A POSTERIORI ERROR ESTIMATES FOR THE CRANK NICOLSON METHOD FOR PARABOLIC EQUATIONS

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1 A POSTERIORI ERROR ESTIMATES FOR THE CRANK NICOLSON METHOD FOR PARABOLIC EQUATIONS GEORGIOS AKRIVIS, CHARALAMBOS MAKRIDAKIS, AND RICARDO H. NOCHETTO Abstract. We derive optimal order a posteriori error estimates for time discretizations by both the Crank Nicolson and the Crank Nicolson Galerkin methods for linear and nonlinear parabolic equations. We examine both smooth and rough initial data. Our basic tool for deriving a posteriori estimates are second order Crank Nicolson reconstructions of the piecewise linear approximate solutions. These functions satisfy two fundamental properties: i they are explicitly computable and thus their difference to the numerical solution is controlled a posteriori, and ii they lead to optimal order residuals as well as to appropriate pointwise representations of the error equation of the same form as the underlying evolution equation. The resulting estimators are shown to be of optimal order by deriving upper and lower bounds for them depending only on the discretization parameters and the data of our problem. As a consequence we provide alternative proofs for known a priori rates of convergence for the Crank Nicolson method.. Introduction In this paper we derive a posteriori error estimates for time discretizations by Crank Nicolson type methods for parabolic partial differential equations p.d.e. s. The Crank Nicolson scheme is one of the most popular time stepping methods; however, optimal order a posteriori estimates for it have not yet been derived. Most of the many contributions in the last years devoted to a posteriori error control for time dependent equations concern the discretization in time of linear or nonlinear equations with dissipative character by the backward Euler method or by higher order discontinuous Galerkin methods; cf., e.g., [2], [7], [8], [2], [9], [9] and [3]. Date: June 6, Mathematics Subject Classification. Primary 65M5, 65M5. Key words and phrases. Parabolic equations, Crank Nicolson method, Crank Nicolson Galerkin method, Crank Nicolson reconstruction, Crank Nicolson Galerkin reconstruction, a posteriori error analysis. The first author was partially supported by a Pythagoras grant funded by the Greek Ministry of National Education and the European Commission. The second author was partially supported by the European Union RTN-network HYKE, HPRN-CT , the EU Marie Curie Development Host Site, HPMD-CT-2-2 and the program Pythagoras of EPEAEK II. The third author was partially supported by NSF Grants DMS and DMS-2467.

2 2 G. AKRIVIS, CH. MAKRIDAKIS, AND R. H. NOCHETTO Let u and U be the exact and the numerical solution of a given problem. In a posteriori error analysis u U ηu, we seek computable estimators ηu depending on the approximate solution U and the data of the problem such that i ηu decreases with optimal order for the lowest possible regularity permitted by our problem, and ii the constants involved in the estimator ηu are explicit and easily computable. In this paper we derive optimal order estimators of various types for the Crank Nicolson and the Crank Nicolson Galerkin time stepping methods applied to evolution problems of the form: Find u : [,T] DA satisfying. { u t+aut = Bt,ut, < t < T, u = u, witha : DA H apositivedefinite, selfadjoint, linearoperatoronahilbertspace H,, with domain DA dense in H, Bt, : DA H,t [,T], a possibly nonlinear operator, and u H. The structural assumption 6. on Bt, implies that problem. is parabolic. A main novel feature of our approach is the Crank Nicolson reconstruction Û of the numerical approximation U. This function satisfies two fundamental properties: i it is explicitly computable and thus its difference to the numerical solution is controlled a posteriori, and ii it leads to an appropriate pointwise representation of the error equation, of the same form as the original evolution equation. Then by employing techniques developed for the underlying p.d.e. we conclude the final estimates. Of course, depending on the stability methods that are used, we obtain different estimators. The resulting estimators are shown to be of optimal order by deriving upper bounds for them, depending only on the discretization parameters and the data of our problem. As a consequence we provide alternative proofs for a priori estimates, depending only on the data and corresponding known rates of convergence for the Crank Nicolson method. The above idea is related to earlier work on a posteriori analysis of time or space discrete approximations of evolution problems [9, 8, 7]. It provides the means to show optimal order error estimates with energy as well as with other stability techniques. An alternative approach for a posteriori analysis of time dependent problems, based on the direct comparison of u and U via parabolic duality, was considered in [2], [7], [2], [9] for p.d.e. s and in [], [] for ordinary differential equations o.d.e. s. In particular Estep and French [] considered the continuous Galerkin method for o.d.e s. Its lowest order representative corresponds to a variant of the Crank Nicolson method the Crank Nicolson Galerkin method considered also in this paper. A posteriori bounds with energy techniques for Crank Nicolson methods for the linear Schrödinger equation were proved by Dörfler [6] and

