Output Bound Approximations for Partial Di erential ... - CiteSeerX

Output Bound Approximations for Partial Dierential Equations; Application to the Incompressible Navier{Stokes Equations? L. Machiels1, J. Peraire2, and A. T. Patera1 1 Department of Mechanical Engineering, Massachusetts Institute of Technology,

Cambridge, MA 02139, USA.

2 Department of Aeronautics and Astronautics, Massachusetts Institute of

Technology, Cambridge, MA 02139, USA.

Abstract. We describe an a posteriori nite element procedure for the ecient computation of lower and upper estimators for functional outputs of semilinear elliptic partial dierential equations. The general theory is presented, and earlier applications of the procedure to a variety of dierent problems|including the Poisson equation, the advection-diusion equation, elasticity problems, the Helmholtz equation, the Burgers equation, and eigenvalue problems|are reviewed. The method is then extended to treat incompressible ow problems, and numerical results are presented for a problem of natural convection in a complex enclosure.

1 Introduction and Motivation In typical design problems, engineers are rarely interested in the entire eld solution; only some selected characteristic metrics|or outputs|of the system are relevant. As an example, we consider the problem of cooling electronic components in a computer by natural convection of air in the enclosure represented in Fig. 1(a), where the temperature is xed on boundary ?0 , a heat

ux q is imposed on segments ?1 ; ?2 , and ?3 , and @ n [3i=0 ?i is isolated. Such model is often the core of a design optimization problem|in the above example, we suppose that, given an input heat R ux q, we wish to determine whether the mean temperature over ?1 , s = ?1 ds, is within an acceptable design interval, Ides = [slo ; sup]. In practice, many dierent uxes q must be tested, so there is a premium on eciency. Much more complex design questions can also be addressed. In a classical nite element design approach, the output of interest is evaluated from an approximate solution of the original problem. In the problem of cooling electronic components, Fig. 1(a), we rst evaluate the output of interest s = S (u ; p ; ) from an approximate eld solution (u ; p ; )| here u is the velocity eld, p is the pressure eld, is the temperature eld, and denotes the diameter of the discretization mesh T . One then veri es whether s 2 Ides , and the heat ux q is accepted or rejected accordingly. The shortcoming of the classical approach resides in the choice of the discretization mesh T . If one chooses an \optimistic" relatively coarse ?

Proceedings of the Istanbul Workshop on Industrial and Environmental Applications of Direct and Large Eddy Numerical Simulation, August, 1998.

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(a)

(b)

q

!

?1

q

!

?2

?0

q

!

?3

g^

(c)

?

Fig. 1. (a): Domain . (b): \Optimistic" coarse mesh, TH . (c): \Conservative" ne mesh, Th .

mesh TH , Fig. 1(b), the calculation is inexpensive but also uncertain since neither sH 2 Ides ) s 2 Ides nor sH 2= Ides ) s 2= Ides . If, instead, one chooses a \conservative" suciently ne mesh Th , Fig. 1(c), then sh s with reasonable certainty, but sh is now very expensive to compute. The purpose of this paper is to present a new approach which oers great promise in reconciling these con icting requirements. We propose to construct a pair of output bound approximations, the estimators s+H and s?H , computed predominantly on the mesh TH and with the following attributes. A1 As H ! h, we have s+H ! sh from above and s?H ! sh from below 8H H . Here H is an unknown threshold discretization parameter; a detailed discussion of H will be given shortly; note that, for linear coercive problems, H = 1. A2 If we de ne the (half) bound gap H = 12 (s+H ? s?H ), we have H jsH ? shj as H ! h, with independent of H . This property guarantees the optimal convergence rate and sharpness of the bounds, provided the eectivity factor is not too large. P A3 The bound gap H admits elemental decomposition H = TH 2TH TH , with TH 0, for all elements TH in TH ; this will serve for adaptive re nement. A4 The work required to evaluate s+H and s?H is substantially smaller that the work necessary for the computation sh, provided H h. Note that attribute A2 is important not only for eciency, but also to ensure the \well-posedness" of the estimator formulation when H < 1.

