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

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1 MATHEMATICS OF COMPUTATION Volue 75, Nuber 54, Pages S Article electronically published on Noveber 3, 5 A POSTERIORI ERROR ESTIMATES FOR THE CRANK NICOLSON METHOD FOR PARABOLIC EQUATIONS GEORGIOS AKRIVIS, CHARALAMBOS MAKRIDAKIS, AND RICARDO H. NOCHETTO Abstract. We derive optial order a posteriori error estiates for tie discretizations by both the Crank Nicolson and the Crank Nicolson Galerkin ethods for linear and nonlinear parabolic equations. We exaine both sooth and rough initial data. Our basic tool for deriving a posteriori estiates are second-order Crank Nicolson reconstructions of the piecewise linear approxiate solutions. These functions satisfy two fundaental properties: i they are explicitly coputable and thus their difference to the nuerical solution is controlled a posteriori, and ii they lead to optial order residuals as well as to appropriate pointwise representations of the error equation of the sae for as the underlying evolution equation. The resulting estiators are shown to be of optial order by deriving upper and lower bounds for the depending only on the discretization paraeters and the data of our proble. As a consequence we provide alternative proofs for known a priori rates of convergence for the Crank Nicolson ethod. 1. Introduction In this paper we derive a posteriori error estiates for tie discretizations by Crank Nicolson type ethods for parabolic partial differential equations p.d.e. s. The Crank Nicolson schee is one of the ost popular tie-stepping ethods; however, optial order a posteriori estiates for it have not yet been derived. Most of the any contributions in the last years devoted to a posteriori error control for tie dependent equations concern the discretization in tie of linear or nonlinear equations with dissipative character by the backward Euler ethod or by higher order discontinuous Galerkin ethods; cf., e.g., [1], [7], [8], [], [9], [19] and [13]. Let u and U be the exact and the nuerical solution of a given proble. In a posteriori error analysis u U ηu, Received by the editor June 1, 4 and, in revised for, February 3, 5. Matheatics Subject Classification. Priary 65M15, 65M5. Key words and phrases. Parabolic equations, Crank Nicolson ethod, Crank Nicolson Galerkin ethod, 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 Coission. The second author was partially supported by the European Union RTN-network HYKE, HPRN-CT--8, the EU Marie Curie Developent Host Site, HPMD-CT-1-11 and the progra Pythagoras of EPEAEK II. The third author was partially supported by NSF Grants DMS and DMS c 5 Aerican Matheatical Society

2 51 G. AKRIVIS, CH. MAKRIDAKIS, AND R. H. NOCHETTO we seek coputable estiators ηu depending on the approxiate solution U and the data of the proble such that i ηu decreases with optial order for the lowest possible regularity peritted by our proble, and ii the constants involved in the estiator ηu are explicit and easily coputable. In this paper we derive optial order estiators of various types for the Crank Nicolson and the Crank Nicolson Galerkin tie-stepping ethods applied to evolution probles of the for: Find u :[,T] DA satisfying 1.1 { u t+aut =Bt, ut, <t<t, u = u, with A : DA H a positive definite, selfadjoint, linear operator on a Hilbert space H,, with doain DA denseinh, Bt, :DA H, t [,T], a possibly nonlinear operator, and u H. The structural assuption 6.1 on Bt, iplies that proble 1.1 is parabolic. A ain novel feature of our approach is the Crank Nicolson reconstruction Û of the nuerical approxiation U. This function satisfies two fundaental properties: i it is explicitly coputable and thus its difference to the nuerical solution is controlled a posteriori, and ii it leads to an appropriate pointwise representation of the error equation, of the sae for as the original evolution equation. Then by eploying techniques developed for the underlying p.d.e. we conclude the final estiates. Of course, depending on the stability ethods that are used, we obtain different estiators. The resulting estiators are shown to be of optial order by deriving upper bounds for the, depending only on the discretization paraeters and the data of our proble. As a consequence we provide alternative proofs for aprioriestiates, depending only on the data and corresponding known rates of convergence for the Crank Nicolson ethod. The above idea is related to earlier work on a posteriori analysis of tie or space discrete approxiations of evolution probles [19, 18, 17]. It provides the eans to show optial order error estiates with energy as well as with other stability techniques. An alternative approach for a posteriori analysis of tie dependent probles, based on the direct coparison of u and U via parabolic duality, was considered in [1], [7], [], [9] for p.d.e. s and in [11], [1] for ordinary differential equations o.d.e. s. In particular Estep and French [1] considered the continuous Galerkin ethod for o.d.e s. Its lowest order representative corresponds to a variant of the Crank Nicolson ethod the Crank Nicolson Galerkin ethod considered also in this paper. A posteriori bounds with energy techniques for Crank Nicolson ethods for the linear Schrödinger equation were proved by Dörfler [6] and for the heat equation by Verfürth []; the upper bounds in [6], [] are of suboptial order. Most of this paper is devoted to linear parabolic equations, naely Bt, ut = ft for a given forcing function f. The general nonlinear proble 1.1 is only briefly discussed in the last section. The paper is organized as follows. We start in Section by introducing the necessary notation, the Crank Nicolson and the Crank Nicolson Galerkin CNG ethods for the linear proble.1. We then observe that the direct use of standard piecewise linear interpolation at the approxiate nodal values see.3, would lead to suboptial estiates as in [6] and []. The Crank Nicolson and Crank Nicolson Galerkin reconstructions Û are, instead,