3 A POSTERIORI ESTIMATES FOR THE CRANK NICOLSON METHOD 3 fortheheat equationbyverfürth[22]; theupperboundsin[6], [22] areofsuboptimal order. Most of this paper is devoted to linear parabolic equations, namely Bt, ut = ftforagivenforcingfunctionf. Thegeneralnonlinear problem.isonlybriefly discussed inthelast section. The paper isorganized asfollows. We startinsection 2 by introducing the necessary notation, the Crank Nicolson and the Crank Nicolson Galerkin CNG methods for the linear problem 2.. We then observe that the direct use of standard piecewise linear interpolation at the approximate nodal values, see 2.3, would lead to suboptimal estimates as in [6] and [22]. The Crank Nicolson and Crank Nicolson Galerkin reconstructions Û are, instead, continuous piecewise quadratic functions which are defined in 2.9 and 2.22, respectively. In Section 3 we estimate Û U. Section 4 is devoted to the a posteriori error analysis for linear equations. Error estimates are obtained by using energy techniques, as well as Duhamel s principle. Both estimators lead to second order convergence rates. Note the interesting similarity of the estimator obtained by Duhamel s principle to those established in the literature by parabolic duality. In Section 5 we discuss the form of estimators in the case of nonsmooth initial data. In Section 6 we finally conclude with the case of nonlinear equations. 2. Crank Nicolson methods for linear equations Most of this paper focuses on the case of a linear equation, { u t+aut = ft, < t < T, 2. u = u, with f : [,T] H. Let = t < t < < t N = T be a partition of [,T], := t n,t n ], and := t n t n. 2.. The Crank Nicolson method. For given {v n } N n= we will use the notation v n := vn v n, v n 2 := 2 vn +v n, n =,...,N. The Crank Nicolson nodal approximations U m DA to the values u m := ut m of the solution u of 2. are defined by 2.2 U n +AU n 2 = ft n 2, n =,...,N, with U := u. Since the error u m U m is of second order, to obtain a second order approximation Ut to ut, for all t [,T], we define the Crank Nicolson approximation U : [,T] DA to u by linearly interpolating between the nodal values U n and U n, 2.3 Ut = U n 2 +t t n 2 U n, t.

4 4 G. AKRIVIS, CH. MAKRIDAKIS, AND R. H. NOCHETTO Let Rt H, 2.4 Rt := U t+aut ft, t, denote the residual of U, i.e., the amount by which the approximate solution U misses being an exact solution of 2.. Now U t+aut = U n +AU n 2 +t t n 2 A U n, t, whence, in view of 2.2, U t+aut = ft n 2 +t t n 2 A U n, t. Therefore, the residual can also be written in the form 2.5 Rt = t t n 2 A U n +[ft n 2 ft], t In. Obviously, Rt is an a posteriori quantity of first order, even in the case of a scalar o.d.e. u t = ft, although the Crank Nicolson scheme yields second-order accuracy. Since the error e := u U satisfies e + Ae = R, applying energy techniques to this error equation, as in [6], [22], leads inevitably to suboptimal bounds Crank Nicolson reconstruction. To recover the optimal order we introduce a Crank Nicolson reconstruction Û of U, namely a continuous piecewise quadratic polynomial in time Û : [,T] H defined as follows. First, we let ϕ : H be the linear interpolant of f at the nodes t n and t n 2, ϕt := ft n 2 + t t n 2 [ft n 2 ft n ], t, and define a piecewise quadratic polynomial Φ by Φt := t ϕsds,t I t n n, i.e., 2.7 Φt = t t n ft n 2 t t n t n t[ft n 2 ft n ]. As will become evident in the sequel, an important property of Φ is that 2.8 Φt n =, Φt n = ft n 2 = We now introduce the Crank Nicolson reconstruction 2.9 Ût := U n t ft n 2 dt. Û of U by t n AUsds+Φt t. We can interpret this formula as the result of formally replacing the constants U n 2 and ft n 2 in 2.2 by their piecewise linear counterparts U and ϕ, and next integrating AU +ϕ from t n to t. Consequently Û t+aut = ϕt, t.

5 A POSTERIORI ESTIMATES FOR THE CRANK NICOLSON METHOD 5 Evaluating the integral in 2.9 by the trapezoidal rule, we obtain 2. Ût = U n 2 t tn A[Ut+U n ]+Φt t, which can also be written as Ût = U n A[t t n U n + 2 t tn 2 U n ]+Φt t. Obviously Ûtn = U n. Furthermore, in view of 2.8 and 2.2, we have Ût n = U n AU n 2 +Φt n = U n + [ AU n 2 +ft n 2 ] = U n + U n = U n. Thus, Û and U coincide at the nodes t,...,t N ; in particular, Û : [,T] H is continuous. Remark 2. Choice of ϕ. Let t. Since ft = f t+f tt t+ok 2 n, t, it is easily seen that the only affine functions ϕ satisfying are the ones of the form Obviously ϕtdt = f t+ 2 sup ft ϕt = Okn 2 t ϕt = f t+ [ f t+o ] t t+ok 2 n. [ f t+o ]{ t n t 2 t n t 2} +Ok 3 n ; therefore, for f t, the requirement ϕtdt = f t leads to t = t n 2 and ϕt = ft n 2 + [ f t n 2 +Okn ] t t n 2. These comments demonstrate the special features of the midpoint method among the one-stage Runge Kutta methods. Furthermore, our choice 2.6 is motivated by the fact that for all affine functions ϕ on we have ϕsds = ϕt n 2, whence the requirement ϕsds = ft n 2, see 2.8, is satisfied if and only if ϕ interpolates f at t n 2. Now, to ensure that ϕt is a second order approximation to ft, we let ϕ interpolate f at an additional point t n, [t n,t n ]; of course, in the case t n, = t n 2, ϕ is the affine Taylor polynomial of f around t n 2. In the sequel we let t n, := t n. In view of 2.9 and 2.6, we have 2. Û 2 t+aut = ft n 2 + t t n 2 [ft n 2 ft n ], t ;

