[PDF] A new error bound for Reduced Basis approximation of parabolic

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A new error bound for Reduced Basis approximation of parabolic partial dierential equations

Karsten Urban

University of Ulm, Institute for Numerical Mathematics, Helmholtzstr. 18,

89081 Ulm (Germany), karsten.urban@uni-ulm.de

Anthony T. Patera

Mechanical Engineering Department, Massachusetts Institute of Technology,

77 Massachusetts Ave., Cambridge, MA 02139-4307 (USA), patera@mit.eduAbstract

We consider a space-time variational formulation for linear parabolic partial dierential equations. We introduce an associated Petrov-Galerkin truth nite element discretization with favorable discrete inf-sup constant: is bounded from below by unity for the heat equation;grows only linearly in time for non-coercive (asymptotically stable) convection operators. The latter in turn permits eective long-timea posteriorierror bounds for reduced basis approximations, in sharp contrast to classical exponentially growing energy estimates.

Resume

Nous considerons une formulation variationnelle espace-temps pour les equa- tions dierentielles paraboliques lineaires. Nous y associons une discretisation par elements nis de Petrov-Galerkin pour laquelle la constante de stabilite inf-suppossede des proprietes agreables :est bornee inferieurement par la constante unite pour l'equation de la chaleur;a une croissance seulement lineaire en temps pour des operateurs de convection non-coercifs (asympto- tiquement stables). Dans le cadre des approximations par bases reduites, cette derniere propriete permet d'obtenir des bornes ecaces pour l'erreur a posteriorien temps long, en net contraste avec les estimateurs d'erreur en energie classiques qui presentent une croissance exponentielle. Keywords:parabolic equations, space-time formulation, inf-sup stability, reduced basis, a posteriori estimation Preprint submitted to C. R. Acad. Sci. Paris, Ser. I October 31, 2011

2000 MSC:35K15, 65M15, 65M60Version francaise abregee

Soit l'equation aux derivees partielles parabolique (1) et sa formulation variationnelle espace-temps :u2 X:=L2(I;V)\H1(I;V0) verie (2) pour l'espace des fonctions testY:=L2(I;V); icib(w;v) :=R

I[h_w(t);v(t)iH+

a(w(t);v(t))]dtetf(v) :=R

Ihg(t);v(t)iHdt. Les normes surXetYsont

denies parkwk2X:=kwk2L

2(I;V)+k_wk2L

2(I;V0)etkvkY:=kvkL2(I;V). On

considereV=H10( ) (operateurs spatiaux de 2e ordre), ou est le domaine spatial. On montre (Proposition 1.1 basee sur [4]) que pour des problemes coercifs, la constante inf-supdebest positive et bornee inferieurement. Nous introduisons ensuite une discretisation par elements nis de Petrov- Galerkin : soitu2 Xveriantb(u;v) =f(v);8v2 Y; avecX:= S t

VhetY=Qt

Vh,= (t;h), ouSt;VhetQtsont respectivement

des elements nis lineaire par morceaux en temps (de pas t) et en espace (de diametreh), et constants en temps. Cette methode concide avec une discretisation de Crank-Nicolson en temps, ainsi nos resultats s'appliquent directement a ce schema standard. Nous denissons une norme modiee (equivalente) surX;jjj jjjXou la partieR

Ikwk2Vdtdek kXest remplacee

par une somme des moyennes dewsur chaque element temporel. Nous montrons (Proposition 2.1) que pour l'equation de la chaleur, la constante inf-sup discreteest bornee inferieurement par la constante unite. Nous considerons maintenant une approximation par bases reduitesuN, comme dans [1]. Nous pouvons construire a la Proposition 3.1 des bornes sur l'erreura posterioripour le champ solutionuNet pour la sortie scalaire s N. L'utilite de ces bornes est tres liee a la dependance deaux parametres du probleme et au temps nalT. La procedure de calcul de ces bornes est similaire a la decomposition hors ligne-en ligne des bases reduites standard. Enn, nous presentons des resultats numeriques pour la constante inf-sup discrete. Pour des operateursnon-coercifsmais asymptotiquement stables (e.g. convection), on observe que le parametre inf-sup cro^t seulement en1T, ou1est la vitesse de deplacement etTle temps nal. L'exemple est un domaine spatial unidimensionnel (0;1) avec l'operateur1(x1=2)uxuxx pour1>22(1<22entra^ne un probleme coercif). Ces estimateurs sont beaucoup plus precis que ceux classiques en energie et predisant une croissance exponentielle ene1T, inutilisables en pratique. 2

