Convergence analyses of Galerkin least - squares methods

Symmetric advective-di usive forms of the Stokes and incompressible Navier-Stokes equations are presented. The Galerkin least-squares method for advective-di usive equations is used for both systems and is related to other stabilized methods previously studied. The presentation reveals that the convergence analysis for advective-di usive equations, as applied before to a linearized form of the compressible Navier-Stokes equations, carries over in a straightforward manner to the Stokes problem and to a linearized form of the incompressible Navier-Stokes equations. ii L.P. Franca and T.J.R. Hughes Preprint, June 1992 1 1. Stokes equations Let be a bounded region of IRN with polygonal or polyhedral boundary , where N denotes the space dimension. The Stokes equations are usually written as: u+rp = f in r u = 0 in (1) where u is the velocity, p is the pressure, is the viscosity (assumed constant, for simplicity) and f is the body force. To simplify the exposition that follows, we adopt homonegeous Dirichlet boundary conditions. The Stokes problem then consists in solving (1) for u and p subjected to: u = 0 on (2) Instead of working with (1) as written, we wish to view it as a symmetric advectivedi usive system, namely, AiV;i = (Kij Vj );i +F (3) where the repeated indices summation convention is adopted for i; j = 1; : : : ;N; and the symbols in (3) are related to those in (1) as follows: V = up (4) Ai = 0N N ei eTi 0 i = 1; : : : ;N (5) Kii = IN 0N 0TN 0 i = 1; : : : ;N (no sum) (6) Kij = 0 i 6= j ; i; j = 1; : : : ;N (7) F = f 0 (8) L.P. Franca and T.J.R. Hughes Preprint, June 1992 2 with 0N being the vector with all N components equal to zero; 0N N , the matrix with all N N components equal to zero; IN , the identity matrix of size N N ; and ei the unit basis vector of the Cartesian system of coordinates with the j-component equal to the Kronecker delta, i.e., (ei)j = ij . By substituting (4)-(8) into (3) one recovers the equations as given in (1). We wish to examine the Stokes problem as the symmetric advective-di usive system given by (3), and relate to the stabilized nite element methods developed for such systems as presented in [20] and references therein which are mainly concerned with applications to the compressible Navier-Stokes equations . We wish also to compare with stabilized methods developed for the Stokes problem employing as the starting point the standard u p form of the equations, eq.(1) (see, e.g., [11,18,19]). Also, the Galerkin method for (3) is related to the Galerkin method for (1) viewed from a mixed method point-of-view. Before continuing let us note that (3) is indeed a symmetric advective-di usive system in that the matrices in (5) and (6) are symmetric, and the di usion matrix K = [Kij ] is positive semi-de nite and, in particular, diagonal with entries equal to and zero. Considering a standard partition Ch of the domain into elements, and parametrized by the mesh size h, we denote by Uh U = H1 0 ( )N and Ph P = L2( ) = IR the nite element spaces for velocity and pressure, respectively, and to be made precise later. We will restrict our arguments to arbitrary piecewise continous polynomials to approximate both variables. Furthermore, we denote by Wh = Uh Ph W = U P the nite element space where we seek an approximate solution for V = fu pgT : L.P. Franca and T.J.R. Hughes Preprint, June 1992 3 We may now state the Galerkin method as: Find Vh 2 Wh such that B(Vh;W) = F (W) 8W 2 Wh (9) where B(V;W) = (AiV;i ; W) + (KijV;j ; W;i) (10) F (W) = (F ; W) (11) with ( ; ) denoting the