6 6 G. AKRIVIS, CH. MAKRIDAKIS, AND R. H. NOCHETTO therefore, the residual ˆRt of Û, 2.2 ˆRt := Û t+aût ft, t, can be written in the form ˆRt = A[Ût Ut] 2.3 { 2 } + ft n 2 + t t n 2 [ft n 2 ft n ] ft, t. We will see later that the a posteriori quantity ˆRt is of second order; compare with The Crank Nicolson Galerkin method. Next we consider the discretization of 2. by the Crank Nicolson Galerkin method. The Crank Nicolson Galerkin approximation to u is defined as follows: We seek U : [,T] DA, continuous and piecewise linear in time, which interpolates the values {U n } N n= given by 2.4 U n +AU n 2 = ftdt, n =,...,N, with U = u. This function U can be expressed in terms of its nodal values, 2.5 Ut = U n 2 +t t n 2 U n, t. For t, U t = U n, and 2.4 takes the form 2.6 U t+au n 2 = ftdt, n =,...,N. Now, ψtdt is the L 2 orthogonal projection of a function ψ onto the space of constant functions on, and Utdt = U n 2; therefore 2.6 yields the pointwise equation for the Crank Nicolson Galerkin approximation 2.7 U t+p AUt = P ft t, with P denoting the L 2 orthogonal projection operator onto the space of constant functions in. Equivalently, as it is customary, this method can be seen as a finite element in time method, [4], [ 2.8 U,v+AU,v ] dt = f,vdt v DA. For a priori results for general Continuous Galerkin methods for various evolution p.d.e s cf. [3, 4, 4, 5]. Let Rt, 2.9 Rt := U t+aut ft,

7 A POSTERIORI ESTIMATES FOR THE CRANK NICOLSON METHOD 7 denote the residual of U. In view of 2.7, the residual can also be written in the form 2.2 Rt = A[Ut P Ut] [ft P ft]. However, this residual is not appropriate for our purposes, since, even in the case of an o.d.e. u t = ft, Rt can only be of first order, although our approximations are piecewise polynomials of degree one Crank Nicolson Galerkin reconstruction. To recover the optimal order Okn 2, we introduce the Crank Nicolson Galerkin reconstruction Û of the approximate solution U, namely the continuous and piecewise quadratic function Û : [,T] H defined by 2.2 Ût := U n Hence t t n [AUs P fs]ds t Ût = U n 2 t tn A[Ut+U n ]+ t t n P fsds, t, with P denoting the L 2 orthogonal projection operator onto the space of linear polynomials in ; that Ût is continuous, namely Ûtn = U n, is a consequence of P f = f. Obviously, Û satisfies the following pointwise equation: 2.23 Û t+aut = P ft t ; compare with 2.7. In view of 2.23, the residual ˆRt, 2.24 ˆRt := Û t+aût ft, of Û can also be written as 2.25 ˆRt = A[ Ût Ut]+[P ft ft], t. ˆRt is an a posteriori quantity and, as we will see in Section 3, it is of second order at least in some cases. 3. Estimation of Û U Inthissectionwewill estimate Û U forboththecrank Nicolson andthecrank Nicolson Galerkin methods; also we will derive representations of Û U that will be useful in the sequel. We let V := DA /2 and denote the norms in H and in V by and, v := A /2 v = Av,v /2, respectively. We identify H with its dual, and let V be the dual of V V H V. We still denote by, the duality pairing between V and V, and by the dual norm on V, v := A /2 v = v,a v /2.

8 8 G. AKRIVIS, CH. MAKRIDAKIS, AND R. H. NOCHETTO 3.. The Crank Nicolson method. From 2. we obtain Ût Ut = U n Ut 2 t tn A[Ut+U n ]+Φt Therefore, in view of 2.2, = t t n U n 2 t tn A[Ut+U n ]+Φt. Ût Ut = t t n [ AU n 2 ft n 2 ] 2 t tn A[Ut+U n ]+Φt = 2 t tn A[Ut U n ]+Φt t t n ft n 2, whence, using 2.7, for t, 3. Ût Ut = t t n t t n 2 A U n [ft n 2 ft n ], from which we immediately see that max t In Ût Ut = Ok2 n The Crank Nicolson Galerkin method. Subtracting 2.5 from 2.22, and utilizing 2.4, for t we obtain 3.2 Now, it is easily seen that 3.3 P ft = Ût Ut = 2 t tn t n ta U n t tn fsds+ 2 k 3 n and 3.2 can be rewritten in the form fsds+ t t n Ût Ut = t t n t n t 2 A U n 6 k 3 n t t n P fsds. fss t n 2 ds, fss t n 2 ds. Therefore, U and Û coincide at the endpoints of, and, consequently, at all nodes t,...,t N. From 3.4 we immediately see that max t In Ût Ut = Ok2 n. Let us write both 3. and 3.4 in the form 3.5 Ût Ut = 2 t tn t n t A U n ρ f,n ; here ρ f,n = ρ CN f,n for the Crank Nicolson method and ρ f,n = ρ CNG f,n Nicolson Galerkin method, respectively, with 3.6 ρ CN f,n := 2 [ ft n 2 ft n ] for the Crank

9 and 3.7 ρ CNG f,n := 2 k 3 n A POSTERIORI ESTIMATES FOR THE CRANK NICOLSON METHOD 9 Consequently, both ρ CN f,n fss t n 2 2 ds = fs ft n kn 3 2 s t n 2 ds. and ρcng f,n depend on the first derivative of f. 4. Smooth data error estimates Let the errors e and ê be defined by e := u U and ê := u Û. Subtracting 2. or 2.23, respectively, from the differential equation in 2., we obtain 4. ê t+aet = R f t, with R f = Rf CN for the Crank Nicolson method and R f = Rf CNG Nicolson Galerkin method, respectively, defined by for the Crank 4.2 Rf CN t := ft { 2 ft n 2 + t t n 2 [ft n 2 ft n ] }, t, and 4.3 R CNG f t := ft P ft, t. We make the following further regularity assumption on 2.2: Ût V, t [,T]. Û, defined in 2.9 and 4.. Energy estimates. Taking in 4. the inner product with êt, we obtain 4.4 Now, and d 2dt êt 2 + Aet,êt = R f t,êt. Aet,êt = et 2 + êt 2 êt et 2 2 Rf t,êt R f t êt 2 ; therefore, 4.4 yields 4.5 d dt êt 2 + et êt 2 Ût Ut 2 +2 R f t 2. We recall that v = A /2 v and v = A /2 v.