Output Bound Approximations

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In our illustrative example, Fig. 1(a), the bound-based design would proceed as follows. Given a heat ux q, we choose an initial mesh TH , and we compute s+H and s?H . Next, we de ne Ib = [s?H ; s+H ]; according to attribute A1 (provided H H ), if Ib 2 Ides , we can accept q; if Ib 2 R nIdes , we can reject q; otherwise, we use property A2, that is, we narrow the bound gap by taking a ner mesh (a smaller H ), and we then repeat the procedure. In the latter case, attribute A3 is important as it allows us to optimally re ne the mesh through an adaptive procedure. Attribute A4 ensures that the complete procedure is much less expensive than directly computing sh . The estimator formulation may be viewed as an implicit Aubin-Nitsche construction. The method is indebted to, but considerably generalizes, earlier nite element a posteriori error estimation techniques in that we obtain quantitative constant-free bounds|contrary to earlier explicit techniques [4]|for the output of interest|contrary to earlier implicit techniques [7,3,2]. Finally, some aspects of the method are also analogous to certain domain decomposition techniques [14,6].

2 Bound Procedure: Semi-Linear Elliptic Equations 2.1 Abstract Procedure We consider here the problem: Find u 2 Y such that

A(u; v) = 0; 8v 2 Y; where Y is a Hilbert space, and A : Y Y ! R is a general form linear in the second argument. We further assume that we are interested in an output s = S (u), where S : Y ! R is a prescribed output functional. We next assume that we are given two nite element subspaces: YH , the coarse space, and Yh , the ne space; we require YH Yh Y . The associated coarse-space and ne-space approximations, uH and uh , exhibit complementary advantages and disadvantages. The ne-space solution, uh 2 Yh , which satis es the discrete problem

A(uh ; v) = 0; 8v 2 Yh ;

(1)

yields a very good approximation, S (uh ), of the exact output s; nevertheless, the eort required to obtain uh will typically be prohibitive. In contrast, the coarse-space solution, uH 2 YH , which satis es the discrete problem

A(uH ; v) = 0; 8v 2 YH ; can be obtained with relatively modest computational eort; nevertheless the delity of the corresponding approximate output, S (uH ), is no longer assured.

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Prior to the description of the bound procedure, we need to introduce the following de nitions and form expansions. We rst de ne E (w; v) = A(uH + w; v) ? A(uH ; v). We next expand E (w; v) = E (w; v) + F (w; w; v) + G(w; v); where E (w; v) is a bilinear form, F (w1 ; w2 ; v) is a trilinear form (symmetric in w1 and w2 ), and G(w; v) is the remainder, linear in v. A similar decomposition applied to the output functional yields S (uh + v) = S (uH )+ L(v)+ M (v; v)+ N (v); where L(v) is linear, M (v1 ; v2 ) is (symmetric) bilinear, and N (v) is the remainder. We then decompose the symmetric part of E (w; v), E s (w; v) = 1 (E (w; v) + E (v; w)), into two components, E s (w; v) = Ecs (w; v) + Eas (w; v), 2 where Ecs (w; v) is positive semi-de nite, that is Ecs (v; v) 0; 8v 2 Y . Finally, we introduce the \broken" spaces, Y^H Y^h Y^ , such that Y = fv 2 Y^ j B (v; q) = 0; 8q 2 Z g, Yh = fv 2 Y^h j B (v; q) = 0; 8q 2 Z g, and YH = fv 2 Y^H j B (v; q) = 0; 8q 2 ZH g. Here ZH Z are additional Hilbert spaces and B : Y^ Z ! R is a bilinear form. The bound procedure then proceeds in ve steps: 1. Compute uH 2 YH as the solution of the primal problem A(uH ; v) = 0; 8v 2 YH : We then de ne the primal residual as Rpr (v) = ?A(uH ; v); 8v 2 Y^ . 2. Compute the output adjoint H 2 YH as the solution of the dual problem E (v; H ) = ?L(v); 8v 2 YH : (2) We then de ne the dual residual as Rdu (v) = ?L(v) ? E (v; H ); 8v 2 Y^ . pr 3. Compute the hybrid uxes, H 2 ZH and Hdu 2 ZH , which satisfy the equations B (v; Hpr ) = Rpr (v); 8v 2 Y^H ; and B (v; Hdu ) = Rdu (v); 8v 2 Y^H :