L2( ) inner product. To have an energy estimate of this formulation, one usually computes the bilinear form with the same argument, i.e., setting W = V, B(V;V) = (AiV;i ; V) + (KijV;j ; V;i) (12) The rst term, by symmetry of the Ai's and by integration by parts, is equal to (AiV;i ; V) = 12(AnV ; V) (13) where ( ; ) denotes the L2( ) inner product and An = 0N N n nT 0 (14) with n being the unit outward normal vector to the boundary . Since V = f0 pgT on ; 8 p 2 Ph, then combining with (14), it follows that the right-hand-side of (13) is equal to zero. Therefore the energy estimate (12) reduces to B(V;V) = kK1=2 ij V;jk20 (15) and, by the de nition of the blocks of K (eqs. (6) { (7) ), no control on pressures is obtained by this argument, (recall that K is positive semi-de nite), since B(V;V) = kK1=2 ij V;jk20; = k 1=2ruk20 = kruk20 : (16) L.P. Franca and T.J.R. Hughes Preprint, June 1992 4 Therefore stability a la Lax-Milgram, i.e., 9C > 0 s.t. B(V;V) CkVkW 8V 2 Wh (17) is not apparent for the Galerkin method approximating this model. Actually, for the Galerkin method, one can prove the more general stability condition sup W2Wh B(V;W) kWkW CkVkW 8V 2 Wh (18) if the nite element spaces Uh and Ph satisfy the inf-sup condition sup u2Uh (r u; q) kuk1 kqk 8 q 2 Ph; (19) (cf. Babu ska [1], Brezzi [3]). It is well-known that condition (19) restricts the possible combinations of nite element spaces Uh and Ph and, in particular, it precludes equallow-order interpolations (see, e.g., [6,14] and references therein for further elaboration). Let us now sketch that (19) and (16) imply (18). First note that from (19) 9uh 2 Uh ; s.t. 8 q 2 Ph (q;r uh) = kqk20 kuhk1 1 kqk0 (20) If our Galerkin nite element method solution fuh; phg of (9) ful lls (19) then (20) holds for q = ph. Now taking W =W = f uh 0gT with uh satisfying (20) with q = ph, we obtain by replacing W in (10): B(Vh;W) = (rph;uh) + (Kij Vh;j ;W;i) (int. by parts) = (ph;r uh) + (Kij Vh;j ;W;i) (by(20)1 and Cauchy-Schwarz) kphk20 kK1=2 ij Vh;jk0kK1=2 ij W;jk0 kphk20 1=2kK1=2 ij Vh;jk0kuhk1 (by(20)2 and (16)) kphk20 kruhk0kphk0 12kphk20 2 2 2 kuhk21 (21) L.P. Franca and T.J.R. Hughes Preprint, June 1992 5 By (16) and by Poincar es inequality, we also have B(Vh;Vh) = kruhk20 C1 kuhk21 (22) Note that by selecting f W = Vh + Wh in the de nition of B( ; ), we obtain by (21) and (22): B(Vh;f W) = B(Vh;Vh + Wh) = B(Vh;Vh) + B(Vh;Wh) C1 kuhk21 + 1 2kphk20 2 2 2 kuhk21 C2(kuhk21 + kphk20) = C2kVhk2W (23) when selecting 0 < < 2C1 2= . On the other hand kf Wk2W = kVh + Whk2W 2kVhk2W + 2 2kWhk2W C2 3kVhk2W or, kf WkW C3kVhkW (24) Therefore by (23) and (24) the stability estimate (18) is established with C = C2=C3. At this point we can draw an interesting observation on why it is worthwhile to write the Stokes problem as a symmetric advective-di usive system of equations. Note that in the process of showing the stability condition (18), a la Babu ska, we have chosen f W = Vh + Wh, i.e., f W could not simply equal to Vh, otherwise no assurance of pressure stability could be claimed (cf. (16)). The fact that f W had to di er from L.P. Franca and T.J.R. Hughes Preprint, June 1992 6 Vh gives us a natural link between mixed methods for Stokes and the Petrov-Galerkin method concept applied to the advective-di usive equations. If we did not know the nature of the matrices of (3), one could show stability for an advective-di usive system of equations based on the Petrov-Galerkin method. Now, by using the particular nature of that system (3) as emanating from a mixed method, we ended up establishing its stability by an argument that appeals