10 G. AKRIVIS, CH. MAKRIDAKIS, AND R. H. NOCHETTO 4... Upper bound. Since ê =, integration of 4.5 from to t T yields t êt 2 + es ês 2 ds max τ t t From 4.6 we easily conclude { êτ 2 + Next, let β be given by Ûs Us 2 ds+2 τ t 4.8 β := then, obviously, t es 2 + } 2 ês 2 ds Ûs Us 2 ds+2 t 2 t 2 dt = 3 ; R f s 2 ds. t R f s 2 ds. t t n 2 t n t 2 dt = βkn 5. With this notation, in view of 3.5, we have t m 4.9 Ût Ut 2 dt β kn A 5 3/2 U n 2 + ρ f,n Lower bound. Obviously, n= Ûs Us es + ês and thus 4. Ûs Us 2 3 es ês 2. In particular, combining the upper and lower bounds, we have t t Ûs Us 2 ds es ês 2 ds 4. 3 t Ûs Us 2 ds+2 t R f s 2 ds. Invoking 4.6, this shows that êt 2 is dominated by t es 2 + ês 2 ds, 2 the energy norm error, plus data oscillation t R fs 2 ds. We will next estimate the lower bounds above in terms of U n and data in analogy to the upper bound in 4.9. To this end we first note that 3.5 yields Ût Ut 2 4 t tn 2 t n t 2 2 A3/2 U n 2 ρ f,n 2,

11 A POSTERIORI ESTIMATES FOR THE CRANK NICOLSON METHOD whence, in view of 4.8, we have 4.2 t m Ûs Us 2 ds β 8 kn 5 A3/2 U n 2 β 4 n= Therefore, 4., 4.9 and 4.2 imply β kn 5 24 A3/2 U n 2 β kn 5 2 ρ f,n 2 n= n= t m 4.3 es ês 2 ds β 2 n= k 5 n A 3/2 U n 2 + ρ f,n 2 kn 5 ρ f,n 2. n= t m +2 R f s 2 ds. Note that error bounds of this type are customary in the a posteriori analysis of elliptic problems in which data oscillation appear with different signs in the upper and lower bounds. On the other hand, if f is constant, then the lower and upper bounds are exactly the same up to a constant: 4.4 β 24 kn 5 A3/2 U n 2 n= t m β 2 es ês 2 ds kn 5 A3/2 U n 2. Let us also note that, in the case of f constant, in view of 3.5, the estimate 4., for t = t m, can be written in the form of 4.4 with the lower bound multiplied by 2 and the upper bound by /2. Remark 4. Optimality of the lower bound. The pointwise lower bound 4.5 Û Us es + ês, s [,T], cannot be expected to be of exactly second order for all s, since Û U vanishes at the nodes t,...,t N. However, we can conlcude from 4.5 the following lower bound in the L H norm, with v L H := max t [,T] vt : n= 4.6 Û U L H e L H + ê L H. For the trivial case that the exact solution u is an affine function, and so is f, then both the Crank Nicolson approximation U and the Crank Nicolson reconstruction Û coincide with u; thus 4.6 is an equality. Next consider the case of u nonaffine. In view of 3., we have k 2 Û Utn 2 = n 4 2 A U n [ ft n 2 ft n ].

12 2 G. AKRIVIS, CH. MAKRIDAKIS, AND R. H. NOCHETTO Now, for smooth data, we have, as, 2 A U n [ ft n 2 ft n ] Au 2[ t n f t n ] = 2 u t n. If u t n, we then have Û U L H; ckn 2 with a positive constant c. This is the generic situation. That thelower bound inthel 2 V normis of thesame formasthe upper bound, in the case of f constant, can be seen from 4.4. Otherwise, let us note that, in view of 3. and 4.8, Û U 2 L 2 V = β N n= k 5 n 2 A U n [ ft n 2 ft n ] 2 and, assuming that the partition is quasiuniform and letting k := max n, Û U 2 L 2 V N cβk4 2 A U n [ ft n 2 ft n ] 2. Now, as k, N n= whence, if u is not affine, n= 2 A U n [ ft n 2 ft n ] 2 2 u 2 L 2 V 4.7 Û U L 2 V Ck 2. Remark 4.2 Alternative estimate in L H. Combining 4. and 4.6 we can replace the factor on the right-hand side of 4.7 by 2, if we only want to estimate 3 the first term on the left-hand side of 4.7. Indeed, in view of 4., from 4.6 we obtain i.e., and thus êt t êt 2 + êt t t 4.8 max τ t êτ Ûs Us 2 ds t es ês 2 ds Ûs Us 2 ds+2 Ûs Us 2 ds+2 t t t Ûs Us 2 ds+2 R f s 2 ds, R f s 2 ds, t, R f s 2 ds.

