information on the discrete velocity space is needed. Consequently, the method is particularly well suited for the discretization of the problem in an adaptive framework as described above. Technically speaking, our term exploits the expansion of the discrete pressures in a wavelet basis (associated with a possibly non-conforming decomposition of the domain into macro-elements), and the existence of a local dual basis. A local right-inverse of the divergence operator is easily built on the space spanned by the latter basis. This operator is used to de ne the test functions for the residual of the momentum equation, thus yielding the stabilization device. The content of the paper is as follows. At rst, we introduce our stabilization term in an abstract setting, and we prove that it implies a uniform inf-sup condition for both the continuous and the discrete velocity-pressure pairs of spaces. This yields optimal a-priori and a-posteriori error estimates. Next, we detail a particular construction of the stabilizing wavelets, based on the Cohen-Daubechies-Feauveau [CDF] biorthogonal spline wavelets on the real line. Finally, we describe the results of some numerical tests which demonstrate the feasibility of the proposed method. The following notation will be used throughout the paper: if, for i = 1; 2, Ni are non-negative functions de ned on sets Ai which may depend on certain parameters, then < N2(a2 ) means the existence of a constant c independent of these parameters N1 (a1) such that N1(a1 ) N2 (a2); 8a1 2 A1 ; 8a2 2 A2 . Moreover, N1 (a1) N2(a2 ) means < N2(a2 ) and N2 (a2) < N1 (a1). N1 (a1)

2 An abstract form of the stabilization method

Given a bounded domain IRd with Lipschitz boundary @ , we want to approximate the Stokes problem submitted to homogeneous boundary conditions: ?U~ + rP = f~ in ; (2.1) r U~ = 0 in ; (2.2) U~ = 0 on @ : (2.3) Let us introduce the function spaces X~ = (H01( ))d for velocities and M ' L2 ( )=IR (so that L2( ) = M span(1)) for pressures. Existence and uniqueness of a solution (U~ ; P ) 2 X~ M follow classically from the assumption f~ 2 X~ 0 = (H ?1( ))d. In the sequel, both the (L2 ( ))d -inner product and the L20( )-inner product will be denoted by (; ); the symbol h; i will indicate the duality pairing between X~ 0 and X~ . Let us equip X~ by the norm k~vkX~ = (r~v; r~v)1=2 and M by the norm kqkM = (q; q)1=2. In order to de ne the stabilizing operator and the approximation spaces, let us assume that M is split as follows: M = M0 M; (2.4) where the complementary spaces M0 and M are closed subspaces of M and satisfy the following conditions:

Stabilized Wavelet Approximations of the Stokes Problem i) M admits a Riesz basis p = f

k

p

X

2rp

4

: 2 rpg, i.e., M = span p with

q p kM kqk`2(rp) ;

(2.5)

for all q = (q )2rp 2 `2(rp); ii) M0 is a (possibly empty) nite dimensional subspace of M ; if M0 6= ;, then a nite dimensional subspace X~ 0 of X~ is associated to it, such that the uniform inf-sup condition (r ~v; q) 9 0 > 0 : qinf sup (2.6) 0 2M0 ~v2X~ 0 k~v kX~ kq kM holds. As a consequence of i) and the Riesz representation Theorem, there exists a dual biorthogonal set ~ p = f ~p : 2 rpg M0?; thus, ( p ; ~p ) = ; ; 8; 0 2 rp; (2.7) 0

0

and any q 2 M can be represented as

q=

X

2rp

q

p

with q = (q; ~p ): 0

Next, we associate a function ~ 2 X~ and a coecient c > 0 to each 2 rp. These functions and coecients are chosen in such a way that the operator S~ (formally) de ned as X X (2.8) 8q = q p ; S~ q := cq ~ ; 2rp

2rp

satis es the following conditions: i) S~ is bounded from M to X~ ; thus, there exists a constant c > 0 such that kS~ qkX~ ckqkM ; 8q 2 M; ii) there exists a constant > 0 such that ?hrq; S~ qi kqk2M ;

8q 2 M;

iii) the following orthogonality relation holds: hrq0; S~ qi = 0; 8q0 2 M0 ; 8q 2 M:

(2.9) (2.10) (2.11)

We shall see in the sequel how such conditions can be ful lled. Finally, let us select (by some adaptive procedure, that we will not detail here) a nite subset p rp; let us set Mp = spanf p : 2 pg and Mp = M0 Mp . Furthermore, let us select a nite dimensional subspace X~ v X~ , containing the subspace X~ 0 de ned in (2.6). Note that X~ v need not contain any of the velocities ~ which enter into the de nition of the stabilizing operator S~ .