3 A POSTERIORI ESTIMATES FOR THE CRANK NICOLSON METHOD 513 continuous piecewise quadratic functions which are defined in.9 and., respectively. In Section 3 we estiate Û U. Section 4 is devoted to the a posteriori error analysis for linear equations. Error estiates are obtained by using energy techniques, as well as Duhael s principle. Both estiators lead to second order convergence rates. Note the interesting siilarity of the estiator obtained by Duhael s principle to those established in the literature by parabolic duality. In Section 5 we discuss the for of estiators in the case of nonsooth initial data. In Section 6 we finally conclude with the case of nonlinear equations.. Crank Nicolson ethods for linear equations Most of this paper focuses on the case of a linear equation, { u t+aut =ft, <t<t,.1 u = u, with f :[,T] H. Let = t <t 1 < <t N = T be a partition of [,T], := t n 1,t n ], and := t n t n The Crank Nicolson ethod. For given {v n } N n= we will use the notation v n := vn v n 1, v n 1 1 := vn + v n 1, n =1,...,N. The Crank Nicolson nodal approxiations U DA tothevaluesu := ut of the solution u of.1 are defined by. U n + AU n 1 = ft n 1, n =1,...,N, with U := u. Since the error u U is of second order, to obtain a second order approxiation Ut tout, for all t [,T], we define the Crank Nicolson approxiation U : [,T] DA tou by linearly interpolating between the nodal values U n 1 and U n,.3 Ut =U n 1 +t t n 1 U n, t. Let Rt H,.4 Rt :=U t+aut ft, t, denote the residual of U, i.e., the aount by which the approxiate solution U isses being an exact solution of.1. Now U t+aut = U n + AU n 1 +t t n 1 A U n, t, whence, in view of., U t+aut =ft n 1 +t t n 1 A U n, t. Therefore, the residual can also be written in the for.5 Rt =t t n 1 A U n +[ft n 1 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 schee yields second-order accuracy. Since the error e := u U satisfies e + Ae = R, applying energy techniques to this error equation, as in [6], [], leads inevitably to suboptial bounds.

4 514 G. AKRIVIS, CH. MAKRIDAKIS, AND R. H. NOCHETTO.. Crank Nicolson reconstruction. To recover the optial order we introduce a Crank Nicolson reconstruction Û of U, naely a continuous piecewise quadratic polynoial in tie Û :[,T] H defined as follows. First, we let ϕ : H be the linear interpolant of f at the nodes t n 1 and t n 1,.6 ϕt :=ft n 1 + t t n 1 [ft n 1 ft n 1 ], t, and define a piecewise quadratic polynoial Φ by Φt := ϕsds, t I t n 1 n, i.e.,.7 Φt =t t n 1 ft n 1 1 t t n 1 t n t[ft n 1 ft n 1 ]. As will becoe evident in the sequel, an iportant property of Φ is that.8 Φt n 1 =, Φt n = ft n 1 = ft n 1 dt. We now introduce the Crank Nicolson reconstruction Û of U by.9 Ût :=U n 1 AUs ds +Φt t. t n 1 We can interpret this forula as the result of forally replacing the constants U n 1 and ft n 1 in. by their piecewise linear counterparts U and ϕ, and next integrating AU + ϕ fro t n 1 to t. Consequently Û t+aut =ϕt t. Evaluating the integral in.9 by the trapezoidal rule, we obtain.1 Ût =U n 1 1 t tn 1 A[Ut+U n 1 ]+Φt t, which can also be written as Ût =U n 1 A[t t n 1 U n t tn 1 U n ]+Φt t. Obviously Ûtn 1 =U n 1. Furtherore, in view of.8 and., we have Ût n =U n 1 AU n 1 +Φt n = U n 1 + [ AU n 1 + ft n 1 ] = U n 1 + U n = U n. Thus, Û and U coincide at the nodes t,...,t N ;inparticular,û :[,T] H is continuous. Reark.1 Choice of ϕ. Let t. Since ft =f t +f tt t +Okn, t, it is easily seen that the only affine functions ϕ satisfying sup ft ϕt = Okn t are the ones of the for ϕt =f t+ [ f t+o ] t t+o. Obviously ϕtdt = f t+ 1 [ f t+o ]{ t n t t n 1 t } + Ok 3 n;

