A POSTERIORI ERROR ESTIMATES FOR THE BDF2 METHOD FOR PARABOLIC EQUATIONS

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1 A POSTERIORI ERROR ESTIMATES FOR THE BDF METHOD FOR PARABOLIC EQUATIONS GEORGIOS AKRIVIS AND PANAGIOTIS CHATZIPANTELIDIS Abstract. We derive optimal order, residual-based a posteriori error estimates for time discretizations by the two step BDF method for linear parabolic equations. Appropriate reconstructions of the approximate solution play a key role in the analysis. To utilize the BDF method we employ one step by both the trapezoidal method or the backward Euler scheme. Our a posteriori error estimates are of optimal order for the former choice and suboptimal for the latter. Simple numerical experiments illustrate this behaviour. Key words. Parabolic equations, BDF method, residual, BDF reconstruction, a posteriori error analysis. AMS subject classifications. Primary 65M5, 65M5; Secondary 65L7. Introduction. In this paper we establish optimal order a posteriori error estimates for time discretizations by the two step BDF method (BDF) for linear parabolic partial differential equations (p.d.e s). We consider initial value problems of the form: Find u : [,T] D(A) satisfying (.) { u (t)+au(t) = f(t), t T, u() = u, with A : D(A) H a positive definite, selfadjoint, linear operator on a Hilbert space (H,, ) with domain D(A) dense in H, forcing term f : [,T] H, and initial value u H. We denote by the norm of H. Let N N,N,k := T/N be the constant time step, t n := nk,n =,...,N, be a uniform partition of [,T], and J n := (t n,t n ]. We define nodal approximations U m D(A) to the values u m := u(t m ) of the solution u of (.) as follows: We set U := u, perform one step with the trapezoidal method to get U and then apply the BDF method to obtain U,...,U N, i.e., the approximations U,...,U N are recursively defined by k U n + U n +AU n = f n, n =,...,N, (.) U +AU / = f /, U = u, with f m := f(t m ). Here we have used the notation v n := k (vn v n ), v n := v n = k (vn v n +v n v n +v n ), v n :=, Computer Science Department, University of Ioannina, 45 Ioannina, Greece (akrivis@cs.uoi.gr). Department of Mathematics, University of Crete, 749 Heraklion-Crete, Greece (chatzipa@math.uoc.gr). The first author s work was co-funded by the European Union in the framework of the project Support of Computer Science Studies in the University of Ioannina of the Operational Program for Education and Initial Vocational Training of the 3 rd Community Support Framework of the Hellenic Ministry of Education, funded by national sources and by the European Social Fund (ESF)

2 G. Akrivis, P. Chatzipantelidis for given v,...,v N. Our goal is to derive optimal order residual based a posteriori error estimates. To define the residual, we need to introduce an approximation U(t) to u(t), for all t [,T]. Since the error u m U m at the nodes is of second order, a natural choice for a second order BDF approximation U : [,T] D(A) to u is the piecewise linear interpolant at the nodal values U m, (.3) U(t) = U n +(t t n ) U n, t J n, n =,...,N. Although U(t) is a second order approximation to u(t), its residual R(t) H, (.4) R(t) := U (t)+au(t) f(t), t J n, i.e., the amount by which the approximate solution U misses being an exact solution of the differential equation in (.), is of first order; see. The error e, e := u U, satisfies the error equation e +Ae = R. Since R(t) is of suboptimal order, applying energy techniques to this error equation leads inevitably to a posteriori estimators of suboptimal order. To recover the optimal order, we shall reconstruct the approximate solution U in an appropriate way. Next, we modify U to construct appropriate reconstructions Û. As we will see later on, several continuous approximations Û are appropriate for our purposes, in the sense that they lead to optimal order residuals. To motivate the construction of Û, let us note that U satisfies the relation (.5) U (t)+au(t) = (t t n )A U n + U n +AU n, t Jn, as we see by writing U in the form U(t) = U n +(t t n ) U n, t J n. We will be lead to Û by replacing the coefficient A U n on the right-hand side of (.5) by appropriate quantities. We first choose a piecewise linear approximation ϕ to f, (.6) ϕ(t) = α n (t t n )+βn, t J n, and introduce the corresponding BDF reconstruction Û of U, namely a piecewise quadratic polynomial in time Û : [,T] D(A) defined in [tn,t n ] by (.7) {Û (t)+au(t) = ϕ(t) in J n, Û(t n ) = U n. Now, obviously, n n Û(t n ) = U n A U(t)dt+ ϕ(t)dt, t n t n and, evaluating the integrals by the mid-point rule, we obtain Û(t n ) = U n +k ( β n AU n ) ; therefore, the continuity requirement Û(tn ) = U n of Û is satisfied, if and only if (.8) β n = U n +AU n.