13 A POSTERIORI ESTIMATES FOR THE CRANK NICOLSON METHOD 3 The following stability estimate for the Crank Nicolson scheme will be useful in the convergence proof: Lemma 4. Stability. Let {U n } N n= be the Crank Nicolson approximations for 2., 4.9 U n +AU n 2 = fn, where either f n = ft n 2 or f n = fsds. Then the following estimate holds for m N: 4.2 A 3/2 U n 2 + A 2 U m 2 A 2 U 2 + A 3/2 fn 2. n= Proof. We apply A to the scheme, A U n +A 2 U n 2 = A fn, n= and take the inner product with 2 A 2 U n U n to obtain 2 A 3/2 U n 2 + A 2 U n 2 A 2 U n 2 = 2 A f n,a 2 U n. Summing here from n = to m, we get 2 A 3/2 U n 2 + A 2 U m 2 = A 2 U 2 +2 n= A f n,a 2 U n, whence the Cauchy Schwarz and the arithmetic geometric mean inequalities yield 2 A 3/2 U n 2 + A 2 U m 2 A 2 U 2 n= + A 3/2 fn 2 + n= n= A 3/2 U n 2, and the proof is complete. From 4.6, 4.9, 4.2 and 3.5 we conclude the following Theorem. We emphasize that the optimal order a priori error estimate 4.23, depending only on the data see below follows from our a posteriori estimate 4.6. This shows, in particular, that the a posteriori estimate is of optimal second order. Theorem 4. Error estimates. Let {U n } N n= be either the Crank Nicolson approximations or the Crank Nicolson Galerkin approximations to the solution of problem 2., e = u U and ê = u Û. The following a posteriori estimate is valid for m N : t m 4.2 êt m 2 + es ês 2 ds β kn 5 2 A3/2 U n 2 +E[f ] n= n=

14 4 G. AKRIVIS, CH. MAKRIDAKIS, AND R. H. NOCHETTO with β given by 4.8 and 4.22 E[f ] := 2 t m R f s 2 ds+ β 2 kn ρ 5 f,n 2. Here R f and ρ f,n are given by 4.2 and 3.6, respectively, for the Crank Nicolson method, and by 4.3 and 3.7 for the Crank Nicolson Galerkin method. Furthermore, if U DA 2 and ft DA 3/2, then the following a priori estimate holds for m N : t m êt m 2 + es ês 2 ds 4.23 β 2 max kn 4 A 2 U 2 + A 3/2 fn +E[f 2 ], n with f n := ft n 2 for the Crank Nicolson method, and f n := fsds for the Crank Nicolson Galerkin method. Remark 4.3 Equivalent upper bound for CNG. In the case of the Crank Nicolson Galerkin method, it is easily seen that 4.23 yields t m êt m 2 + es ês 2 ds 4.24 β t m 2 max kn 4 A 2 U 2 + A 3/2 fs 2 ds +E[f ]. n Remark 4.4 Estimate for E[ f ]. As a by-product of interpolation theory, we realize that the following optimal order estimate is valid for the error E[f ] in the forcing term f: E[f ] C kn 2 f s 2 + f s 2 ds. n= 4.2. Estimates via Duhamel s principle. We first rewrite the relation 4. in the form n= n= 4.25 ê t+aêt = RÛt+R f t with 4.26 RÛt := A[Ut Ût]. We will use Duhamel s principle in Let E A t be the solution operator of the homogeneous equation 4.27 v t+avt =, v = w,

15 A POSTERIORI ESTIMATES FOR THE CRANK NICOLSON METHOD 5 i.e., vt = E A tw. It is well known that the family of operators E A t has several nice properties, in particular it is a semigroup of contractions on H generated by the operator A. Duhamel s principle applied to 4.25 yields 4.28 êt = t E A t s [ RÛs+R f s ] ds. In the sequel we will use the smoothing property cf., e.g., Crouzeix [5], Thomée [2] 4.29 A l E A tw C A t l m Am w l m. In addition, note that A and E A commute, i.e., AE A tw = E A taw. In particular, 4.29 can also be written in the form 4.3 E A ta l w C A t l m Am w, l m, whence E A tw C A t m A m w. Next, using 4.28 for t = t n, we have êt n EA t n s [ RÛs+R f s ] ds and thus t n C A êt n C A +C A tn EA t n s [ RÛs+R f s ] ds A RÛs ds+ca Rf s t n s I ds n We now recall 4.26 and 3.5, namely, [ A RÛs+R t n f s ] ds s A RÛs ds+ca Rf s t n s I ds n +C A sup [ A RÛs+R f s ] s [,t n ] t n t n s ds A RÛs = Us Ûs = 2 t tn t n t A U n ρ f,n, to obtain a bound for the first two terms on the right-hand side of 4.3, 4.33 A RÛs ds+ Rf s t n s I k 2 ds n A U n +E [ ;f], n 4

16 6 G. AKRIVIS, CH. MAKRIDAKIS, AND R. H. NOCHETTO with 4.34 E [ ;f] := k2 n ρf,n +kn max Rf s. 4 s In addition, using again 4.32, we have 4.35 with sup [ A RÛs+R f s ] t n s [,t n ] 8 lntn max j n t n s ds k 2 j A U j +ln tn E 2 [[,t n ];f] 4.36 E 2 [[,t n ];f] := 8 max k 2 j ρ f,j + max sup A R f s. j n j n s I j We have thus proved Theorem 4.2 Error estimates. Let {U n } N n= be either the Crank Nicolson or the Crank Nicolson Galerkin approximations to the solution of problem 2.. Then, with the notation of Theorem 4., the following a posteriori estimate is valid for n N : êt n 8 C A 2k 2n A U n +ln tn max k 4.37 n j n k2 j A U j + C A E [ ;f]+ln tn E 2 [[,t n ];f], with C A the constant of 4.29 for l =, m =, and the terms involving f are defined in 4.34 and Furthermore, if U DA 2,ft DA 3/2, and k := max j n k j 2+ln t n /2, the following a priori estimate holds for n N : 4.38 êt n n /2 } 8 C A k { A 2 2 U 2 + k j A 3/2 fj 2 + max A f j j n j= +C A E [ ;f]+ln tn E 2 [[,t n ];f]. Proof. It only remains to prove It immediately follows from 4.9 that A U n A 2 U n 2 + A fn max A 2 U n+, A 2 U n + A f n, and 4.38 thus results from 4.37 in light of 4.2.