5 A POSTERIORI ESTIMATES FOR THE CRANK NICOLSON METHOD 515 therefore, for f t, the requireent ϕtdt = f t leadsto t = t n 1 and ϕt =ft n 1 + [ f t n 1 +Okn ] t t n 1. These coents deonstrate the special features of the idpoint ethod aong the one-stage Runge Kutta ethods. Furtherore, our choice.6 is otivated by the fact that for all affine functions ϕ on we have ϕsds = ϕt n 1, whence the requireent ϕsds = ft n 1 see.8, is satisfied if and only if ϕ interpolates f at t n 1. Now, to ensure that ϕt is a second order approxiation to ft, we let ϕ interpolate f at an additional point t n, [t n 1,t n ]; of course, in the case t n, = t n 1,ϕis the affine Taylor polynoial of f around t n 1. In the sequel we let t n, := t n 1. In view of.9 and.6, we have.11 Û t+aut =ft n 1 + t t n 1 [ft n 1 ft n 1 ], t ; therefore, the residual ˆRt of Û,.1 ˆRt := Û t+aût ft, t, canbewritteninthefor ˆRt =A[Ût Ut].13 { + ft n 1 } + t t n 1 [ft n 1 ft n 1 ] ft, t. We will see later that the a posteriori quantity ˆRt is of second order; copare with The Crank Nicolson Galerkin ethod. Next we consider the discretization of.1 by the Crank Nicolson Galerkin ethod. The Crank Nicolson Galerkin approxiation to u is defined as follows: We seek U :[,T] DA, continuous and piecewise linear in tie, which interpolates the values {U n } N n= given by.14 U n + AU n 1 1 = ftdt, n =1,...,N, with U = u. This function U can be expressed in ters of its nodal values,.15 Ut =U n 1 +t t n 1 U n, t. For t,u t = U n, and.14 takes the for.16 U t+au n 1 1 = ftdt, n =1,...,N. 1 Now, ψtdt is the L orthogonal projection of a function ψ onto the space of constant functions on, and Utdt = U n 1 ; therefore.16 yields the pointwise equation for the Crank Nicolson Galerkin approxiation.17 U t+p AUt =P ft t,

6 516 G. AKRIVIS, CH. MAKRIDAKIS, AND R. H. NOCHETTO with P denoting the L orthogonal projection operator onto the space of constant functions in. Equivalently, as it is custoary, this ethod can be seen as a finite eleent in tie ethod, [4], [.18 U,v+AU, v ] dt = f,vdt v DA. For a priori results for general Continuous Galerkin ethods for various evolution p.d.e. s cf. [3, 4, 14, 15]. Let Rt,.19 Rt :=U t+aut ft, denote the residual of U. In view of.17, the residual can also be written in the for. 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 approxiations are piecewise polynoials of degree one..4. Crank Nicolson Galerkin reconstruction. To recover the optial order Okn, we introduce the Crank Nicolson Galerkin reconstruction Û of the approxiate solution U, naely, the continuous and piecewise quadratic function Û :[,T] H defined by.1 Ût :=U n 1 [AUs P 1 fs] ds t n 1 t. Hence. Ût =U n 1 1 t tn 1 A[Ut+U n 1 ]+ P 1 fsds, t, t n 1 with P 1 denoting the L orthogonal projection operator onto the space of linear polynoials in ;thatût is continuous, naely, Ûtn =U n, is a consequence of P 1 f = f. Obviously, Û satisfies the following pointwise equation:.3 Û t+aut =P 1 ft t ; copare with.17. In view of.3, the residual ˆRt,.4 ˆRt := Û t+aût ft, of Û can also be written as.5 ˆRt =A[ Ût Ut] + [P 1 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 soe cases.

7 A POSTERIORI ESTIMATES FOR THE CRANK NICOLSON METHOD Estiation of Û U In this section we will estiate Û U for both the Crank Nicolson and the Crank Nicolson Galerkin ethods; also we will derive representations of Û U that will be useful in the sequel. We let V := DA 1/ and denote the nors in H and in V by and, v := A 1/ v =Av, v 1/, 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 nor on V, v := A 1/ v =v, A 1 v 1/ The Crank Nicolson ethod. Fro.1 we obtain Ût Ut =U n 1 Ut 1 t tn 1 A[Ut+U n 1 ]+Φt Therefore, in view of., = t t n 1 U n 1 t tn 1 A[Ut+U n 1 ]+Φt. Ût Ut =t t n 1 [ AU n ] 1 ft n 1 1 t tn 1 A[Ut+U n 1 ]+Φt = 1 t tn 1 A[Ut U n ]+Φt t t n 1 ft n 1, whence, using.7, for t, Ût Ut =t t n 1 t n t A U n 1 [ft n 1 ft n 1 ], fro which we iediately see that ax Ût Ut = Ok t In n. 3.. The Crank Nicolson Galerkin ethod. Subtracting.15 fro., and utilizing.14, for t we obtain Ût Ut = 1 t tn 1 t n ta U n 3. t tn 1 Now, it is easily seen that 3.3 P 1 ft = 1 fsds + 1 kn 3 and 3. can be rewritten in the for fsds + t t n Ût Ut =t t n 1 t n t A U n 6 t n 1 P 1 fsds. fss t n 1 ds, k 3 n fss t n 1 ds. Therefore, U and Û coincide at the endpoints of, and, consequently, at all nodes t,...,t N. Fro 3.4 we iediately see that ax t In Ût Ut = Ok n. Letuswriteboth3.1and3.4inthefor 3.5 Ût Ut = 1 t tn 1 t n t A U n ρ f,n ; here ρ f,n = ρ CN f,n for the Crank Nicolson ethod and ρ f,n = ρ CNG f,n for the Crank Nicolson Galerkin ethod, respectively, with 3.6 ρ CN f,n := [ ft n 1 ft n 1 ]