1. Space-time formulation

We rst formulate a general linear parabolic equation. Consider Hilbert spacesV ,!H ,!V0and an operatorA2 L(V;V0),hAv;wiH=a(v;w) for v;w2V. SettingI:= [0;T],T >0 and giveng2L2(I;V0), we look foru such that _u(t) +Au(t) =g(t) inV0; t2I;u(0) = 0;(1) and an associated output of the forms:=R

I`(u(t))dtfor some`2V0. We

restrict ourselves to LTI systems even though some of our results can be extended to the more general situation. For the derivation of the space-time variational form of (1), we introduce the trial spaceX:=L2(I;V)\H1(I;V0) with normkwk2X:=kwk2L

2(I;V)+

k_wk2L

2(I;V0)and the test spaceY:=L2(I;V) with normkvkY:=kvkL2(I;V).

Note thatX= (L2(I)

V)\(H1(I)

V0) andY=L2(I)

Vwhich

will allow for a tensor product discretization. Then denitionsb(w;v) :=R

I[h_w(t);v(t)iH+a(w(t);v(t))]dtandf(v) :=R

Ihg(t);v(t)iHdtyield the

space-time variational formulation u2 X:b(u;v) =f(v);8v2 Y:(2) The well-posedness of (2) has been shown (under suitable assumptions) in [4, Theorem 5.1]. The approach of [4] can also yield an estimate for the inf-sup constant := infw2Xsupv2Yb(w;v)kwkXkvkY. We dene%:= sup06=2VkkHkkVanda:= inf

2Vsup 2Va( ;)kkVk kV; we then claim

Proposition 1.1.Assume that there existMa<1, >0, and2Rsuch thatja( ;)j Mak kVkkV(continuity) anda(;) +kk2Hkk2V (Garding) for allv;w2V. Then,LB:=(%2)minf1;M2agmaxf1;(a)1gp2 Remark 1.2.Note thatLBdoes not depend on the nal time. However, the estimate is only meaningful if%2, i.e., if the system is coercive. In the non-coercive case,(1)is often transformed via^u(t) :=etu(t)to obtain a coercive problem; however, this yields an inf-sup bound that behaves aseT | clearly unsuitable for long-time integration. 3

2. Petrov-Galerkin Truth Approximation

LetX X,Y Ybe nite dimensional subspaces andu2 X

the discrete approximation of (2), i.e.,b(u;v) =f(v)8v2 Y,s=RT

0`(u(t))dt. Henceforth, we concentrate on the caseH=L2(

),V= H 10( ). LetX=St

Vh,Yh=Qt

Vh,= (t;h), whereSt,Vhare

piecewise linear andQtpiecewise constant nite elements with respect to triangulationsTspace hin space andTtimet fti1(i1)t < tit t i;1irgin time for t:=T=r. This coincides with the Crank{Nicolson (CN) scheme if a trapezoidal approximation of the right-hand side temporal integration is used; hence, we can derive error bounds for the CN scheme via our space-time formulation. Towards that end we introduce a dierent norm onX: Forw2 Xand I i:= (ti1;ti), set wi:= (t)1R I iw(t)dt2Vand w:=Pr i=1Ii wi2 L

2(I;V), whereIiis the characteristic function onIi; then, setjjjwjjj2X:=

k_wk2L

2(I;V0)+kwk2L

2(I;V). It can be proven thatkkXandjjjjjjXare equivalent.

We may now demonstrate

Proposition 2.1.SetAu:=u; then we have the inf-sup lower bound := infw2Xsupv2Yb(w;v)jjjwjjjXkvkY1. Proof.(Sketch) Note that= 0,= 1, andMa= 1 for our choicekwkV krwkL2( ). Let 06=w2 X. Setz:=A1 h_w(i.e.,a(z;h) =h_w;hiH,

8h2Vh) andv:=z+ w. Then,b(w;v) jjjwjjj2X+kw(T)k2H

jjjwjjjXkvkYsince (z;w)Y=12 kw(T)k2Handkvk2Y=R IkA1 h_wk2Vdt+ tPr i=1kwik2V+kw(T)k2H=jjjwjjj2X+kw(T)k2H.3. The Reduced Basis Method (RBM) Now, let= (1;2)2 D:=R2be a parameter vector andA=A() := u+1(x)ru+2u, i.e., a diusion-convection-reaction operator with convection eld. We then introduce a standard Reduced Basis (RB) ap- proximation for the Crank{Nicolson interpretation of our discrete problem, u

N()2 Xt;N=St

VN; hereVNis an RB space provided for exam-

ple by the POD-Greedy procedure of [1]. The RB output is then given by s N=RT

0`(uN(t))dt. (It is possible, alternatively, to consider a space{time

RB approximation as well [5].) It is then simple [3] to demonstrate 4 Proposition 3.1.The RB error satisesjjjuuNjjjX krNkY0=LB, where r N(v) :=f(v)b(uN;v);8v2 Y, andLBis a lower bound fordened in Proposition 2.1. Furthermore,jssNj k`kV0pTkrNkY0=LB. The utility of thesea posteriorierror bounds is critically dependent on the dependence ofas a function of the parameterand nal timeT,(;T).