17 A POSTERIORI ESTIMATES FOR THE CRANK NICOLSON METHOD 7 Remark 4.5 Alternative bound for CNG. In the case of the Crank Nicolson Galerkin method, it is easily seen that 4.38 yields êt n t n /2 } 8 C Ak { A 2 2 U 2 + A 3/2 fs ds 2 + max 4.39 s t n Afs +C A E [ ;f]+ln tn E 2 [[,t n ];f]. 5. Estimates for initial data with reduced smoothness In this section our objective is the derivation of a posteriori error estimates in the case of initial data with reduced smoothness. Since the initial value problem 2. can be split into one with homogeneous initial data and one with homogeneous equation, and we are mainly concerned with the effect of the initial data, we focus on the homogeneous initial value problem { u t+aut =, < t < T, 5. u = u. A typical a priori nonsmooth data estimate reads see [2] 5.2 ut n U n C k t n s U, s being the order of the method at hand and k the constant time step, t n = nk. It is well known that estimates of this type are available for strongly A stable schemes, such as the Runge Kutta Radau IIA methods, and, in particular, the backward Euler method. Similar estimates are not available for A stable schemes; a way of securing such estimates for A stable schemes is to start the time stepping procedure with a few steps of a strongly A stable scheme of order s and then proceed with the main scheme. For instance, for the Crank Nicolson method it suffices to perform the first two steps with the backward Euler scheme and subsequently proceed with the Crank Nicolson method. The use of the Euler scheme as starting procedure for the Crank Nicolson method in order to establish a priori error estimates for nonsmooth initial data was suggested by Luskin and Rannacher see [2], Theorems 7.4 and 7.5. In the a posteriori error analysis we have to derive estimates with reduced regularity requirements on the initial data, but in a form that allows efficient monitoring of the time steps. To this end we will establish estimates that are the a posteriori analogs of Theorems 7.4 and 7.5 of [2] for the following modification of the scheme: We let U := u and define the approximations U m to u m := ut m, m =,...,N, by 5.3i 5.3ii U n +AU n =, n =,2, U n +AU n 2 =, n = 3,...,N.

18 8 G. AKRIVIS, CH. MAKRIDAKIS, AND R. H. NOCHETTO Note first that, even for U H, due to the fact that U and U 2 are backward Euler approximations, we have U DA and U 2 DA 2 ; then, obviously, U n DA 2 for n 2. We now proceed to the definition of the reconstruction U. 5.. Reconstruction. Given the nodal approximations U,...,U N, we define the approximation Ut to ut, t [,T], in the intervals I and I 2 by interpolating by piecewise constants at their right ends and at the other subintervals by linearly interpolating between the nodal values, i.e., 5.4i 5.4ii Ut = U n, t, n =,2, Ut = U n +t t n U n, t, n = 3,...,N cf Furthermore, we let the reconstruction Û of U be given by 5.5 Ût := U n i.e., t t n AUsds, t, 5.6i 5.6ii Ût = U n t t n AU n, t, n =,2, Ût = U n 2 t tn A[Ut+U n ], t, 3 n N. We observe that 5.6i coincides with the continuous piecewise linear reconstruction of[9], whereas5.6ii agrees with the continuous piecewise quadratic reconstruction 2. for f =. In view of 5.3i, we obtain 5.7i Ut Ût = tn tau n, t, n =,2. Furthermore, in view of 5.3ii, 5.7ii Ut Ût = 2 t tn t n ta U n, t, 3 n N see 3.. Note that, even for U H\DA, Û is well defined; this would not be the caseif U andu 2 were Crank Nicolson approximationsbecause then U n / DA for any n. It immediately follows from 5.5 that 5.8 ê t+aet =, < t < T; compare with 4. for f =. Next, we will briefly discuss some of the estimators we obtain by applying the energy method or Duhamel s principle to the above error equation.

19 A POSTERIORI ESTIMATES FOR THE CRANK NICOLSON METHOD Estimate I: Energy method. Taking in 5.8 the inner product with êt, and recalling that Aet,êt = et 2 + êt 2 êt et 2, 2 we obtain 5.9 êt 2 + t ês 2 + es 2 ds cf Now, in view of 5.7i, therefore 5. Furthermore, 5. t 2 t Ûs Us 2 ds, t T, Ût Ut 2 = t n t 2 A 3/2 U n 2, n =,2; Ûs Us 2 ds = 3 t m = 3 t 2 Ûs Us 2 ds β 2 k 3 A 3/2 U 2 +k 32 A 3/2 U 2 2 k 3 A /2 U 2 +k 32 A /2 U 2 2. m n=2 k 5 n A 3/2 U n 2, with β as in 4.8. We thus deduce the upper a posteriori estimate t êt 2 + ês 2 + es 2 ds k 3 A /2 U 2 +k 32 A /2 U β 2 m n=2 k 5 n A3/2 U n 2 ; compare with 4.2 for f =. Proceeding as in subsection 4..2 we also get a sharp lower bound. Note that the above estimate holds provided that U DA /2. Further reasonable error control based on this estimate requires us to balance the terms k A /2 U, k 2 A /2 U 2 and k 2 n A3/2 U n Estimate II: Duhamel principle. We next modify arguments of Sections 4.2 to obtain the final result êt n { 8 C A 2kn A U 2 n 5.3 +ln tn max k U,k 2 U 2, max k 2 k j A U j }, n 2 j n which could be compared with 4.37 for f =. Note that the above estimate holds, provided thatu H. Further reasonableerror control based onthis estimate requires us to balance the terms k U,k 2 U 2 and k 2 n A U n.