3 A posteriori estimates for the BDF method 3 As we already mentioned, there are several appropriate choices for α n ; see (.6). (We recall that α n := A U n corresponds to U; cf. (.5).) In the sequel we will consider two particular choices: The first choice is α n := f n, n =,...,N, and the second α := U +A U,α := U +A U,andα n := f n k 3 U n,n = 3,...,N.Aswe will see later on, the second choice corresponds to the three point reconstruction, i.e., the reconstruction is, for t [,t ], the quadratic interpolant of U,U and U, and, for n 3, the restriction to J n of the quadratic polynomial interpolating U n,u n and U n. As in [6,, 3], we consider the error functions e and ê, (.9) e := u U, ê := u Û. Once an appropriate reconstruction Û is in place, the derivation of a posteriori error estimates is elementary; cf. [, 3]. Let V := D(A / ), be the norm of V, v := A / v, V be the (topological) dual of V and its norm, v := A / v. As we will see in, the following upper and lower error bounds are valid, for t [,T], [ τ ( max ê(τ) + e(s) + ] τ t ê(s) ) ds (.) (.) 3 Û(s) U(s) ds+ Û(s) U(s) ds f(t) ϕ(t) ds, ( e(s) + ê(s) ) ds. In the sequel we will refer to the upper bound on the right-hand side of (.) as the estimator E. The above idea is related to earlier work on a posteriori analysis of time or space discrete approximations of evolution equations [6,, 3, 5]. It provides the means to establish optimal order error estimates with energy as well as with other stability techniques. In these references single step time stepping schemes were considered; the present work is devoted to a multistep scheme that is quite popular in the computations of parabolic equations. The paper is organized as follows: In we present two appropriate reconstructions and establish the upper and lower estimates (.) and (.). In 3 we show that the estimator E is of optimal order. In 4 we consider the scheme k U n + U n +AU n = f n, n =,...,N, (.) U +AU = f, U = u ; the only difference to (.) is that in this case U is computed by the backward Euler scheme. This choice for U is indeed more natural in the a priori error analysis. Since the backward Euler method is applied only once (in particular, a finite number of times, independent of the time step k), it is well known that the method (.) yields second order approximations U m to u m. Unfortunately, as illustrated in 4, our approach leads to a suboptimal a posteriori estimator for the scheme (.). Finally, in 5 we present numerical results that illustrate the theoretical results of 4 and demonstrate the effectivity of our upper and lower a posteriori error estimates.

4 4 G. Akrivis, P. Chatzipantelidis. A posteriori error estimates. In this section we present two appropriate BDF reconstructions and establish the estimates (.) and (.) for the errors e and ê. First, we show that the residual R(t) H of U is of first order. Indeed, from the differential equation in (.) and the definition (.4) of the residual, we have (.) u +Au = f, U +AU = f +R; consequently, (.) R = (u U) A(u U). The second term on the right-hand side of (.) is of second order; however, the first term can be at most of first order, since u is approximated by a piecewise constant function U (and this is valid for any choice of piecewise linear function, not only for the specific approximation U); negative norms in time are excluded from this discussion. A concrete example illustrating that R(t) is of first order might be instructive here: In the case of the initial value problem for an o.d.e. u (t) = f(t), (.4) yields R(t) = U n f(t), t J n. Let now f be an affine function, f(t) = ct+d, c, and assume that U = u. Since both the trapezoidal scheme and the BDF method integrate this o.d.e. exactly, we have U n = u n, n =,...,N, and thus R(t) = u n f(t) = k = k n n t n u (s)ds k t n [ f(s) f(t) ] ds = k c = c ( t n t ), t Jn, n n t n f(t)ds t n (s t)ds whence the order of the residual is equal to one. It is obvious from (.) that to obtain a second order residual by a piecewise polynomial function Û, we should allow Û to be piecewise quadratic. We require two fundamental properties from the approximatesolution Û: it should be continuous and its residual should be of second order. Next, we introduce two appropriatereconstructions, Û and Ũ; they are associated to two piecewise linear functions ˆϕ and ϕ, given in (.3) and in (.9), (.) in the sequel, respectively. First choice: Motivated by the discussion in the Introduction (see, in particular, (.6) and (.8)) and the fact that f n is a second order approximation to f (t n ), our first choice is based on the piecewise linear approximation ˆϕ to f, (.3) ˆϕ(t) = (t t n ) f n + U n +AU n, t Jn. We then let Û be given by (.7), with ϕ replaced by ˆϕ. Obviously, (.4) Û(t) = U n A t n U(s)ds+ t n ˆϕ(s)ds, t J n.