We brie

y comment on the computational implications of this space-time error bound. As always, our model problem (as is the case here) must honor the \ane-in-functions-of-parameter" condition to permit eective oine{ online computation of the reduced basis approximation anda posteriorier- ror bounds. The current formulation introduces the further complication of the space{time norms. In fact, calculation of the dual normk kY0(through the corresponding Riesz representation [3]) does not couple dierent time steps/elements, and hence the additional online diculty is relatively minor. (The oine stage for computation of the inf-sup lower bound by the Succes- sive Constraint Method [3, 6] does require special treatment, in particular due to the normjjj jjjX.)

4. Numerical Results

We report numerical results for the Crank{Nicolson scheme for various choices of the parameters1,2as well as for dierent time steps tand uniform mesh sizesh. For simplicity, we consider the univariate case (in space) = (0;1) and choose(x) =x12 . Let us denote by(;T) the numerical value for the truth inf-sup corresponding to parameterand nal timeT. We start by conrming the quality of the estimate in Proposition

2.1. Thus, we choose1=2= 0; for several values ofT,hand twe

invariantly obtain 1:000, indicating that our estimate is in fact sharp. Next, we investigate the case of convection,2= 0, in which caseais coercive only for1<22. We get((50;0);1:0) = 0:052,((50;0);2:0) =

0:028 and((100;0);1:0) = 0:020,((100;0);2:0) = 0:010. These numbers

are mostly invariant for suciently smallhand t. The overall behavior is((1;0);T)(1T)1. NoteT=O(1) is eectively a \long time" in convective units, 1=1. We emphasize that although the problem is non- coercive, the problem is asymptotically stable in the sense that all eigenvalues ofa( ;) =h ;iHlie in the left-hand plane; this stability is re ected in the inf-sup behavior. In contrast, a standard energy approach [2] gives eective inf-sup constants on the order ofe1T(here about 1010). Hence, 5 the traditional method fails whereas our new bound re ects the truetime- coupledproperties of the system. Finally, we consider the case1= 0 which gives rise to an asymptotically unstable (and non-coercive) system for2<2. This means that any error estimatemustgrow exponentially with the nal timeT. We observe this for our estimator as well, e.g.((0;20);1:0) = 5:62105(order ofe2T). Acknowledgements.A.T.P. was supported by OSD/AFOSR/MURI Grant FA9550-09-1-0613 and ONR Grant N00014-11-1-0713; K.U. by the Deutsche Forschungsgemeinschaft (DFG) under Ur-63/9 and GrK1100. The authors are grateful to Sylvain Vallaghe for helpful comments. This paper was written while K.U. was Visiting Professor at MIT.

References

[1] B. Haasdonk and M. Ohlb erger.Reduced basis metho dfor nite v ol- ume approximations of parametrized linear evolution equations. M2AN

Math. Model. Numer. Anal.42(2008), 277-302.

[2] D.J. Knezevic, N.C. Nguy en,and A.T. P atera.Reducedbasis appro xi- mation and a posteriori error estimation for the parametrized unsteady Boussinesq equations. Math. Mod. Meth. Appl. Sci.,21no. 7 (2011),

1415-1442.

[3] G. Rozza, D.B.P .Huynh, and A. T. P atera.Reducedbasis appro xima- tion and a posteriori error estimation for anely parametrized elliptic coercive partial dierential equations | Application to transport and continuum mechanics. Arch. Comp. Meth. Eng.,15no. 3 (2008), 229- 275.
[4] C. Sc hwaband R. Stev enson.Space-time adaptiv ew aveletmetho dsfor parabolic evolution problems. Math. Comp.78(2009), 1293-1318. [5] K. Steih a ndK. Urb an.Spac e-timereduced basis metho dsfor time- periodic parabolic problems.In preparation. [6] S. V allaghe,A. Le-Hy aric,M. F ouquemberg,and C. Prud'homme. A successive constraint method with minimal oine constraints for lower bounds of parametric coercivity constant.C.R. Acad. Sci. Paris, submit- ted 2011. 6quotesdbs_dbs17.pdfusesText_23

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