8 518 G. AKRIVIS, CH. MAKRIDAKIS, AND R. H. NOCHETTO and 3.7 ρ CNG f,n := 1 kn 3 fss t n ds = fs ft n kn 3 s t n 1 ds. Consequently, both ρ CN f,n and ρcng f,n depend on the first derivative of f. 4. Sooth data error estiates Let the errors e and ê be defined by e := u U and ê := u Û. Subtracting.11 or.3, respectively, fro the differential equation in.1, we obtain 4.1 ê t+aet =R f t, with R f = Rf CN for the Crank Nicolson ethod and R f = Rf CNG for the Crank Nicolson Galerkin ethod, respectively, defined by 4. Rf CN t :=ft { ft n 1 + t t n 1 [ft n 1 ft n 1 ] }, t, and 4.3 Rf CNG t :=ft P 1 ft, t. We ake the following further regularity assuption on Û, defined in.9 and.1: Ût V, t [,T] Energy estiates. Taking in 4.1 the inner product with êt, we obtain 4.4 Now, and therefore, 4.4 yields 1 d dt êt + Aet, êt = R f t, êt. 1 Aet, êt = et + êt êt et Rf t, êt R f t êt ; d 4.5 dt êt + et + 1 êt Ût Ut + R f t. We recall that v = A 1/ v and v = A 1/ v Upper bound. Since ê =, integration of 4.5 fro to t T yields êt + es + 1 ês ds 4.6 Ûs Us ds + R f s ds. Fro 4.6 we easily conclude that { τ êτ ax τ t es + 1 ês ds Ûs Us ds + } R f s ds.

9 A POSTERIORI ESTIMATES FOR THE CRANK NICOLSON METHOD 519 Next, let β be given by 4.8 β := then, obviously, 1 t 1 t dt = 1 3 ; t t n 1 t n t dt = βkn 5. With this notation, in view of 3.5, we have 4.9 Ût Ut dt β kn 5 A 3/ U n + ρ f,n Lower bound. Obviously, Ûs Us es + ês and thus 4.1 Ûs Us 3 es + 1 ês. In particular, cobining the upper and lower bounds, we have Ûs Us ds es + 1 ês ds Ûs Us ds + R f s ds. Invoking 4.6, this shows that êt is doinated by es + 1 ês ds, the energy nor error, plus data oscillation R f s ds. We will next estiate the lower bounds above in ters 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 1 4 t tn 1 t n t 1 A3/ U n ρ f,n, whence, in view of 4.8, we have 4.1 Ûs Us ds β kn 5 A 3/ U n β kn 5 ρ f,n. 8 4 Therefore, 4.11, 4.9 and 4.1 iply β kn 5 A 3/ U n β kn 5 ρ f,n es + 1 ês ds β kn 5 A 3/ U n + ρ f,n + R f s ds. Note that error bounds of this type are custoary in the a posteriori analysis of elliptic probles in which data oscillation appear with different signs in the upper

10 5 G. AKRIVIS, CH. MAKRIDAKIS, AND R. H. NOCHETTO and lower bounds. On the other hand, if f is constant, then the lower and upper bounds are exactly the sae up to a constant: β kn 5 A 3/ U n es ês ds 4.14 β kn 5 A 3/ U n. Let us also note that, in the case of f constant, in view of 3.5, the estiate 4.11, for t = t, can be written in the for of 4.14 with the lower bound ultiplied by and the upper bound by 1/. Reark 4.1 Optiality of the lower bound. The pointwise lower bound 4.15 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 fro 4.15 the following lower bound in the L H-nor, with v L H := ax t [,T ] vt : 4.16 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 approxiation U and the Crank Nicolson reconstruction Û coincide with u; thus 4.16 is an equality. Next consider the case of u nonaffine. In view of 3.1, we have Û Utn 1 k = n 1 4 A U n 1 [ ft n 1 ft n 1 ]. Now, for sooth data, we have, as, 1 A U n 1 [ ft n 1 ft n 1 ] 1 [ Au t n 1 f t n 1 ] = 1 u t n 1. If u t n 1, we then have Û U L H; ckn with a positive constant c. This is the generic situation. That the lower bound in the L V -nor is of the sae for as the upper bound, in the case of f constant, can be seen fro Otherwise, let us note that, in view of 3.1 and 4.8, N Û U L V = β kn 5 1 A U n 1 [ ft n 1 ft n 1 ] k n and, assuing that the partition is quasi-unifor and letting k := ax n, Û U L V N 1 cβk4 A U n 1 [ ft n 1 ft n 1 ]. k n Now, as k, N 1 A U n 1 [ ft n 1 ft n 1 ] 1 k n u, L V whence, if u is not affine, 4.17 Û U L V Ck.