5 A posteriori estimates for the BDF method 5 Since both U and ˆϕ are affine in J n, Û is quadratic in J n. In fact, it is easily seen that (.5) Û(t) = U(t)+ (t tn )(t t n ) (f n AU n ) t J n. Obviously, Û coincides with U at the nodes tm ; in particular, Û is continuous. Also, relation (.5) yields (.6) Û(t) = U(t)+ (t tn )(t t n )Û t J n. Let us also note, for later use, the relation between ˆϕ and f; we will denote by I f the piecewise linear interpolant of f at the nodes t,t,...,t N. First, for n =, we have i.e., in view of (.), ˆϕ(t) = (t t ) f + U +AU, (.7) ˆϕ(t) = (t t ) f +f = (I f)(t) t (t,t ). Furthermore, for n we have whence, in view of (.), i.e., thus ˆϕ(t) = U n +AU n k AU n +(t t n ) f n + k f n = U n +AU n + k (f n AU n )+(t t n ) f n, ˆϕ(t) = f n k U n + k (f n AU n )+(t t n ) f n, ˆϕ(t) = f n +(t t n ) f n + k (f n AU n U n ); (.8) ˆϕ(t) = (I f)(t)+ k (f n AU n U n ), t (t n,t n ), n. Therefore, using again (.), for n 3, (.9) ˆϕ(t) = (I f)(t)+ k 4 3 U n, t (t n,t n ). Remark. (Regularityof Û). AnaturalquestioniswhetherÛ(t)belongstoany space containing the approximations U,...,U N. We will see that this is indeed the case, provided u () is contained in the same space; in particular, in the applications, Û(t) satisfies the same boundary conditions as U,...,U N. First, let n 3; then, (.) yields (f n AU n ) = U n + k 3 U n.

6 6 G. Akrivis, P. Chatzipantelidis Therefore, from (.5) we obtain (.) Û(t) = U(t)+ (t tn )(t t n ) ( U n + k 3 U n) t J n, n 3, and conclude that Û(t) belongs, for t [t,t], to any space containing the approximations U,...,U N. (Note that, for t [t,t], the assumption that u () is contained in the same space is not needed.) Furthermore,sinceU iscomputedbythetrapezoidalmethod, the secondrelation in (.) yields i.e., (f AU ) = U (f AU ), (.) f AU = U u (). Therefore, (f AU ) = U + k U [ U u () ] k andweeasilyconcludefrom(.5)thatû(t) belongs,fort J,toanyspacecontaining u () and the approximations U,U and U. Finally, combining (.5) with (.), we see that Û(t) belongs, for t J, to any space containing u () and the approximations U,U. Second choice: The three-point reconstruction. In J J, we let the reconstruction Ũ be the quadratic interpolant of U, U and U ; for n 3, we let Ũ in J n be the restriction to J n of the quadratic interpolant of U n, U n and U n, i.e., Ũ(ti ) = U i, i = n,n,n. It is easily seen that (.) Ũ(t) = U n +(t t n ) U n + (t tn )(t t n ) U n t J n, n. The three-point reconstruction was proposed in [4] for the trapezoidal scheme. Let us note that, since U(t) = U n +(t t n ) U n, t J n, we have (.3) Ũ(t) = U(t)+ (t tn )(t t n ) U n t J n. Obviously, i.e., Ũ (t) = U n +(t t n ) U n, t J n, (.4) Ũ (t) = U n +(t t n ) U n + k U n, t J n. Therefore, Ũ (t)+au(t) = U n +(t t n ) U n + k U n +AU n +(t t n )A U n,

7 whence, in view of (.), A posteriori estimates for the BDF method 7 (.5) Ũ (t)+au(t) = f n +(t t n ) ( U n +A U n), t J n, n. Furthermore, for n 3, using again (.), we have i.e., Ũ (t)+au(t) = f n +(t t n ) ( U n +AU n) = f n +(t t n ) ( f n k U n) = f n +(t t n ) f n k (t tn ) 3 U n, (.6) Ũ (t)+au(t) = (I f)(t) k (t tn ) 3 U n, t J n, n 3. Finally, (.7) Ũ(t) = U(t)+ t(t t ) U t J, and we easily see that (.8) Ũ (t)+au(t) = (t t ) ( U +A U ) +f, t J. Remark.. Let (.9) ϕ(t) := (t t ) ( U +A U ) +f, t J, and (.) ϕ(t) := f n +(t t n ) ( U n +A U n), t J n, n ; see (.5) and (.8). Then, the three-point reconstruction Ũ could be alternatively defined by (.7), with ϕ replaced by ϕ. Remark.3. From (.6) and (.9), we immediately obtain, for n 3, (.) ϕ(t) = ˆϕ(t) k ( t t n ) 3 U n, t J n. In particular, in view of (.3), (.) ϕ(t) = ( t t n )( f n k 3 U n) + U n +AU n, t Jn. Furthermore, for n =, it is easily seen from (.) that (.3) ϕ(t) = ( t t 3 )( U +A U ) + U +AU 3, t J. Finally, for t J, ϕ is given in (.9). Remark.4 (Regularity of Ũ). Obviously, Ũ(t) belongs to any space containing the approximations U,...,U N, for all t [,T]. In the remaining part of this section, we let Û be either one of the two reconstructions defined above and ϕ stand for either ˆϕ or ϕ, depending on the corresponding