4. Compute the \reconstructed" errors, e^pr 2 Y^h and e^du 2 Y^h , 2Ecs (êpr; v) = Rpr (v) ? B (v; Hpr ); 8v 2 Y^h ; (3) s du du du ^ 2Ec (ê ; v) = R (v) ? B (v; H ); 8v 2 Yh : (4) 5. Evaluate the lower and upper bound approximations: sH = S (uH ) Ecs (ê ; e^), where 2 R+ and e^ = e^pr 1 e^du. The choice of will aect the bound gap H = 21 (s+H ? s?H ) = Ecs (êpr ; e^pr) + 1 Ecs (êdu; e^du). Since e^pr and e^du do not depend on , we can readily nd the which minimizes the bound gap, s s edu ; e^du ) (5) = EEcs(^ (êpr ; e^pr) ; c

yieldingpthe optimized bounds sH = S (uH ) ? 2Ecs (êpr ; e^du) H , with H = 2 Ecs (êpr ; e^pr)Ecs (êdu; e^du).


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We have shown in earlier papers [13,11] that, if we de ne e = uh ? uH , sH = S (uh) D I ; (6) with D = Ecs (e ? e^ ; e ? e^); (7) I = Eas (e; e) F (e; e; H ) M (e; e) + F (e; e; e) G(e; H ) N (e) + G(e; e): Here D 0 is a positive semi-de nite term and I is an inde nite term. The essential point is that, for a large and important class of problems|semilinear elliptic partial dierential equations|,there exists a splitting, E s = Ecs + Eas , such that, relative to D , the corrections in I involve weaker norms or higher power of the error e; therefore, D dominates all other deviations of sH from S (uh) for H suciently small (H < H ). We conclude that s?H and s+H approach S (uh) from below and above, respectively. Note that for linear coercive problems I = 0, and we obtain bounds for all H (H = 1).

2.2 Basic Spaces

We now introduce dierent concrete functional spaces which will serve as building blocks for de ning the spaces Y , YH , Yh , Y^ , Y^H , Y^h , Z , and ZH in the instantiations of the general procedure. Let 2 Rd (d = 1; 2) be open with Lipschitz boundary. We rst set X = H01 ( ) [1], and we de ne the nite element spaces

X(p) = fv 2 X j vjT 2 Pp (T ); 8T 2 T g; where = h or H , p = 1; 2, Th and TH are two triangulations chosen such that XH(p) Xh(p) (Th is a re nement of TH ), T denotes an element in T , and Pp(T ) is the space of all polynomials of degree p de ned on T . We

next de ne the \broken" spaces X^ = fv 2 L2 ( ) j vjTH 2 H 1 (TH ); 8TH 2 TH g; and, for = h or H , X^(p) = fv 2 L2( ) j vjTH 2 H 1 (TH ); 8TH 2 TH ; vjT 2 Pp(T ); 8T 2 T g:

Note that v 2 X^ may be discontinuous over @TH ; 8TH 2 TH . The \broken" meshes corresponding to the spaces X^H and X^h are represented in Figs. 2(a) and (b). Given the set of edges of TH , ? (TH ), we de ne the hybrid ux spaces Q = L2 (? (TH )), and

Q(Hp) = fv 2 Q j vj 2 Pp( ); 8 2 ? (TH )g:

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(Note that for R1 ; vj 2 R.) Finally, we de ne the continuity form, b : X^ Q ! R X Z b(v; ) = [v] j ds;

2? (TH )

where [v] denotes the jump in v across when 2 and the trace of v when

2 @ . For this de nition of b(v; ) the following equivalence conditions are satis ed: X = fv 2 X^ j b(v; ) = 0; 8 2 Qg, and XH = fv 2 X^H j b(v; ) = 0; 8 2 QH g. (a)

(b)

Fig. 2. (a): Space X^H . (b) Space X^h .