to the construction of Petrov-Galerkin methods, i.e., by selecting f W = Vh + Wh 6= Vh. Next, we wish to examine stabilized methods for (3). Let us investigate a Galerkin least-squares (GLS) method for advective-di usive equations as suggested in [20]. The GLS method may be stated as: Find Vh 2 Wh such that BGLS(Vh;W) = FGLS(W) 8W 2 Wh (25) where BGLS(V;W) = B(V;W) + X K2Ch(LV; LW)K (26) FGLS(W) = F (W) + X K2Ch(F; LW)K (27) and LV = AiV;i (Kij V;j);i (28) To complete the de nition of the method, we need to set up the matrix of coe cients . Let us suggest a simple design which is the di usive limit of the incompressible Navier-Stokes case, discussed in the next section: = 1IN 0N 0TN 2 (29) L.P. Franca and T.J.R. Hughes Preprint, June 1992 7 with 1 = mk h2K 4 (30) 2 = 2 mk (31) and mk = min 1 3 ; 2Ck (32) Ck X K2Ch h2Kk vk20;K krvk20 v 2 Wh (33) The constant Ck is the maximum value that satis es (33) and can be computed with an associated eigenvalue problem as explicity done in [16] for some examples. In [16] we nd suitable de nitions for hK to make the computations feasible. One may employ those de nitions of hK together with the computations of Ck to complete the de nitions (30){(31). Before continuing, note that this GLS method, up to the speci c design of , is the method suggested by Douglas-Wang [8]. To see this in detail the reader might note that in the advective term of the \Galerkin operator" we have a term (rp ; v), which, for a Dirichlet problem, and by integration by parts, equals (p ; r v), and therefore it has an opposite sign to the other advective term of the Galerkin operator which is (r u ; q). Adding a least-squares form of the equilibrium equation recovers the Douglas-Wang method, as reviewed in detail in [10,12,13]. Here we see that for the symmetric advective-di usive system form of the Stokes problem, a recipe for designing a GLS method is closely linked to the Douglas-Wang method [8], which in turn, is a variation on a similar strategy proposed earlier by Hughes and Franca [18]. Let us now sketch a convergence analysis from the advective-di usive system pointof-view. First, the stability is immediate: L.P. Franca and T.J.R. Hughes Preprint, June 1992 8 LEMMA 1.1 (Stability) Assuming constant > 0 then for all V 2 Wh BGLS(V;V) = kK1=2 ij V;jk20 + X K2Ch k 1=2LVk20;K def: = jjjVjjj2 (34) Proof From (15) and (26) the result follows immediately. Convergence will be established in the norm given by (34), which can be written in terms of fu ; pg as follows: jjjVjjj2 = kK1=2 ij V;jk20 + X K2Ch k 1=2LVk20;K = kruk20 + X K2Ch k 1=2 1 (rp u)k20;K + X K2Ch k 1=2 2 r uk20;K 8V = fu; pgT 2 Wh (35) Convergence in this norm, with 2 = 0, is proven in [8]. However one can also get an estimate for rp out of this norm as follows: jjjVjjj2 = kruk20 + X K2Ch k 1=2 1 rpk20;K + X K2Ch k 1=2 1 uk20;K 2 X K2Ch( 1rp ; u)0;K + X K2Ch k 1=2 2 r uk20;K kruk20 + 1 2 X K2Ch k 1=2 1 rpk20;K X K2Ch 1 2k uk20;K + X K2Ch k 1=2 2 r uk20;K (36) Noting that by de nition of 1, eq. (30), X K2Ch 1 2k uk20;K = X K2Ch mk h2K 4 k uk20;K (by 32) 2 Ck X K2Ch h2Kk uk20;K (by 33) 2 kruk20 (37) L.P. Franca and T.J.R. Hughes Preprint, June 1992 9 Substituting (37) in (36) implies jjjVjjj2 2 kruk20 + 12 X K2Ch k 1=2 1 rpk20;K + X K2Ch k 1=2 2 r uk20;K (38) Therefore (38) states that an error estimate obtained in terms of jjj jjj sufces even if we are interested in an error estimate for the pressure in terms of PK2Ch h2Kkrpk20;K 1=2. Error estimates for the Galerkin least-squares type methods in the norm kruk20 +PK2Ch h2Kkrpk20;K 1=2 are obtained in the early works of [5,18,19]. In [13] these