8 8 G. Akrivis, P. Chatzipantelidis choice of Û. Once an appropriate reconstruction Û is in place, the rest of the analysis is elementary as the following result illustrates; cf. [, 3]. Theorem. (Error estimates). The upper and lower error bounds (.) and (.) are valid for the errors e = u U and ê = u Û, for t [,T]. Proof. Subtracting the differential equation in (.7) from the one in (.), we obtain (.4) ê (t)+ae(t) = f(t) ϕ(t). Taking in (.4) the inner product with ê(t) and using the identity Ae(t),ê(t) = e(t) + ê(t) Û(t) U(t), we arrive at (.5) Now, and (.5) yields d dt ê(t) + e(t) + ê(t) = Û(t) U(t) + f(t) ϕ(t),ê(t). f(t) ϕ(t),ê(t) f(t) ϕ(t) + ê(t), d dt ê(t) + e(t) + ê(t) Û(t) U(t) + f(t) ϕ(t), whence, since ê is continuous and ê() =, ( ê(t) + e(s) + ê(s) ) ds (.6) Û(s) U(s) ds+ f(s) ϕ(s) ds. This easily leads to the upper bound (.). Furthermore, obviously, (Û U)(s) e(s) + ê(s), and thus (.7) 3 (Û U)(s) 3 [ e(s) + ê(s) ] ; integrating in [, t], we obtain the lower bound (.). Combining (.) and (.), we immediately conclude ( Û(s) U(s) ds e(s) + ê(s) ) ds Û(s) U(s) ds+ f(s) ϕ(s) ds. 3. Optimality of the estimator. In this section we show that the estimator E on the right-hand side of (.) is of optimal order, i.e., of order O(k 4 ), for both reconstructions Û and Ũ described in ; see first and second choice in. We will present the details for the three-point estimator Ũ. For the other estimator, Û, the proof goes along the same lines; we will briefly discuss this case in Remark 3.. Let (3.) E := T Ũ(t) U(t) dt and E := T f(t) ϕ(t) dt with ϕ as described in detail in and Ũ the corresponding, three-point reconstruction. In the sequelweshowthat both E ande areofordero(k 4 ); thus, the estimator E = E +E onthe right-handside of (.) isofoptimal orderforthis reconstruction.

9 i.e., (3.) A posteriori estimates for the BDF method Optimality of E. First, let n. Then, using (.3), we obtain Ũ(t) U(t) dt = (t n t) (t t n ) dt U n J n 4 J n = 4 k5 (s ) s ds U n, J n Ũ(t) U(t) dt = k5 U n, n. Similarly, in view of (.7), we have (3.3) Ũ(t) U(t) dt = J k5 U. We readily conclude from (3.), (3.) and (3.3) that (3.4) E = k4 k [ U + N U n ]. We will estimate E. We begin with some preparatory estimates for e and e ; actually, to estimate E we will only need an estimate for e, but since an estimate for e will be needed in the next subsection to show that E is also of optimal order, we will provide here estimates for both e and e. Let E denote the consistency error of the first step by the trapezoidal scheme, n=3 (3.5) E := u +Au / f /, and E denote the consistency error of the BDF scheme in the second step, (3.6) E := k u + u +Au f. It is well known and easily seen that, under obvious regularity assumptions, (3.7) E + E Ck. Now, we have e + k Ae = ke ; taking here the inner product with e and using elementary inequalities, we obtain whence e + k e k E e k E + k 4 e, (3.8) e + k 4 e k E.

10 G. Akrivis, P. Chatzipantelidis Combining (3.8) with (3.7), we arrive at the desired estimate for e, (3.9) e +k e Ck 5. Similarly, we have e + 3 e +kae = ke ; taking here the inner product with e, we obtain whence 3 e +k e = k E,e + e,e k E + k e + e + e, (3.) e +k e k E + e. Combining (3.) with (3.7) and (3.9), we arrive at the desired estimate for e, (3.) e +k e Ck 5. We shall next use these preparatory estimates to estimate E. It is well known from the a priori error analysis for the BDF method that, for sufficiently smooth u, (3.) e n +k n e l C ( e + e +k 4), n =,...,N, l= with e m := u m U m ; cf. [7], []. In particular, in view of (3.9), (3.3) k Now, whence (3.4) k n= N e l Ck 4. l= N N N k U n k e n +k u n n= n= n= C N N k 3 e n +k u n, n= N U n C k 3 e +C N k 3 e n +C(u). n= Using (3.3) in (3.4), we conclude (3.5) k n= N U n C k 3 e +C. n=