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We consider the following consistently stabilized Galerkin discretization of problem (2.1)-(2.3): nd ~u 2 X~ v and p 2 Mp such that (2.12) (r~u; r~v) ? (r ~v; p) = hf~;~vi; 8~v 2 X~ v ; ~ (2.13) (r ~u; q) + hres; ~ S qMi = 0; 8q 2 Mp ; where > 0 is a suitable stabilization parameter, res ~ = f~ + ~u ? rp 2 X~ 0 is the residual of the momentum equation, and pressures are split as q = q0 +qM 2 M0 M according to (2.4). In order to study this problem, let us introduce the bilinear form B : (X~ M )2 ! IR de ned as B [(w~ ; r); (~v; q)] = (rw~ ; r~v) ? (r ~v; r) + (r w~ ; q) + hw~ ? rr; S~ qM i; (2.14) as well as the linear form F : X~ M ! IR de ned as F (~v; q) = hf~;~vi ? hf~; S~ qMi:

(2.15)

Then, problem (2.12)-(2.13) can be rewritten as: nd (~u; p) 2 X~ v Mp such that B [(~u; p); (~v; q)] = F (~v; q); 8(~v; q) 2 X~ v Mp : Note that the exact problem (2.1)-(2.3) can be equivalently written as: nd (U~ ; P ) 2 X~ M such that B [(U~ ; P ); (~v; q)] = F (~v; q); 8(~v; q) 2 X~ M: Thanks to condition (2.9), both forms B and F are continuous on their spaces of de nitions. The following result will guarantee existence and uniqueness of the solution of the above variational problems.

Proposition 2.1 Let satisfy the inequality min (1; 4c2 2+ 2 );

(2.16)

; sup k(w~ ; rB)[(k w~ ; r)k;((~v~v;; qq))]k (w~ ;r)2XM (~v;q)2X~ M X~ M X~ M

(2.17)

0

where 0 = 0 if M0 = ;, 0 = 1 if M0 6= ;. Then, there exists a constant > 0 (independent of ) such that

inf~

where ~X M equals X~ M or X~ v Mp , and the norm in X~ M is scaled as k(~v; q)k2X~ M = k~vk2X~ + kqk2M .


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Proof. Taking (~v; q) = (w~ ; r) 2 ~X M and recalling (2.9)-(2.11), one gets

B [(w~ ; r); (w~ ; r)] = (rw~ ; rw~ ) + hw~ ; S~ rMi ? hrrM; S~ rM i kw~ k2X~ + krMk2M ? ckw~ kX~ krMkM 2 (1 ? 2c )kw~ k2X~ + 2 krMk2M : This gives the desired result if M0 = ;. From now on, let us assume that M0 6= ;. Recalling (2.6), let ~v0 2 X~ 0 ~X be such that (r ~v0; r0) 0 k~v0kX~ kr0kM ; with k~v0 kX~ = kr0 kM ,

> 0 to be de ned. Then, B [(w~ ; r); (?~v0; 0)] 0 kr0k2M ? kw~ kX~ kr0kM ? krMkM kr0kM 20 kr0k2M ? 2 kw~ k2X~ ? 2 krMk2M : 0

0

Summing up, we obtain 2 B [(w~ ; r); (w~ ? ~v0 ; r)] (1 ? 2c ? 2 )kw~ k2X~ 0 2

+( ? )krMk2M + 0 kr0k2M : 2 0 2 Now, we choose = 0 . The conclusion follows taking into account (2.4). 8 Proposition 2.1 allows us to establish classical abstract a priori and a posteriori error estimates between the solutions of problems (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 has been made explicit.

Proposition 2.2 Under the conditions (2.5)-(2.6) on the pressure spaces, (2.9)-(2.11) on the stabilizing operator S~ and (2.16) on the stabilization parameter , the following a priori estimate holds:

kU~ ? ~ukX~ + 1=2 kP ? pkM
0g: Let us make the following


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Hypothesis 3.8 For any (i; j ) 2 I , there exists ~v (i;j) 2 X~ v such that

k

~v (i;j)

kX~