results are consolidated and generalized for discontinuous pressures, and estimates are obtained in the \natural norm" of the model, i.e., kuk21 + kpk20 1=2. We also need the following interpolation estimate : LEMMA 1.2 : Assume that the solution V = fu pgT of (3) satis es u 2 Hk+1( )N \ H1 0 ( )N and p 2 H l+1( ) \ L20( ) where k and l are the order of the interpolations used for velocity and pressure, respectively. Denoting by H = f u pgT ; u = e uh u and p = e ph p, the interpolation errors, then we have: k 1=2Hk20 + kK1=2 ij H;jk20 + X K2Ch k 1=2 LHk20;K C h2kjuj2k+1 + 1 h2l+2jpj2l+1 Proof: k 1=2Hk20 = kf 1=2 1 u 1=2 2 pgT k20 = kf 4 mk h2K 1=2 u mk 2 1=2 pgTk20 = 4 mk k 1 hK uk20 + mk 2 k pk20 L.P. Franca and T.J.R. Hughes Preprint, June 1992 10 C h2kjuj2k+1 + 1 h2l+2jpj2l+1 (39) kK1=2 ij H;jk20 = kr uk20 C h2kjuj2k+1 (40) X K2Ch k 1=2 LHk20;K = X K2Ch k 1=2 1 (r p u)k20;K + k 1=2 2 r uk20 2 X K2Ch mk h2K 4 kr pk20;K + 2 X K2Ch mk h2K 4 k uk20;K + 2 mk kr uk20 C h2kjuj2k+1 + 1 h2l+2jpj2l+1 (41) The last inequalities of (39){(41) above follow from standard interpolation theory. We are now ready to establish convergence. THEOREM 1.3 Assuming constant , the solution Vh of GLS converge to V, solution of (3), as follows jjjVh Vjjj2 C h2kjuj2k+1 + 1 h2l+2jpj2l+1 Proof : Let Eh = Vh e Vh and E = Eh +H = Vh V then, jjjEhjjj2 = BGLS(Eh ; Eh) (by Lemma 1.1) = BGLS(H ; Eh) (by consistency) jB(H ; Eh) + X K2Ch(LH; LEh)0;K j = j(AiH;i ; Eh) + (KijH;j ; Eh;i) + X K2Ch(LH; LEh)0;K j (by (10)) = j (H ; AiEh;i) + X K2Ch (H ; (KijEh;j);i)0;K + 2(KijH;j ; Eh;i) L.P. Franca and T.J.R. Hughes Preprint, June 1992 11 + X K2Ch(LH; LEh)0;K j (by integration-by-parts) = j X K2Ch( 1=2H ; 1=2LEh)0;K + 2(K1=2 ij H;j ; K1=2 ij Eh;j) + X K2Ch( 1=2LH ; 1=2LEh)0;K j (by (28)) k 1=2Hk20 + 14 X K2Ch k 1=2LEhk20;K + 2kK1=2 ij H;jk20 + 1 2kK1=2 ij Eh;jk20;K + X K2Ch k 1=2LHk20;K + 1 4 X K2Ch k 1=2LEhk20;K 12 jjjEhjjj20 + k 1=2Hk20 + 2kK1=2 ij H;jk20 + X K2Ch k 1=2LHk20;K 12 jjjEhjjj20 + C h2kjuj2k+1 + 1 h2l+2jpj2l+1 (by Lemma 1.2) (42) Also from Lemma 1.2, jjjHjjj2 = kK1=2 ij H;jk20;K + X K2Ch k 1=2LHk20;K C h2kjuj2k+1 + 1 h2l+2jpj2l+1 (43) and therefore the result follows from (42) and (43) combined with the triangle inequality. Remark The proof of Theorem 1.3 is closely related to the convergence proof of the GLS method in [20] applied to advective-di usive equations. Up to the interpolation estimate in Lemma 1.2, it is nowhere apparent that the system dealt with above is actually the Stokes problem. A speci c design of that gives an optimal rate of convergence for the velocity and pressure elds satisfying the Stokes equations is L.P. Franca and T.J.R. Hughes Preprint, June 1992 12 the only ingredient that distinguishes this system from any other advective-di usive system, e.g., the compressible Navier-Stokes equations. 2. Incompressible Navier-Stokes Equations In this section, we generalize the results in the previous section to account for the advection term of the incompressible Navier-Stokes equations. The presentation and analysis that follows will be restricted to the steady-state, linearized, \frozen-coe cient" form of the incompressible Navier-Stokes equations, given by: (ru)a+rp u = f in r u = 0 in (44) where is the density and a is a given velocity eld (= u in the nonlinear incompressible Navier-Stokes case) which is assumed to be divergence-free, i.e., r a = 0, a.e.. Most of the formalism that follows (including the symmetric advective-di usive form of the equations and the de niton of the GLS method) can be used for the nonlinear model by replacing a by u, except for the convergence