11 A posteriori estimates for the BDF method Combining (3.5) with the estimate (3.9), we obtain N (3.6) k U n C, n= and we arrive at the desired optimal order estimate for E, (3.7) E ck Optimality of E. We split E in the form E = E, +E, with (3.8) E, := k f(t) ϕ(t) dt, E, := T k First, for t J, using (.9) and (.), we have f(t) ϕ(t) dt. i.e., ϕ(t) = ( t t ) ( U +A U )+f ( = f + t t ) f + ( t t) ( U +A U f ), (3.9) ϕ(t) = (I f)(t)+ ( t t ) ( U +A U f ), t J. Thus, easily, (3.) f(t) ϕ(t) Ck + k U +A U f, t J. Now, using (.) and (.), we have k ( U +A U f ) = k U k (f AU ) = (f U AU ) (f AU )+ (f Au ) = (f AU ) (f AU )+ (f Au ) U = (f Au ) (f Au )+ (f Au )+Ae Ae + e u = [ u (t ) u (t )+ u () u ] +Ae Ae + e, and we easily conclude that (3.) k U +A U f Ck + e + e + e. Using here the estimates (3.9) and (3.), we obtain (3.) Now (3.) yields k U +A U f Ck 3. k f(t) ϕ(t) dt Ck 5 +k 3 U +AU f,

12 G. Akrivis, P. Chatzipantelidis whence, in view of (3.), (3.3) k f(t) ϕ(t) dt Ck 4. Furthermore, for n =, in view of (.), we have f(t) ϕ(t) = f(t) [ f +(t t ) f ] (t t ) ( U +A U f ) whence, easily, = [ f(t) (I f)(t) ] (t t ) ( U +A U f ), t J, (3.4) f(t) ϕ(t) Ck +k U +A U f, t J. Now, whence k (f AU U ) = k (f AU ) k U = k (f AU )+ U +AU f = U +AU 3 3 f = U +AU 3 3 [ Au u (t )+u (t ) ] = e Ae 3 + [ u [ u (t )+u (t ) ]], (3.5) k (f AU U ) e + e 3 +Ck. Estimating the first two terms on the right hand side of (3.5) by (3.9) and (3.), we get (3.6) k (f AU U ) Ck 3. Now (3.4) yields k whence, in view of (3.6), (3.7) k f(t) ϕ(t) dt Ck 5 +k 3 (f AU U ), k k f(t) ϕ(t) dt Ck 4. From (3.3) and (3.7) we obtain the desired optimal order estimate for E,, (3.8) E, Ck 4. To estimate E,, we first note that, in view of (.6), f(t) ϕ(t) = [ f(t) (If)(t) ] + k (t tn ) 3 U n, t J n,

13 n 3. Therefore, whence A posteriori estimates for the BDF method 3 T E, f(t) (If)(t) dt+ k k N n=3 (3.9) E, Ck 4 + k4 N 6 k 3 U n. Therefore, it suffices to show (3.3) k n=3 N 3 U n C. n=3 But (3.3) follows from the stability estimate (3.3) A 3 U n +k n j=5 3 U j C [ A 3 U 3 + A 3 U 4 +k J n (t t n ) dt 3 U n, n ] A 3 f j, n = 5,...,N, cf. [7], [], and the fact that U,...,U 4 are third order approximations to u,...,u 4, respectively, in the norm A and approximations of order 5/ in the norm. Remark 3.. Here we briefly sketch the proof of the optimality of the estimator for the reconstruction Û given by (.4). With notation analogous to (3.), with Û and ˆϕ instead of Ũ and ϕ, respectively, it is easily seen that in this case j=5 (3.3) E = E, +E, with (3.33) E, := k4 k( (f AU ) + (f AU ) ) E, := k4 N k U n + k 3 U n. n=3 We will now use (3.9) and (3.) to estimate E,. First, in view of (.) and (.), we have whence (f AU ) = k[ U u () ], (3.34) (f AU ) = k e + k[ u u () ]. Under obvious regularity requirements, the second term on the right-hand side of (3.34) can be easily estimated; we conclude k (f AU ) 4 k 3 e +Ck.

14 4 G. Akrivis, P. Chatzipantelidis Therefore, in view of (3.9), (3.35) k (f AU ) c. Using (3.35) and the analogous estimate for (f AU ), we can easily see that E, Ck 4. Also, combining (3.6) with (3.3) we see that E, Ck 4. Therefore, E Ck 4. Furthermore, in view of (3.8) and (.9), we easily obtain E := T f(t) ˆϕ(t) dt Ck 4 + k4 N 8 k 3 U n ; using here (3.3), we conclude E Ck 4, i.e., the estimator E is of optimal order. 4. Starting with the backward Euler scheme. Since we want to have a second order approximation U to u, the first choice that comes to mind is to define U by performing one step with the backward Euler method. Unfortunately, for this choice our estimator is of suboptimal order for both reconstructions. We illustrate this with two elementary examples. Example 4.. Let us first consider the initial value problem n=3 (4.) { u (t) = t, t, u() =. It is an easy task to derive a posteriori error estimates for (4.) and we will not dwell upon this. Our purpose here is to study the order of our estimator E for this concrete example, cf. (3.). We first perform one step of the backward Euler method and get U = k. Subsequently, we apply the BDF method to obtain the approximations U,...,U N, (4.) Un U n + 3 Un = k n, n =,...,N, U =, U = k. ThesolutionU n of (4.)canbeeasilydetermined; sincetheorderofthebdfmethod is two, and the exact solutionuof (4.) is a polynomialofdegreetwo, u,u,...,u N is a particular solution of the inhomogeneous difference equation. Using also the general solution of the corresponding homogeneous equation and the given starting values U and U, we obtain (4.3) U n = Therefore, for the error u n U n we have (4.4) u n U n = 3 [ 3 ( ) ] +n 3 n k, n =,...,N. ( 3 n ) k, n =,...,N; in particular, U n are second order approximations to u n. Furthermore, (4.3) yields (4.5) k 3 U n = 3n, n = 3,...,N.