20 2 G. AKRIVIS, CH. MAKRIDAKIS, AND R. H. NOCHETTO 6. Error estimates for nonlinear equations In this section we consider the discretization of.. We assume that Bt, can be extended to an operator from V into V. A natural condition for. to be locally of parabolic type is the following local one-sided Lipschitz condition: 6. Bt,v Bt,w,v w λ v w 2 +µ v w 2 v,w T u in a tube T u,t u := {v V : min t ut v }, around the solution u, uniformly in t, with a constant λ less than one and a constant µ. With Ft,v := Av Bt,v, it is easily seen that 6. can be written in the form of a Gårding type inequality, 6.2 Ft,v Ft,w,v w λ v w 2 µ v w 2 v,w T u. Furthermore, in order to ensure that an appropriate residual is of the correct order, we will make use of the following local Lipschitz condition for Bt, : 6.3 Bt,v Bt,w L v w v,w T u with a constant L, not necessarily less than one. Here the tube T u is defined in terms of the norm of V for concreteness. The analysis may be modified to yield a posteriori error estimates under analogous conditions for v and w belonging to tubes defined in terms of other norms, not necessarily the same for both arguments. In the sequel the estimates are proved under the assumption that Ût,Ut T u, for all t [,T]. This assumption can, in some cases, be verified a posteriori under conditional assumptions on Û and U. Thus the final result will hold, pending on a condition that Û and U may or may not satisfy. However the validity of this condition can be computationally verified. The derivation of these bounds requires the use of fine properties of the specific underlying p.d.e., as was done in [6, 8], and therefore goes beyond the scope of the present paper. We refer to [3] for existence and local uniqueness results for the continuous Galerkin approximations, in particular for the Crank Nicolson Galerkin approximations, as well as for a priori error estimates. Concrete examples of parabolic equations satisfying 6. and 6.3 in appropriate tubes are given in [] and [2]. 6.. The Crank Nicolson Galerkin method. We recall that this method for. consists of seeking a function U : [,T] V, continuous and piecewise linear, such that U = u and 6.4 U n +AU n 2 = Bs,Usds, where U n := Ut n,u n 2 := 2 U n + U n = Ut n 2. The Crank Nicolson Galerkin approximate solution U can be expressed in terms of its nodal values U n and U n, 6.5 Ut = U n 2 +t t n 2 U n, t.

21 A POSTERIORI ESTIMATES FOR THE CRANK NICOLSON METHOD 2 In view of 2.22, we let the Crank Nicolson Galerkin reconstruction Û of U be 6.6 Ût := U n A i.e., t t n Usds+ t 6.7 Ût = U n 2 t tn A[Ut+U n ]+ It immediately follows from 6.6 that t n P Bs,Usds, t 6.8 Û t+aut = P Bt,Ut. Using 6.4, it easily follows from 6.5 and 6.7 that 6.9 Ut Ût = 2 t tn t n ta U n + t tn Bs,Usds t n P Bs,Usds, t. t t n P Bs,Usds; therefore, again using 6.4, we have Ut Ût = 2 t tn t n t A 2 U n 2 ABs,Usds 6. t + t tn Bs,Usds P Bs,Usds, t n t. Hence, in view of 3.3, Ut Ût = 2 t tn t n t A 2 U n 2 ABs,Usds k 6. n I n + 6 knt t n t n t Bs,Uss t n 3 2 ds, t, from which we immediately see that max t In Ut Ût = Ok2 n The Crank Nicolson method. The Crank Nicolson approximations U m DA to the nodal values u m := ut m of the solution u of. are defined by 6. U n +AU n 2 = Bt n 2,U n 2, n =,...,N, with U := u. As before, we define the Crank Nicolson approximation U to u by linearly interpolating between the nodal values U n and U n, 6.2 Ut = U n 2 +t t n 2 U n, t.

22 22 G. AKRIVIS, CH. MAKRIDAKIS, AND R. H. NOCHETTO Let b : H be the linear interpolant of B,U at the nodes t n and t n 2, 6.3 bt = Bt n 2,U n t t n 2 [Bt n 2,U n 2 Bt n,u n ], t. Inspired by 2.9, we define the Crank Nicolson reconstruction 6.4 Ût := U n A i.e., t t Û of U by Usds+ bsds, t, t n t n 6.5 Ût =U n 2 t tn A[Ut+U n ]+t t n Bt n 2,U n 2 + t t n t n t[bt n 2,U n 2 Bt n,u n ], t. Note that 6.5 reduces to 2. in the case B is independent of u. It immediately follows from 6.4 that 6.6 Û t+aut = bt, t. Furthermore, it is easily seen that, for t, 6.7 Ut Ût = { 2 t tn t n t A U n 2 [ Bt n 2,U n 2 Bt n,u n ]} Error estimates. We now derive a posteriori error estimates for both the Crank Nicolson Galerkin and the Crank Nicolson method. Let e := u U and ê := u Û. The following estimates hold under the assumption that Ût,Ut T u for all t [,T]. Crank Nicolson Galerkin method. Subtracting6.8 from the differential equation in., we obtain 6.8 ê t+aet = Bt,ut P Bt,Ut, which we rewrite in the form 6.9 ê t+aet = Bt,ut Bt,Ut+R U t, with 6.2 R U t = Bt,Ut P Bt,Ut. Now, Bt,ut Bt,Ut,êt = Bt,ut Bt,Ut,et + Bt,ut Bt,Ut,Ut Ût