analysis which is more technical. The linear model then consists in solving (44) for u and p subjected to: u = 0 on (45) As in the previous section, we wish to view (44) as a symmetric advective-di usive system, namely, AiV;i = (KijVj);i + F (46) where the symbols in (46) are the same as given in eqs. (4), (6)-(8), and the advective matrices Ai's are now given by: Ai = aiIn ei eTi 0 i = 1; : : : ;N (47) L.P. Franca and T.J.R. Hughes Preprint, June 1992 13 where ai is the ith-component of a. By substituting (4), (6)-(8), and (47) in (46) one recovers the equations as given in (44).It is rather simple to repeat the de nition of the Galerkin method as given by eqs. (9)-(11) and verify that the same energy estimate result, given by eq. (16), applies here. Therefore, for the Galerkin method, a more general stability condition, as given by (18), should be pursued to establish well-posedness for the velocity-pressure pair. However, for the Galerkin method per se, it will not be possible to show stability for high advection, i.e., roughly speaking when jaj =l, where l is a characteristic length scale. The Galerkin method in this case can be stable for specially constructed spaces, as recently shown in [2,4]. We will not delve further into the Galerkin method for high advection, instead, we pursue the GLS method that simultaneously overcomes the highadvection and the pressure-velocity compatibility limitations of the Galerkin method. The GLS method may be stated as: Find Vh 2 Wh such that BGLS(Vh;W) = FGLS(W) 8W 2 Wh (48) where BGLS(V;W) = B(V;W) + X K2Ch(LV; LW)K (49) FGLS(W) = F (W) + X K2Ch(F; LW)K (50) and LV = AiV;i (Kij V;j);i (51) This is form-identical to eqs. (25)-(28), which in turn is the method in [20] for advectivedi usive equations. However, there are \hidden di erences", incorporated in the de nitions of the matrices, given by (4), (6)-(8), and (47), and the design of the coe cient L.P. Franca and T.J.R. Hughes Preprint, June 1992 14 matrix, which in this case is given by: = 1IN 0N 0TN 2 (52) with 1 = hk 2jajp (53) 2 = jajphk 1 (54) = ReK , 0 ReK < 1 1 , ReK 1 (55) ReK = mkjajphK 2 = (56) jajp = 8<: PNi=1jaijp 1=p , 1 p <1 max i=1;Njaij , p =1 (57) mk = min 1 3 ; 2Ck (58) Ck X K2Ch h2Kk vk20;K krvk20 v 2 Wh (59) The formulae (53)-(59) de ning, 1 and 2 are practically the same as introduced in [10] for the advective-di usive model and applied to the incompressible Navier-Stokes equations in [9]. The di erence here is that in (54) we have 1 as opposed to in [9]. These formulas are improvements over the years from the original works of Hughes and Brooks [7,17] and the later works [20] and references therein. The main ingredient is the importance of distinguishing di usive dominated regimes (0 ReK < 1) from advective-dominated regimes (ReK 1) through (55). Note that it follows immediately from the de nitions above that For ReK 1 : 1 = hk 2jajp (60) 2 = jajphk (61) L.P. Franca and T.J.R. Hughes Preprint, June 1992 15 For 0 ReK < 1 : 1 = mkh2K 4 = (62) 2 = 2 = mk (63) Therefore 1 and 2 are O(h) for advective-dominated regions; and 1 = O(h2) and 2 = O(1) for di usion-dominated regions. As far as the method (48)-(51) is concerned, when written for velocity and pressure, the reader will recognize for low-order interpolations the methods studied in [15,25] and for more general interpolations and further applications [9,23,24]. The convergence analysis in this case will be very close to what we presented in the previous section. First, the stability given by Lemma 1.1 (eq. (34)) remains unchanged. Next, the interpolation estimate established in Lemma 1.2 changes