15 i.e., A posteriori estimates for the BDF method 5 Now, for f(t) := t, we obviously have I f = f, and (.6) yields, for n 3, f(t) ϕ(t) = k (t tn ) 3 U n, t (t n,t n ), (4.6) f(t) ϕ(t) = (t t n ) 3 n, t (tn,t n ). Thus, we have E, = k f(t) ϕ(t) dt = 4 3 k3 N n=3 9 n, and conclude that E, is of suboptimal order O(k 3 ). Remark 4.. Using the first reconstruction for the initial value problem (4.) and the scheme (4.), we get E, = k f(t) ˆϕ(t) dt = k 3 N n=3 9 n, and conclude that also in this case E, is of suboptimal order O(k 3 ). Remark 4.. Let us discretize the initial value problem (4.) by combining the BDFmethod with thetrapezoidalscheme: We startwith theexactinitial valueu =, perform one step with the trapezoidal scheme to compute U and subsequently apply the BDF method to obtain U,...,U N. It is then easily seen that U n = u n, i.e., U n = n k, n =,,...,N. Therefore, we have 3 U n =,n = 3,...,N, whence E, =. It is also easily seen that both ˆϕ and ϕ coincide with f in the interval [,k]; we conclude that E vanishes. Furthermore, it is readily seen that both reconstructions Û and Ũ of U considered in this paper coincide in this case with the exact solution u. Therefore, ê = and the a posteriori estimate (.6) holds as an equality for this problem, e(s) ds = Û(s) U(s) ds = Ũ(s) U(s) ds, t [,]. Example 4.. Let us also consider the initial value problem u + u =, t, (4.7) u() =. The BDF method for problem (4.7) is i.e., 3 Un U n + Un + kun =, (4.8) (3+k)U n 4U n +U n =.

16 6 G. Akrivis, P. Chatzipantelidis We will first determine the approximationsu n, in terms ofthe starting approximation U ; we will use the exact value U := and in the sequel will consider two cases, when U is given by the backward Euler or the trapezoidal methods, respectively. For the sake of brevity we will use the notation α := k. Since the roots of the characteristic polynomial ρ, ρ(z) := (3+k)z 4z+, of the difference equation (4.8) are we have z := +α 3+k and z := α 3+k, (4.9) U n = c (z ) n +c (z ) n, n, with constants c and c depending only on the starting approximations U and U. From the relations c +c = U (= ) and c z +c z = U, we obtain (4.) c = U z z z and c = U z z z, and conclude that (4.) U n = U z z z z n U z z z z n, n. Now, according to (.), we have ϕ(t) = (t t n ) ( U n + U n), t J n, and conclude easily ϕ(t) = t tn k [ (+k)u n (4+k)U n +U n ], t J n, n, whence, in view of (4.9), (4.) ϕ(t) = t tn k i= c i [ (+k)z i (4+k)z i + ] z n i, t J n, n. Using the relation (3+k)z i 4z i + =, we easily see that (+k)z i (4+k)z i + = [ 4( z i ) k(+z i ) ] z i, i =,; thus, we rewrite (4.) in the form (4.3) ϕ(t) = t tn k i= c i [ 4( zi ) k(+z i ) ] z n i, t J n, n. In the sequel we will distinguish two cases: In the first case U is computed by the backward Euler method and in the second by the trapezoidal scheme. First case: Starting with the backward Euler method. Performing one step with the backward Euler method for the initial value problem (4.7), we obtain U = /(+k). Therefore, in view of (4.), in this case we have (4.4) c = (+α)+αk α(+k) and c = ( α) αk. α(+k)

17 Now, it is easily seen that (4.5) Using (4.5), from (4.3) we easily obtain A posteriori estimates for the BDF method 7 c [ 4( z ) k(+z ) ] = α α(+k) k, c [ 4( z ) k(+z ) ] = +α α(+k) k. (4.6) ϕ(t) = t tn [ ( α)z n (+α)z n ], t Jn, n. 4α(+k) Furthermore, using (.9) we get (4.7) ϕ(t) = [ +k (+k) 3+k (t t/ )+ k ], t J. From (3.), (4.7) and (4.6), we easily obtain i.e., E = (4.8) E = with (4.9) E = z k 3 [+ (+k) + 8(+k) 3 3+k 3α N n= [ ( α)z n (+α)z n ] ], k 3 [+ (+k) [ + ( α) 8(+k) 3 3+k 3α E E +(+α) ] ] E 3 z N (z z ) N z, E = kz z, E 3 = z z z z N z. Our next task is to determine the order of the terms E,E and E 3. Let us start with E. In view of k /, from z z = /(3+k) we obtain /7 z z /3 and conclude easily that E k 7 7 [ (z z ) N ] 4 5 k( 3 ) = 8 5 k and summarizing, we have E k 3 3 = k; (4.) 8 5 k E k. Concerning E 3, from z = ( α)/(3+k) we obtain /3 z /3 and conclude E = 4 5