2.3 Instantiations Linear Coercive. Our rst example is the Poisson equation with homogeneous Dirichlet boundary conditions: ?r u = f in ; uj@ = 0. We shall start by the identi cation of theRvarious abstract spaces and forms of the general formulation: A(w; v) = rw rv ? fv dA, Y = X , E = E = R s Ec (w; v) = rw rv dA, F = G = 0, Y = X , Y^ = X^ , ZH = QH , 2

(1)

(1)

(1)

and B = b. Next, we review the dierent steps of the bound procedure. Steps 1 and 2 (coarse primal and dual equilibria) are straightforward. For the hybrid ux equilibration, Step 3, we adopt (and adapt) the procedure developed in the context of energy-norm implicit Neumann subproblem indicators [7]. The solvability of the hybrid ux equations follows the equilibria obtained in Steps 1 and 2 [17]. This construction of equilibrated hybrid


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uxes ensures the solvability of the local Neumann subproblems for the error reconstruction in Step 4. Then the optimal lower and upper bounds are computed in StepR 5. The bound error expression, Eqs. (6) and (7), reads sH = S (uh) jr(e ? e^)j2 dA. The advection-diusion equation is an example of nonsymmetric linear coercive problem: ? r2 u + U ru = f in ; uj@ = 0, > 0, and r U = 0 in . The following identi cation shows R that nonsymmetric problems fall readily into the formalism: A ( w; v ) =

rw rv + (U rw)v ? fv dA, R Y = X , E R= E (w; v) = rw rv + (U rw)v dA, F = G = 0, E s = Ecs (w; v) = rw rv dA, Y = X(1), Y^ = X^(1) , ZH = Q(1) H , and BR = b. The identi cation yields the bound error expression sH = S (uh ) jr(e ? e^ )j2 dA, identical to that for the Poisson problem. We now illustrate the use of product spaces, which permits extension to vector-valued problems and, as will be seen later, to \many- eld" problems like natural convection. We treat the linear two-dimensional elasticity system, k ? @x@ (Eijkl @u @x ) = fi in ; ui j@ = 0; j

l

where Eijkl is the elasticity constitutive tensor. For this problem the spaces are de ned by the following products: Y = X X , Y = X(1) X(1), Y^ = (1) X^(1) X^(1), and ZH = Q(1) H QH . The forms are given by Z k @vi ? fi vi dA; A(w; v) = Eijkl @w @x l @xj

Z k @vi E = E = Ecs (w; v) = Eijkl @w @x @x dA;

l

j

F = G = 0, and B (v ; ) = b(vi ; i ). The bound error expression reads Z sH = S (uh) Eijkl @ (ei@x? eî ) @ (ek@x? e^k ) dA: j l

Advection-diusion and elasticity results are presented in [17]. Dierent outputs, such as the average of the eld variable over parts of the domain (advection-diusion), or the averaged force or displacement over parts of the domain boundary, are considered. The results demonstrate the convergence of the upper and lower bounds from above and below, sH =sh ! 1, the optimality of the convergence rate, jsH =sh ? 1j / CH 2 (P1 elements), and, for a mesh that barely resolves the solution, jsH =sh ? 1j 10%. In [19], the local bound gap contributions are used in an adaptive mesh re nement strategy. The procedure is easily generalized to treat other examples of linear coercive problems such as stabilized linear advection (a hyperbolic problem, in fact) [11]. Linear Noncoercive. We consider the Helmholtz equation with homogeneous Dirichlet boundary conditions: ?r2 u ? k2 u = f in ; uj@ = 0. The

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R identi cation of spaces and forms yields A(w; v) = rw rv ? k2 wv ? fv dA, R Y = X , ER = E = E s (w; v) = rw Rrv ? k2 wv dA, F = G = 0, Ecs (w; v) = rw rv dA, Eas (w; v) = ?k2 wv dA, Y = X(1), Y^ = X^(1) , ZH = Q(1) H , and B = b. Another variant of the method for the Helmholtz equation is found in [18]. The complex Helmholtz problem in two-dimensional exterior domains (scattering wave problem) has been treated in [22]; other examples of linear noncoercive problems include the calculation of bound approximations for sensitivity derivatives [8]. Nonlinear Noncoercive. We now discuss the nonlinear Burgers equation (in one space dimension): ?uxx + uux = f in =]0; 1[, with u(0) = 0 = u(1) = 0; Rhere > 0 is the viscosity. This equation leads to the form A(w; v) = 01 wx vx + wwx v ? fv dxR . For this problem, we consider the following (nonlinear) output: S (v) = 01 vx x ? 21 v2 x dx, where 2 C 1 ( ) is such that (0) = 0; (1) = 1. For u suciently smooth, we have S (u) = ux (1). We de ne the functional spaces as Y = X , Y = X(1) , Y^ = X^(1) , and ZRH = Q(1) H . We can now proceed with theR form expan1 sions: E (w; v) = 0R wx vx ? uH wvx dx, F (w1 ; w2 ; v) = ? 21 R 01 w1 w2 vx dx, G(w) = 0, L(w) = 01 wx x ? wuH x dx, M (w1 ; w2 ) = ?R12 01 w1 w2 x dx, and N (w) = R0. The coercive splitting yields Ecs (w; v) = 01 wx vx dx and Eas (w; v) = 21 01 (uH )x wv dx. Another variant of the method is found in [18]. Other results on the Burgers equation with adaptivity are presented in [11]. We now apply the framework to the symmetric eigenvalue problem: ?r2 u = u in ; uj@ = 0. We rst de ne Y = X R, Y = X(1) R, YR^ = X^(1) R, and ZH = Q(1) H . We then write the form A((w; ); (v; )) = rw rv ? 2 wv + w dA ? . This forms leads to E = E + F , G = 0, E ((w; ); (v; )) = R r w r v ? H wv ? uH v + 2uH w dA, and F ((w; )1 ; (w; )2 ; (v;R )) = R