to accommodate the present design of , namely, eqs. (52)-(59) : LEMMA 2.1 : Assume that the solutionV = fu pgT of (44) satis es u 2 Hk+1( )N \ H1 0 ( )N and p 2 H l+1( ) \ L20( ). Employing the same notation for the interpolation errors, as in Lemma 1.2, we have: k 1=2Hk20 + kK1=2 ij H;jk20 + X K2Ch k 1=2LHk20;K C X K2Ch H(ReK 1) sup x2K jajp h2k+1 K juj2k+1;K + sup x2K jaj 1 p h2l+1 K jpj2l+1;K +H(1 ReK ) h2k K juj2k+1;K + h2l+2 K jpj2l+1;K (64) where H( ) is the Heaviside function given by H(x y) = 0 , x < y 1 , x > y (65) L.P. Franca and T.J.R. Hughes Preprint, June 1992 16 Proofk 1=2Hk20 = k 1=2 1 uk20 + k 1=2 2 pk20 = X K2Ch"H(ReK 1) 2jajp hK 1=2 u 20;K+ p (jajphK)1=2 20;K +H(1 ReK ) 4 = mkh2K 1=2 u 20;K+ mk 2 = 1=2 p 20;K # C X K2Ch"H(ReK 1) sup x2K jajp 1 h1=2 K u 20;K + sup x2K jaj 1 p 1 h1=2 K p 20;K +H(1 ReK ) 1 hK u 20;K + k pk20;K # C X K2Ch"H(ReK 1) sup x2K jajp h2k+1 K juj2k+1;K + sup x2K jaj 1 p h2l+1 K jpj2l+1;K +H(1 ReK ) h2k K juj2k+1;K + h2l+2 K jpj2l+1;K # kK1=2 ij H;jk20 = kr uk20 = X K2Ch"H(ReK 1)k 1=2r uk20;K +H(1 ReK) kr uk20;K# = X K2Ch"H(ReK 1)k mkjajphk 2ReK 1=2r uk20;K +H(1 ReK ) kr uk20;K# L.P. Franca and T.J.R. Hughes Preprint, June 1992 17 C X K2Ch"H(ReK 1) sup x2K jajp h2k+1 K juj2k+1;K +H(1 ReK ) h2k K juj2k+1;K# X K2Ch k 1=2 LHk20;K = X K2Ch k 1=2 1 ( (r u)a+r p u)k20;K + k 1=2 2 r u)k20 = X K2Ch"H(ReK 1) hK 2jajp 1=2( (r u)a+r p u) 20;K + k(jajphK)1=2r u)k20;K +H(1 ReK ) mkh2K 4 = 1=2( (r u)a +r p u) 20;K + 2 = mk 1=2r u) 20;K # C X K2Ch"H(ReK 1) sup x2K jajp h2k+1 K juj2k+1;K + sup x2K jaj 1 p h2l+1 K jpj2l+1;K +H(1 ReK ) h2k K juj2k+1;K + h2l+2 K jpj2l+1;K # The convergence can now be established: THEOREM 2.2 Assuming constant and , the solution Vh of GLS converge to V, L.P. Franca and T.J.R. HughesPreprint, June 199218the solution of (46), as followsjjjVh Vjjj2 C XK2Ch H(ReK 1) supx2K jajp h2k+1K juj2k+1;K+ supx2K jaj 1p h2l+1K jpj2l+1;K+H(1 ReK) h2kK juj2k+1;K +h2l+2K jpj2l+1;KProof The proof is exactly like the proof of Theorem 1.3, except that in the last linesof (42) and (43) the reader has to replace the rates of convergence by those given inLemma 2.1 to obtain the result stated above.Remark Symmetric-hyperbolic systems, also known as Friedrichs' systems, are special casesof symmetric advective-di usive systems in which the di usion term (i.e., K) isabsent. Johnson et al. [22] where the rst to perform a convergence analysis ofcertain stabilized methods for Friedrichs' systems (so called \SUPG" or \streamlinedi usion" methods). Hughes et al. obtained error estimates for SUPG formulationsof advective-di usive systems in [21].3. ConclusionIt is rather remarkable that the convergence analyses of GLS for the linearizedmodel of the incompressible Navier-Stokes equations and for the Stokes equations areexactly the same, except for the interpolation estimate result to take into account thedi erent designs of the stability matrix . This is where the advantage of adopting theframework of symmetric advective-di usive systems become apparent. Unrelated setsof equations , such as the Stokes problem, incompressible Navier-Stokes equations, com-pressible Navier-Stokes equations, etc., all t in the same abstract setting of symmetric L.P. Franca and T.J.R. HughesPreprint, June 199219advective-di usive systems. And for frozen-coe cient linearized models, they can allbe tackled by Theorem 1.3, up to the details of the interpolation estimates.References[1] I. Babuska, \The nite element method with Lagrangian multipliers", Numer.Math. 20 (1973) 179{192.[2] C. Baiocchi, F. Brezzi and L.P. 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