18 8 G. Akrivis, P. Chatzipantelidis and E [ () N ] ( 4) 5 = 9 7 ; thus (4.) 5 7 E Furthermore, since k /, we have ( )k +α and thus (4.) From (4.) and (4.) we obtain (4.3) 3 + (+α) 4. 5 (3 7 + ) (+α) E Finally, as far as the order of E is concerned, we first note that (4.4) k z 3 k, whence consequently, since k /, (4.5) Therefore, 3 k 9 k z k; 8 k z k. z z 8 k = 8 k and z z 3 4 k = 3 4k, i.e., (4.6) 3 4k z z 8 k. Using now (4.4) and the fact that ( ) N lim = e /3 N 3N and ( ) N lim = e / N N we easily conclude, in view of (4.6), that E is of exactly order minus one with respect to k. Furthermore, we have whence (4.7) k α k, 4 k ( α) k.

19 A posteriori estimates for the BDF method 9 Now, (4.7) and the previous discussion leads to the conclusion that ( α) E is exactly of first order. Summarizing, E is exactly of third order. Next, we show that E is of fourth order. Indeed, first, in this case (3.4) takes the form (4.8) E = k4 4 k[ U + Now, in view of (4.9), (4.9) U n = k i= Since (3+k)z i 4z i + =, we have and, using (4.4), we easily see that therefore, (4.9) takes the form (4.3) U n = Hence, (4.8) yields i.e., E = (4.3) E = N U n ]. n=3 c i (z i z i +)z n i. z i z i + = [( z i ) kz i ]z i, c i (z i z i +)α(+k) = ( ) i+ k z i, i =,; ( z n z n ). α(+k) k 4 [ 4(α) (+k) k (z z ) + N n= ( z n z n ) ], k 4 [ (α) ] 4(α) (+k) (3+k) k +(ke E +ke 3 ) with E,E and E 3 as in (4.9). Thus, we easily conclude that E is of fourth order; see the discussion following (4.9). Second case: Starting with the trapezoidal method. Here, we will consider the discretization of (4.7) by first performing one step with the trapezoidal method and subsequently applying the BDF method to compute U,...,U n. It is easily seen that U = (4 k)/(4+k), whence, in view of (4.), (4.3) c = 4(+α) ( α)k k α(4+k) and c = 4( α) (+α)k k. α(4+k) Also, the analogous calculation to the one leading to (4.5) yields in this case [ c 4( z ) k(+z ) ] = α(4+k) k3, (4.33) [ c 4( z ) k(+z ) ] = α(4+k) k3.

20 G. Akrivis, P. Chatzipantelidis Using (4.33), from (4.3) we easily obtain (4.34) ϕ(t) = (t tn )k ( z n z n ), t Jn, n. 4α(4+k) Furthermore, using (.9) we get k (4.35) ϕ(t) = (4+k)(3+k) (t t/ ), t J. From (3.), (4.35) and (4.34), we easily obtain i.e., E = (4.36) E = k 5 [ 4α (4+k) 4α N (3+k) + n= ( z n z n ) ], k 4 [ 4α k ] 4α (4+k) (3+k) +(ke E +ke 3 ) with E,E and E 3 as in (4.9). In view of (4.), (4.) and the fact that ke is of zeroth order, we conclude that E is of optimal order in this case, namely of order exactly four. 5. Numerical experiments. In this section we present numerical results for Example 4. for both methods (.) and (.). Our numerical calculations justify the theoretical results of 4 and illustrate the effectivity of our a posteriori error estimators in a simple case. First, we use the three-point reconstruction Ũ of the approximation U; cf. (.). In Tables 5. and 5. we state the values of the parts E and E of (the square of) our a posteriori error estimator as well as their orders in the cases U is computed by the trapezoidal method and the backward Euler scheme, respectively. It is clearly seen that while all other quantities are of optimal order four (since they estimate the square of the error), part E in the case of the backward Euler scheme is of reduced order three; this confirms the theoretical results in 4. For the computation of E and E we employed the Gauss Legendre quadrature formula with three nodes in each subinterval J n ; notice that the integrand, as a polynomial of degree four, is integrated exactly by this formula. Furthermore, we employed the same quadrature formula to approximate the errors e(s) ds and ê(s) ds in the estimates (.) and (.). Also, we denote by Err the square of the L norm (in time) of the errors, Err = ( e(s) + ê(s) ) ds, and by Err the sum of Err and the discrete maximum norm (in time) of ê, Err = max ê(t n ) + Err. The lower and upper estimators are E /3 and E +E, respectively; see (.) and (.). We present the results of this computation as well as their effectivity indices Eff i, Eff := Lower estimator Err and Eff := Upper estimator Err in Tables 5.3 and 5.4, again for the trapezoidal and the backward Euler schemes, respectively. We graphically demonstrate the effectivity indices in log-log scale (with the base of the logarithms equal to two) in Figure 5.. Finally, we state the corresponding results for the reconstruction Û given in (.5) in Tables and in Figure 5..