11 A POSTERIORI ESTIMATES FOR THE CRANK NICOLSON METHOD 51 Reark 4. Alternative estiate in L H. Cobining 4.1 and 4.6 we can replace the factor 1 on the right-hand side of 4.7 by 3, if we only want to estiate the first ter on the left-hand side of 4.7. Indeed, in view of 4.1, fro 4.6 we obtain êt Ûs Us ds êt + es + 1 ês ds i.e., and thus êt 3 Ûs Us ds + Ûs Us ds + R f s ds, R f s ds, 4.18 ax τ t êτ 3 Ûs Us ds + R f s ds. The following stability estiate for the Crank Nicolson schee will be useful in the convergence proof: Lea 4.1 Stability. Let {U n } N n= be the Crank Nicolson approxiations for.1, 4.19 U n + AU n 1 = f n, where either f n = ft n 1 or f n = 1 fsds. Then the following estiate holds for N: 4. A 3/ U n + A U A U + A 3/ f n. Proof. We apply A to the schee A U n + A U n 1 = A f n, and take the inner product with A U n U n 1 toobtain A 3/ U n + A U n A U n 1 = A f n,a U n. Suing here fro n =1to, we get A 3/ U n + A U = A U + A f n,a U n, whence the Cauchy Schwarz and the arithetic-geoetric ean inequalities yield A 3/ U n + A U A U and the proof is coplete. + A 3/ f n + A 3/ U n,

12 5 G. AKRIVIS, CH. MAKRIDAKIS, AND R. H. NOCHETTO Fro 4.6, 4.9, 4. and 3.5 we conclude the following Theore. We ephasize that the optial order apriorierror estiate 4.3, depending only on the data see below, follows fro our a posteriori estiate 4.6. This shows, in particular, that the a posteriori estiate is of optial second order. Theore 4.1 Error estiates. Let {U n } N n= be either the Crank Nicolson approxiations or the Crank Nicolson Galerkin approxiations to the solution of proble.1, e = u U and ê = u Û. The following a posteriori estiate is valid for N: 4.1 êt + es + 1 ês ds β kn 5 A 3/ U n + E[ f ], with β given by 4.8 and 4. E[ f ]:= R f s ds + β k n ρ 5 f,n. Here R f and ρ f,n are given by 4. and 3.6, respectively, for the Crank Nicolson ethod, and by 4.3 and 3.7 for the Crank Nicolson Galerkin ethod. Furtherore, if U DA and ft DA 3/, then the following a priori estiate holds for N: êt + es + 1 ês ds 4.3 β ax n k 4 n A U + A 3/ f n + E[ f ], with f n := ft n 1 for the Crank Nicolson ethod, and f n := 1 fsds for the Crank Nicolson Galerkin ethod. Reark 4.3 Equivalent upper bound for CNG. In the case of the Crank Nicolson Galerkin ethod, it is easily seen that 4.3 yields êt + es + 1 ês ds 4.4 β ax k 4 n n A U + A 3/ fs ds + E[ f ]. Reark 4.4 Estiate for E[ f ]. As a by-product of interpolation theory, we realize that the following optial order estiate is valid for the error E[ f ]intheforcing ter f: E[ f ] C kn f s + f s ds. 4.. Estiates via Duhael s principle. We first rewrite the relation 4.1 in the for 4.5 ê t+aêt =RÛt+R f t with 4.6 RÛt :=A[Ut Ût].

13 A POSTERIORI ESTIMATES FOR THE CRANK NICOLSON METHOD 53 We will use Duhael s principle in 4.5. Let E A t be the solution operator of the hoogeneous equation 4.7 v t+avt =, v = w, i.e., vt =E A tw. It is well known that the faily of operators E A t hasseveral nice properties, in particular, it is a seigroup of contractions on H generated by the operator A. Duhael s principle applied to 4.5 yields 4.8 êt = E A t s [ RÛ s+r f s ] ds. In the sequel we will use the soothing property cf., e.g., Crouzeix [5], Thoée [1] 4.9 A l 1 E A tw C A t l A w, l. In addition, note that A and E A coute, i.e., AE A tw = E A t Aw. In particular, 4.9 can also be written in the for 4.3 E A t A l 1 w C A t l A w, l, whence E A tw C A t A w. Next, using 4.8 for t = t n, we have êt n EA t n s [ RÛs+R f s ] ds and thus 4.31 n 1 + C A êt n C A 1 n 1 + C A EA t n s [ RÛs+R f s ] ds A 1 t n RÛs ds + C A R f s s I ds n 1 1 A 1[ t n RÛ s+r f s ] ds s A 1 t n RÛs ds + CA Rf s s I ds n + C A sup s [,t n 1 ] A 1[ RÛs+R f s ] We now recall 4.6 and 3.5, naely, n 1 1 t n s ds. 4.3 A 1 RÛ s =Us Ûs =1 t tn 1 t n t A U n ρ f,n, to obtain a bound for the first two ters on the right-hand side of 4.31, A 1 t n RÛs ds + Rf s s I k ds n A U n + E1 [ ; f], n 4 with 4.34 E 1 [ ; f] := k n ρ f,n + ax R f s. 4 s