1 ( 1Rw2 v + 2 w1 v) dA. The splitting is de ned by Ecs = rw w w ? 1 2 2

s rv dA and Ea = ?H wv + 21 (uH v + uH w) dA. The eigenvalue problem is treated in [13].

2.4 Discussion of Attributes Attribute A1. For coercive linear problems, in view of Eqs. (6) and (7), bounds are obtained uniformly for all H . An alternative proof and interpretation can be obtained by a duality argument [16,17,19]. For noncoercive problems, we require an additional hypothesis|a variation on the standard Aubin-Nitsche relationship between the L2 and H 1 discretization error|which applies generally to second-order equations that satisfy a Garding inequality [20]. More precisely, we assume that there exists three positive constants, CD1 ; CD2 and CI , such that (say for the lower bound and P1 elements) CD1 H 2 D? CD2 H 2 and jI ? j CI H 4 ; then,


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s?H = psh ? D? ? I ? sh ? CD1 H 2 + CI H 4 sh , for H H where H = CD1 =CI . (A weaker hypothesis is introduced in [21].)

We now show that the noncoercive examples presented above ful ll this hypothesis. For the Helmholtz equation, weR have sH = S (uh) D I , where D = Ecs (Re ? e^ ; e ? e^ ) = jr(e ? e^ )j2 dA, and I = Ea (e; e) = ? k2 e2 dA. We see that j Ea (e; e)j C kek2L2 vanishes as O(H 4 )|much faster than Ecs (e ? e^; e ? e^), which should tend to zero as O(H 2 ). It then follows that s?H and s+H will approach S (uh ) from below and above for H suciently small, H < H . For the Burgers equation, note thatR F (e; e; e) = 0; therefore, we R 1 we rst 2 nd sH = S (uh ) 0 (ex ? e^x ) dx + 21 01 (uH + H + )x e2 dx. Again, we see that the inde nite terms appear only in the weaker norm, and should be dominated by the gradient as H becomes small. Note that for both the Burgers equation and the Helmholtz equation the singular-perturbation prefactor ( and k?2 , respectively) unfortunately favors the inde nite terms; nevertheless, the corresponding numerical singular-perturbation parameter (for instance, the grid Peclet number) will be small as soon as the coarse space even roughly resolves the structure of the solution, and thus the bound property will be preserved except perhaps on very crude meshes. If we consider the eigenvalue problem with the output being the eigenvalue (see [13] forR the more generalR case), we have sH = h R S (v; ) = jr(e ? e^ )j2 dA H e2 dA. Since jr(e ? e^ )j2 dA is O(H 2 ) and since from a priori results [23] we know that kek2L2 = O(H 4 ); the estimators sH will approach h from above and below as H ! h. In all cases studied in [13], upper and lower bounds are obtained, except in one case for a very coarse discretization. As expected, the convergence rate is O(H 2 ) (for P1 elements). We conclude that, even if H is not known a priori, bounds are obtained once the solution is marginally resolved. In numerical examples, we have almost always observed H = 1, and thus the uncertainty associated with H is not an important practical issue, though it constitutes a real theoretical issue. Finally, H is a threshold parameter (bounds are obtained for all H < H ); therefore, even if H is not known, the method present a signi cant advantage over previous explicit a posteriori estimation procedures in which estimators involve unknown constants and functions. There is another source of uncertainty associated with the choice of the ne mesh Th , which should be selected ne enough to ensure that js ? sh j is negligible (recall that s is the exact solution). In practice, the mesh Th is chosen conservatively by estimating a priori the size of the smallest structure anticipated and ensuring that Th provides for an accurate representation of such structures. Attribute A2. The bounds produced by the procedure are sharp. The bound gap converges at an optimal rate: H CH 2 (proved for linear coercive and certain noncoercive equations, for P1 elements, in [12]), and