23 A POSTERIORI ESTIMATES FOR THE CRANK NICOLSON METHOD 23 and, in view of 6. and 6.3, elementary calculations yield 6.2 Bt,ut Bt,Ut,êt λ êt 2 +µ et 2 +L et Û Ut. Similarly, Bt,ut Bt,Ut,êt = Bt,ut Bt, Ût,êt and + Bt,Ût Bt,Ut,êt 6.22 Bt,ut Bt,Ut,êt λ êt 2 +µ êt 2 +L êt Û Ut. Summing 6.2 and 6.22, we obtain 6.23 Now, and 2 Bt,ut Bt,Ut,êt λ êt 2 + et 2 +µ êt 2 + et 2 +L êt + et Û Ut. et 2 êt + Û Ut 2 2 êt 2 +2 Û Ut 2 L êt + et Û Ut ε 2 êt + et + L2 Û Ut 2 2 2ε ε êt 2 + et 2 + L2 2ε Û Ut 2. Consequently, from 6.23, we obtain 2 Bt,ut Bt,Ut,êt λ êt 2 + et 2 +3µ et µ Û Ut 2 +ε êt 2 + et 2 + L2 Û Ut 2 2ε for any positive ε. Taking in 6.9 the inner product with êt we obtain 6.25 d dt êt 2 + et 2 + êt 2 = Ût Ut 2 +2 Bt,ut Bt,Ut,êt +2 R U t,êt, whence, in view of 6.24, 6.26 d dt êt 2 + λ 2ε êt 2 + et 2 3µ êt L2 2ε Û Ut 2 +2µ Û Ut 2 + ε R Ut 2.

24 24 G. AKRIVIS, CH. MAKRIDAKIS, AND R. H. NOCHETTO We thus easily obtain the desired a posteriori error estimate via Gronwall s lemma t êt 2 + λ 2ε e 3µt s ês 2 + es ds t [ + e 3µt s L 2 Û Us 2 2ε +2µ Û Us 2 + ] ε R Us 2 ds. Crank Nicolson method. Subtracting 6.6 from the differential equation in., we obtain ê t+aet = Bt,ut bt. Therefore, 6.27 is valid for the Crank Nicolson method as well, this time with R U t := Bt,Ut bt. References. G. Akrivis, M. Crouzeix, and Ch. Makridakis, Implicit explicit multistep finite element methods for nonlinear parabolic problems, Math. Comp MR g: G. Akrivis, M. Crouzeix, and Ch. Makridakis, Implicit explicit multistep methods for quasilinear parabolic equations, Numer. Math MR e: G. Akrivis and Ch. Makridakis, Galerkin time stepping methods for nonlinear parabolic equations, M 2 AN Math. Mod. Numer. Anal MR f: A. K. Aziz and P. Monk, Continuous finite elements in space and time for the heat equation, Math. Comp MR d: M. Crouzeix, Parabolic Evolution Problems. Unpublished manuscript, W. Dörfler, A time- and spaceadaptive algorithm for the linear time dependent Schrödinger equation, Numer. Math MR g: K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem, SIAM J. Numer. Anal MR m: K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. IV. Nonlinear problems, SIAM J. Numer. Anal MR i: K. Eriksson, C. Johnson, and S. Larsson, Adaptive finite element methods for parabolic problems. VI. Analytic semigroups, SIAM J. Numer. Anal MR d:6528. D. Estep and D. French, Global error control for the continuous Galerkin finite element method for ordinary differential equations, RAIRO Math. Mod. Numer. Anal MR k:6579. C. Johnson, Error estimates and adaptive time step control for a class of one step methods for stiff ordinary differential equations, SIAM J. Numer. Anal MR a: C. Johnson, Y. Y. Nie, and V. Thomée, An a posteriori error estimate and adaptive timestep control for a backward Euler discretization of a parabolic problem, SIAM J. Numer. Anal MR g: C. Johnson and A. Szepessy, Adaptive finite element methods for conservation laws based on a posteriori error estimates, Comm. Pure Appl. Math MR b:7684

25 A POSTERIORI ESTIMATES FOR THE CRANK NICOLSON METHOD O. Karakashian and Ch. Makridakis, A space time finite element method for the nonlinear Schrödinger equation: the continuous Galerkin method, SIAM J. Numer. Anal MR h: O. Karakashian and Ch. Makridakis, Convergence of a continuous Galerkin method with mesh modification for nonlinear wave equations, Math. Comp MR g: O. Lakkis and R. H. Nochetto, A posteriori error analysis for the mean curvature flow of graphs, SIAM J. Numer. Anal MR Ch. Makridakis and R. H. Nochetto, Elliptic reconstruction and a posteriori error estimates for parabolic problems, SIAM J. Numer. Anal MR k: Ch. Makridakis and R. H. Nochetto, A posteriori error analysis for higher order dissipative methods for evolution problems. Submitted for publication. 9. 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 MR k: R. H. Nochetto, A. Schmidt, and C. Verdi, A posteriori error estimation and adaptivity for degenerate parabolic problems, Math. Comp MR i: V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, Berlin, 997. MR m: R. Verfürth, A posteriori error estimates for finite element discretizations of the heat equation, Calcolo MR f:653 Computer Science Department, University of Ioannina, 45 Ioannina, Greece address: akrivis@ cs.uoi.gr Department of Applied Mathematics, University of Crete, 749 Heraklion-Crete, Greece and Institute of Applied and Computational Mathematics, FORTH, 7 Heraklion-Crete, Greece URL: address: makr@ math.uoc.gr, makr@ tem.uoc.gr Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MARYLAND 2742 URL: address: rhn@ math.umd.edu