14 54 G. AKRIVIS, CH. MAKRIDAKIS, AND R. H. NOCHETTO In addition, again using 4.3, we have 4.35 with sup A 1 [ RÛ s+r f s ] s [,t n 1 ] 1 8 ln tn n 1 1 t n s ds ax k j A U j +ln tn E [[,t n 1 ];f] 1 j n E [[,t n 1 ];f] := 1 8 We have thus proved ax k j ρ f,j + ax sup A 1 R f s. 1 j n 1 1 j n 1 s I j Theore 4. Error estiates. Let {U n } N n= be either the Crank Nicolson or the Crank Nicolson Galerkin approxiations to the solution of proble.1. Then, with the notation of Theore 4.1, the following a posteriori estiate is valid for n N: êt n 1 8 C A A U n +ln tn ax k 4.37 n j n 1 k j A U j + C A E 1 [ ; f]+ln tn E [[,t n 1 ];f], with C A the constant of 4.9 for l =1,=, and the ters involving f are defined in 4.34 and Furtherore, if U DA,ft DA 3/, and k := ax j n k j +ln t n 1/, the following a priori estiate holds for n N: 4.38 êt n 1 8 C A k { A U + n j=1 k j A 3/ f j 1/ + ax A f } j 1 j n + C A E 1 [ ; f]+ln tn E [[,t n 1 ];f]. Proof. It only reains to prove It iediately follows fro 4.19 that A U n A U n 1 + A f n ax A U n+1, A U n + A f n, and 4.38 thus results fro 4.37 in light of 4.. Reark 4.5 Alternative bound for CNG. In the case of the Crank Nicolson Galerkin ethod, it is easily seen that 4.38 yields 4.39 n êt n 1 8 C Ak { 1/ } A U + A 3/ fs ds + ax Afs s t n + C A E 1 [ ; f]+ln tn E [[,t n ];f]. 5. Estiates for initial data with reduced soothness In this section our objective is the derivation of a posteriori error estiates in the case of initial data with reduced soothness. Since the initial value proble.1 can be split into one with hoogeneous initial data and one with hoogeneous equation, and we are ainly concerned with

15 A POSTERIORI ESTIMATES FOR THE CRANK NICOLSON METHOD 55 the effect of the initial data, we focus on the hoogeneous initial value proble { u t+aut =, <t<t, 5.1 u = u. A typical a priori nonsooth data estiate reads see [1], s k 5. ut n U n C t n U, s being the order of the ethod at hand and k the constant tie step, t n = nk. It is well known that estiates of this type are available for strongly A- stable schees, such as the Runge Kutta Radau IIA ethods, and, in particular, the backward Euler ethod. Siilar estiates are not available for A-stable schees. A way of securing such estiates for A-stable schees is to start the tie-stepping procedure with a few steps of a strongly A-stable schee of order s 1 and then proceed with the ain schee. For instance, for the Crank Nicolson ethod it suffices to perfor the first two steps with the backward Euler schee and subsequently proceed with the Crank Nicolson ethod. The use of the Euler schee as a starting procedure for the Crank Nicolson ethod in order to establish a priori error estiates for nonsooth initial data was suggested by Luskin and Rannacher see [1], Theores 7.4 and 7.5. In the a posteriori error analysis we have to derive estiates with reduced regularity requireents on the initial data, but in a for that allows efficient onitoring of the tie steps. To this end we will establish estiates that are the a posteriori analogs of Theores 7.4 and 7.5 of [1] for the following odification of the schee: We let U := u and define the approxiations U to u := ut,=1,...,n, by 5.3i U n + AU n =, n =1,, 5.3ii U n + AU n 1 =, n =3,...,N. Note first that, even for U H, due to the fact that U 1 and U are backward Euler approxiations, we have U 1 DAandU DA ; then, obviously, U n DA for n. We now proceed to the definition of the reconstruction U Reconstruction. Given the nodal approxiations U,...,U N, we define the approxiation Ut tout, t [,T], in the intervals I 1 and I 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 Ut =U n, t,,, 5.4ii Ut =U n +t t n U n, t, n =3,...,N 1 cf..3. Furtherore, we let the reconstruction Û of U be given by 5.5 Ût :=U n 1 i.e., 5.6i t n 1 AUs ds, t, Ût =U n 1 t t n 1 AU n, t,,, 5.6ii Ût =U n 1 1 t tn 1 A[Ut+U n 1 ], t, 3 n N.