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the the eectivity factor, = H =jsh ? sH j, has been measured for various applications, typically 1 10. Attribute A3. Since the cost of computing the bounds is essentially a function of the number of elements TH in TH , it is desirable to construct optimized triangulations that maximize the bound accuracy (minimize the bound gap) for a given number of degrees of freedom. As shown in [19], the bound gap H can P be expressed as a sum of local elemental positive contributions: H = TH 2TH TH , with TH 0. We can, therefore, implement a simple adaptive strategy. Starting from an initial grid, TH0 , with bound gap 0H , we generate a sequence of triangulations fTHk ; k = 1; 2; :::g with corresponding bound gaps fkH ; k = 1; 2; :::g, such that each triangulation THk is obtained by re nement of the selected triangles THk?1 such that Tk?H1 > maxTH Tk?H1 , for a speci ed parameter 0 < < 1. The approach ensures that, for a suciently large k, kH targ , where targ > 0 is a gap target. Attribute A4. We recall that uh and S (uh) are the eld variable and the output that are eectively indistinguishable from the truth|and correspondingly expensive. Our lower and upper bounds are only interesting if they can be obtained at considerably less expense than computation of S (uh )| and preferably at only slightly greater expense than computation of S (uH ), the coarse approximation. From our de nitions of Xh(p) and X^h(p) and the identi cations of Yh and Y^h for the dierent semilinear elliptic problems considered in Section 2.3, we see that Eqs. (3) and (4) correspond to many local, uncoupled, small, Neumann subproblems, whereas Eq. (1) corresponds to a single, large, global problem. It follows that Eqs. (3) and (4) present a smaller bandwidth|yielding substantial savings in both memory and computational time of direct solution strategies|and a smaller condition number|yielding faster convergence in iterative methods. In addition, Eqs. (3) and (4) are symmetric, positive-de nite, and completely decoupled|the latter enabling straightforward parallel implementations. Furthermore, for nonlinear equations, the dual problem, Eq. (2), and the subproblems, Eqs. (3) and (4), are linear, leading to extra savings compared to the original ne mesh problem. Relative savings are reduced as the global solver improves.

3 Output Bounds for Incompressible Flow Problems We rst consider the Stokes equations @u @u @p i ? @x@ @x + @xj + @x = fi in ; j

j

i

i

@ui @xi = 0 in ; ui j@ = 0;

where f is a prescribed function, u is the velocity eld, and p is the pressure eld. We de ne the function spaces X = H01 ( ) H01 ( ), Y = X L2( )=R,


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and A((w ; r); (v ; q)) : Y Y ! R, Z @w @w @v @vi ? f v ? q @wi dA: i A((w; r); (v ; q)) = + @xj @xi ? p @x i i @x @xi j i j i

Then the weak formulation of the Stokes problem is: Find (u; p) 2 Y such that

A((u; p); (v; q)) = 0; 8(v; q) 2 Y: We also specify an output functional S : Y ! R, and, as usual, two discretization meshes T , = h; H |Th being a re nement of TH . We choose the Crouzeix-Raviart nite element spaces for the velocity and the pressure [5],

X(2+) = fv 2 H01 ( ) H01 ( ) j vi jT 2 P2(T ) B3 (T ); 8T 2 T g; M = fp 2 L2 ( )=R j pjT 2 P1(T ); 8T 2 T g; where B3 (T ) denotes the space of bubble functions of degree 3 on T . The \broken" spaces X^2+ and M^ are obtained as usual by relaxing the continuity across the edges of the coarse mesh. Note that, since the pressure is discontinuous across element boundaries, we have M = M^ . Then, we set Y = X(2+) M and Y^ = X^(2+) M^ . Finally, we de ne the hybrid ux (2) spaces ZH = Q(2) H QH and the bilinear form B (v ; ) = b(vi ; i ). We now proceed with the usual form expansion; we have E = E s = s s where Ec + Eas Ec;V Z @w @w @v i s Ec ((w; r); (v ; q)) = + @xj @xi dA; @x j i j