21 A posteriori estimates for the BDF method Table 5. Three-point reconstruction: Order of E and E when starting with the trapezoidal method N E Order E Order 9.448e e e e e e e e e e e e e e e e e e e e Table 5. Three-point reconstruction: Order of E and E when starting with the backward Euler scheme N E Order E Order 3.44e e e e e e e e e e e e e e e e e e e e.998 REFERENCES [] G. Akrivis and M. Crouzeix, Linearly implicit methods for nonlinear parabolic equations, Math. Comp. 73 (4) [] G. Akrivis, Ch. Makridakis, and R. H. Nochetto, A posteriori error estimates for the Crank Nicolson method for parabolic equations, Math. Comp. 75 (6) [3] G. Akrivis, Ch. Makridakis, and R. H. Nochetto, Optimal order a posteriori error estimates for a class of Runge Kutta and Galerkin methods, Numer. Math. 4 (9) [4] A. Lozinski, M. Picasso and V. Prachittham, An anisotropic error estimator for the Crank- Nicolson method: Application to a parabolic equation, SIAM J. Sci. Comput. 3 (9) [5] Ch. Makridakis and R. H. Nochetto, A posteriori error analysis for higher order dissipative methods for evolution problems. Numer. Math. 4 (6) [6] R. H. Nochetto, G. Savaré, and C. Verdi, A posteriori error estimates for variable time step discretizations of nonlinear evolution equations, Comm. Pure Appl. Math. 53 () [7] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. nd ed., Springer-Verlag, Berlin, 6.

22 G. Akrivis, P. Chatzipantelidis Table 5.3 Three-point reconstruction: Lower and upper estimators of the error and effectivity indices when starting with the trapezoidal method N 3 E Err Err E +E Eff Eff 3.494e e 6.875e e e e e e e e e 7 7.e e.33e 9.937e e e.3568e 7.58e 3.736e e e 4.576e.3e e e e.759e e e 4.839e 3 8.6e e 6.99e 5.366e e e 7.34e 6 7.7e e Table 5.4 Three-point reconstruction: Lower and upper estimators of the error and effectivity indices when starting with the Euler method N 3 E Err Err E +E Eff Eff.338e e e e e e e 5.47e e e e 6.869e e.793e e e e.459e e e e 9.354e.899e e e e.874e 6.45e e e.65e 7.547e e 6.346e e e e 7.469e e 4.84e Table 5.5 Reconstruction (.5): Order of E and E when starting with the trapezoidal method N E Order E Order.4e e e e e e e e e e e e e e e e e e e e

23 A posteriori estimates for the BDF method 3 Table 5.6 Reconstruction (.5): Order of E and E when starting with the Euler method N E Order E Order 9.349e 6.74e e e e e e e e e e e e e e e e e e e.9985 Table 5.7 Reconstruction (.5): Lower and upper estimators of the error and effectivity index when starting with the trapezoidal method N 3 E Err Err E +E Eff Eff e e e e e e e e e 8 3.6e e e e.79e 9.93e e e.3487e 7.5e.3679e e 8.54e 4.569e.548e e e e 9.677e e e 4.836e e e 6.98e 5.365e e e 7.3e 6 7.4e 6.39e Table 5.8 Reconstruction (.5): Lower and upper estimators of the error and effectivity index when starting with the Euler method N 3 E Err Err E +E Eff Eff 3.6e e e e e e e e e e e 6.3e e.74e e 7.653e e.4486e e 8.746e e e.89e 9.637e e e.87e 3.344e e e.65e 4.73e e e e e e 7.469e e e

24 4 G. Akrivis, P. Chatzipantelidis Fig. 5.. Three-point reconstruction: Log-log graphs of the effectivity indices, of upper and lower estimator, when starting with the Euler and the trapezoidal method, respectively 5 Lower estimator (Euler) Upper estimator (Euler) Lower estimator (trapezoidal) Upper estimator (trapezoidal) Effectivity index Degrees of freedom Fig. 5.. Reconstruction (.5): Log-log graphs of the effectivity indices, of upper and lower estimator, when starting with the Euler and the trapezoidal method, respectively 5 Lower estimator (Euler) Upper estimator (Euler) Lower estimator (trapezoidal) Upper estimator (trapezoidal) Effectivity index Degrees of freedom