16 56 G. AKRIVIS, CH. MAKRIDAKIS, AND R. H. NOCHETTO We observe that 5.6i coincides with the continuous piecewise linear reconstruction of [19], whereas 5.6ii agrees with the continuous piecewise quadratic reconstruction.1 for f =. Inviewof5.3i, we obtain 5.7i Ut Ût = tn tau n, t,,. Furtherore, in view of 5.3ii, 5.7ii Ut Ût = 1 t tn 1 t n ta U n, t, 3 n N see 3.1. Note that, even for U H\DA, Û is well defined; this would not be the case if U 1 and U were Crank Nicolson approxiations because then U n / DA for any n. It iediately follows fro 5.5 that 5.8 ê t+aet =, <t<t; copare with 4.1 for f =. Next, we will briefly discuss soe of the estiators we obtain by applying the energy ethod or Duhael s principle to the above error equation. 5.. Estiate I: Energy ethod. Taking in 5.8 the inner product with êt, and recalling that 1 Aet, êt = et + êt êt et, we obtain 5.9 êt + ês + es ds Ûs Us ds, t T cf Now, in view of 5.7i, Ût Ut =t n t A 3/ U n, n =1, ; therefore Ûs Us ds = 1 k A 3/ U 1 + k A 3 3/ U = 1 k A 1/ U 1 + k A 3 1/ U. Furtherore, Ûs Us ds β k t n A 5 3/ U n, n= with β as in 4.8. We thus deduce the upper a posteriori estiate êt + ês + es ds 1 k A 1/ U 1 + k A 3 1/ U β 1 k n A 5 3/ U n ; n= copare with 4.1 for f =. Proceeding as in Subsection 4.1. we also get a sharp lower bound. Note that the above estiate holds, provided that U DA 1/. Further reasonable error control based on this estiate requires us to balance the ters k 1 A 1/ U1,k A 1/ U and kn A 3/ Un.

17 A POSTERIORI ESTIMATES FOR THE CRANK NICOLSON METHOD Estiate II: Duhael principle. We next odify arguents of Sections 4. to obtain the final result êt n 1 { 8 C A kn A U n ln tn ax k 1 k U 1,k U, ax k j A U j }, n j n 1 which could be copared with 4.37 for f =. Note that the above estiate holds, provided that U H. Further reasonable error control based on this estiate requires us to balance the ters k 1 U 1,k U and kn A U n. 6. Error estiates for nonlinear equations In this section we consider the discretization of 1.1. We assue that Bt, canbeextendedtoanoperatorfrov into V. A natural condition for 1.1 to be locally of parabolic type is the following local one-sided Lipschitz condition: 6.1 Bt, v Bt, w,v w λ v w + µ v w v, w T u in a tube T u,t u := {v V :in t ut v 1}, around the solution u, uniforly in t, with a constant λ less than one and a constant µ. With F t, v :=Av Bt, v, it is easily seen that 6.1 can be written in the for of a Gårding-type inequality, 6. F t, v F t, w,v w 1 λ v w µ v w v, w T u. Furtherore, in order to ensure that an appropriate residual is of the correct order, we will ake 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 ters of the nor of V for concreteness. The analysis ay be odified to yield a posteriori error estiates under analogous conditions for v and w belonging to tubes defined in ters of other nors, not necessarily the sae for both arguents. In the sequel the estiates are proved under the assuption that Ût,Ut T u, for all t [,T]. This assuption can, in soe cases, be verified a posteriori under conditional assuptions on Û and U. Thus the final result will hold, pending on a condition that Û and U ay or ay not satisfy. However, the validity of this condition can be coputationally verified. The derivation of these bounds requires the use of fine properties of the specific underlying p.d.e., as was done in [16, 18], and therefore goes beyond the scope of the present paper. We refer to [3] for existence and local uniqueness results for the continuous Galerkin approxiations, in particular for the Crank Nicolson Galerkin approxiations, as well as for a priori error estiates. Concrete exaples of parabolic equations satisfying 6.1 and 6.3 in appropriate tubes are given in [1] and [] The Crank Nicolson Galerkin ethod. We recall that this ethod for 1.1 consists of seeking a function U :[,T] V, continuous and piecewise linear, such that U = u and 6.4 U n + AU n 1 1 = Bs, Usds,

18 58 G. AKRIVIS, CH. MAKRIDAKIS, AND R. H. NOCHETTO where U n := Ut n,u n 1 := 1 U n + U n 1 = Ut n 1. The Crank Nicolson Galerkin approxiate solution U can be expressed in ters of its nodal values U n 1 and U n, 6.5 Ut =U n 1 +t t n 1 U n, t. In view of., we let the Crank Nicolson Galerkin reconstruction Û of U be 6.6 Ût :=U n 1 A i.e., t n 1 Us ds Ût =U n 1 1 t tn 1 A[Ut+U n 1 ]+ It iediately follows fro 6.6 that t n 1 P 1 Bs, Usds, 6.8 Û t+aut =P 1 Bt, Ut. Using 6.4, it easily follows fro 6.5 and 6.7 that t n 1 P 1 Bs, Usds, t. Ut Ût = 1 t tn 1 t n ta U n t tn 1 Bs, Usds P 1 Bs, Usds; t n 1 therefore, again using 6.4, we have Ut Ût =1 t tn 1 t n t A U n 1 1 ABs, Usds t tn 1 Bs, Usds P 1 Bs, Usds, t n 1 t. Hence, in view of 3.3, Ut Ût =1 t tn 1 t n t A U n 1 1 ABs, Usds k 6.1 n I n + 6 kn 3 t t n 1 t n t Bs, Uss t n 1 ds, t, fro which we iediately see that ax t In Ut Ût = Ok n. 6.. The Crank Nicolson ethod. The Crank Nicolson approxiations U DA to the nodal values u := ut ofthesolutionu of 1.1 are defined by 6.11 U n + AU n 1 = Bt n 1,U n 1, n =1,...,N, with U := u. As before, we define the Crank Nicolson approxiation U to u by linearly interpolating between the nodal values U n 1 and U n, 6.1 Ut =U n 1 +t t n 1 U n, t. Let b : H be the linear interpolant of B,U at the nodes t n 1 and t n 1, 6.13 bt =Bt n 1,U n 1 + t t n 1 [Bt n 1,U n 1 Bt n 1,U n 1 ], t.