Z @w @v i i s Ea ((w; r); (v ; q)) = ? p @x + q @x dA:

i

i

s indicates that E s is Here Ecs is coercive over H01 ( ) H01 ( ), and Ec;V coercive if we take w and v in V , where Z @v i V = fv 2 H01 ( ) H01 ( ) j @x q dA = 0; 8q 2 L2( )g:

i

We de ne e = uh ? uH and = ph ? pH , and we consider Eqs. (6) and (7). We see that Eas ((e; ); (e; )) = O(H 4 ) is not subdominant; therefore, bounds cannot be obtained for this problem by strictly following the general procedure. Nevertheless, uniform bounding properties are recovered for this problem if we replace, in Step 4 of the bound procedure, (uH ; ph) 2 YH by (~uH ; p~H ) 2 Yh , where u~ H is chosen such that Z @ u~ H;i @x q dA = 0; 8q 2 Mh :

i

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An appropriate choice of (~uH ; p~H ) is to take the Mh -incompressible \Stokes projection" of (uH ; pH ) with the additional constraint that u~ H j@TH = uH j@TH . Since we use discontinuous pressure, these projections are elemental and, thus, inexpensive. With this new formulation, we have two possible strates , or we use E s in the local Neumann subproblems. gies: either we use Ec;V c Since the former choice yields Mh-incompressible reconstructed errors, it is arguably more accurate. For earlier results on the Stokes problem, with a slightly dierent approach, see [15]. The extension of the above formulation to the Navier-Stokes equations, @u @u @ @ui + @p = f in ; ? @x @x i + @xj + uj @x i @x j

j

i

is obtained if we de ne

j

i

@ui = 0 in ; u j = 0; i @

@xi

A((w; r); (v; q)) = Z

@wi + @wj @vi ? u u @vi ? p @vi ? f v ? q @wi dA: (8) @xj @xi @xj i j @xj @xi i i @xi

The general form expansion combines the approaches for the Burgers equation and the Stokes equations. Again, to guarantee subdominance of the inde nite terms in Eqs. (6) and (7), we use the projection (~uH ; p~H ). To retain the solvability of the local Neumann subproblems, Eqs. (3) and (4), the nonlinear terms in the primal problem are written in their conservative form, and (~uH ; p~H ) must be replaced by (uH ; pH ) in the convective (L2 ) parts of Rdu . Optimal asymptotic bounds are obtained since the perturbations kuH ? u~ H kH 1 , kuH ? u~ H kL2 , and kpH ? p~H k can be shown to be suciently small. The natural convection problem,

@ui + @p = ? g^ in ; ? @@xui + uj @x i @x 2

2

j

j

i

@ui =0 in ; @xi 2 1 @ @ =0 in ; ? @x2 + uj @x j j

(9)

where g^ is the unit vector indicating the direction of gravity, is the Prandtl number, is the Grashof number, and is the temperature, requires us to incorporate the temperature equation in the formulation. This is readily achieved using a product space formulation since the inde nite error term due to the Boussinesq coupling in the bound error expression involves only weak norms. Moreover, the weak coupling allows us to introduce two parameters


13

u and ; these parameters can be optimized independently (with relations

similar to Eq. (5)), yielding sharper bounds. For a detailed presentation of the formulation for this problem, see [10]. We will now present numerical results for a simple natural convection test case. The geometry of the problem is shown in Fig. 3(a). The boundary @ = 1 on ? , and @ = 0 otherwise. conditions are uj@ = 0, = 0 on ?0 , @n I @n The Prandtl number and the Grashof number are = 1 and = 30; 000. ForR this choice of parameters, the Nusselt number is Nu = 2:49 (Nu?1 = 1 ?I ?I ds); for a pure conduction problem in this geometry, Nu = 1, and thus Nu = 2:49 indicates the existence of velocity and thermal boundary layers near ?I and ?0 . The elements of the coarsest mesh, TH0 , (represented on Fig. 3(a)) are chosen so that the diameter of the mesh, H0 , is approximately the size of the thickness of the boundary layers. Figures 3(b) and 3(c) show the convergence of the bounds for a nonlinear output, Z S (u) = juj2 dA;