19 A POSTERIORI ESTIMATES FOR THE CRANK NICOLSON METHOD 59 Inspired by.9, we define the Crank Nicolson reconstruction Û of U by 6.14 Ût :=U n 1 A i.e., t n 1 Us ds + t n 1 bsds, t, Ût =U n 1 1 t tn 1 A[Ut+U n 1 ]+t t n 1 Bt n 1,U n t t n 1 t n t[bt n 1,U n 1 Bt n 1,U n 1 ], t. Note that 6.15 reduces to.1 in the case B is independent of u. It iediately follows fro 6.14 that 6.16 Û t+aut =bt, t. Furtherore, it is easily seen that, for t, Ut Ût 6.17 = 1 { t tn 1 t n t A U n [ Bt n 1,U n 1 Bt n 1,U n 1 ]} Error estiates. We now derive a posteriori error estiates for both the Crank Nicolson Galerkin and the Crank Nicolson ethod. Let e := u U and ê := u Û. The following estiates hold under the assuption that Ût,Ut T u for all t [,T]. Crank Nicolson Galerkin ethod. Subtracting 6.8 fro the differential equation in 1.1, we obtain 6.18 ê t+aet =Bt, ut P 1 Bt, Ut, which we rewrite in the for 6.19 ê t+aet =Bt, ut Bt, Ut + R U t, with 6. R U t =Bt, Ut P 1 Bt, Ut. Now, Bt, ut Bt, Ut, êt = Bt, ut Bt, Ut,et + Bt, ut Bt, Ut,Ut Ût, and, in view of 6.1 and 6.3, eleentary calculations yield 6.1 Bt, ut Bt, Ut, êt λ êt + µ et + L et Û Ut. Siilarly, Bt, ut Bt, Ut, êt = Bt, ut Bt, Ût, êt + Bt, Ût Bt, Ut, êt and 6. Bt, ut Bt, Ut, êt λ êt + µ êt + L êt Û Ut. Suing 6.1 and 6., we obtain Bt, ut Bt, Ut, êt λ êt + et µ êt + et + L êt + et Û Ut.

20 53 G. AKRIVIS, CH. MAKRIDAKIS, AND R. H. NOCHETTO Now, and et êt + Û Ut êt + Û Ut L êt + et Û Ut ε L êt + et + Û Ut ε ε êt + et + L ε Û Ut. Consequently, fro 6.3, we obtain Bt, ut Bt, Ut, êt λ êt + et +3µ et 6.4 +µ Û Ut + ε êt + et + L Û Ut ε for any positive ε. Taking in 6.19 the inner product with êt, we obtain d 6.5 dt êt + et + êt = Ût Ut + Bt, ut Bt, Ut, êt + R U t, êt, whence, in view of 6.4, d dt êt +1 λ ε êt + et 3µ êt L Û Ut +µ Û ε Ut + 1 ε R Ut. We thus easily obtain the desired a posteriori error estiate via Gronwall s lea êt +1 λ ε e 3µt s ês + es ds 6.7 e 3µt s[ L 1+ Û Us ε +µ Û Us + 1 ] ε R Us ds. Crank Nicolson ethod. Subtracting 6.16 fro the differential equation in 1.1, we obtain ê t+aet =Bt, ut bt. Therefore, 6.7 is valid for the Crank Nicolson ethod as well, this tie with R U t :=Bt, Ut bt. References 1. G. Akrivis, M. Crouzeix, and Ch. Makridakis, Iplicit-explicit ultistep finite eleent ethods for nonlinear parabolic probles, Math. Cop MR g:6588. G. Akrivis, M. Crouzeix, and Ch. Makridakis, Iplicit-explicit ultistep ethods for quasilinear parabolic equations, Nuer. Math MR17188 e: G. Akrivis and Ch. Makridakis, Galerkin tie-stepping ethods for nonlinear parabolic equations, M ANMath.Mod.Nuer.Anal MR f: A. K. Aziz and P. Monk, Continuous finite eleents in space and tie for the heat equation, Math. Cop MR d:65189

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