corresponding to the kinetic energy of the ow. For this problem, bounds are always observed on all the meshes considered. In Fig. 3(b), we have represented the normalized lower and upper bounds s+H =sh and s?H =sh, and the normalized \coarse" mesh output, sH =sh , for dierent meshes TH . The mesh diameters vary from H = H0 to H = H0 =6; Th = TH0 =6 is the ne mesh. Figure 3(c) demonstrates that the convergence rate of the bound gap is optimal, O(H 4 ); the measured eectivity factor is 7:3 7:7. We next present results obtained for the model problem, 1(a). More precisely, Eqs. (9) are solved in the domain with the following boundary con@ = q on segments ? ; ? ; and ? ; ditions: uj@ = 0; = 0 on segment ?0 ; @n 1 2 3 @ = 0 otherwise. The output of interest is s = 1 R ds. For this case @n j?1 j ?1 the (nondimensional) ux q = 1, and the Prandtl and Grashof numbers are = 1 and = 50; 000; the resulting streamlines are shown in Fig. 4(a). In this example an automatic adaptive strategy is used to re ne the grid and \optimally" narrow the bound gap. The eectiveness of the adaptive procedure is summarized in Table 1, in which TH , TH , and TH denotes the successive adapted meshes corresponding to Figures 4(b), (c) and (d), respectively; the mesh TH=2 is a uniform re nement of TH . Table 1 shows that a reduction of the relative bound gap from 8% for mesh TH to < 1% for mesh TH is achieved by optimal re nement with a nal mesh which contains only slightly more than twice the number of elements of the mesh TH . In conclusion, for a problem of natural convection involving the full nonlinear incompressible Navier-Stokes and energy equation at moderate Reynolds number, we have obtained new nite element error indicators which are constant-free, accurate, relevant (providing bounds directly for the quantities of interest), and ecient; these estimators can greatly reduce the numerical uncertainty associated with the use of inexpensive meshes. 0

00

00

14

Machiels et al. (b) 1.03 +

s /sh

1.02 1.01

(a)

1

sH/sh

0.99

−

s /sh

0.98 0.97

?I

0

0.04

(c)

?0

0.08

0.12

H

0.5

g^

?

0

log(∆H)

−0.5 −1 −1.5 −2 −1.6

−1.4 −1.2 log(H)

−1

Fig. 3. Natural convection: rst test case. (a) Domain and mesh TH0 . (b) Conver gence of the bounds sH . (c) Convergence of the bound gap H . (a)

(b)

(c)

(d)

Fig. 4. Natural convection: model problem. (a) Streamlines. (b) Mesh TH . (c) Mesh

TH . (d) Mesh TH 0

00

.


15

TH TH TH TH=2 0

00

# of elements 264 416 628 1056 s? 0:253 0:270 0:273 0:272 s+ 0:296 0:282 0:279 0:286 H 0:0212 0:0057 0:0026 0:0036 Table 1. Natural convection: model problem.

Finally, although the capabilities of the method have now been demonstrated in a variety of dierent contexts, some outstanding areas of investigation remain. In the application to incompressible uid ow problems, the method should be extended to treat time-dependent problems [9], threedimensional problems, and linear stability problems; the extension to compressible ow problems also presents interesting challenges. Moreover, recently, a new output bound result which generalizes the present approach has been formulated [21] that provides a framework for applications in iteration error estimation and control|stopping criteria for iterative solution methods for systems of linear equations, for instance. Finally, the eciency of the method (its implementation on parallel computers) and its concrete use in design optimization of industrial (non-academic) systems should be demonstrated. Acknowledgments. This work was supported by NASA Grants NAG11978, NAG1-1587, and NAG4-105, DARPA and ONR Grant N00014-91-J1-1889, and AFOSR Grant F49620-97-1-0052. L.M. was partially supported by Fulbright and BAEF Fellowships. We would like to acknowledge our longstanding and very fruitful collaboration with Prof. Y. Maday of Universite Pierre et Marie Curie, Paris, France.

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