A posteriori error analysis for the Crank-Nicolson method for linear Schrödinger equations

DOI:10.1051/m2an/2010101 www.esaim-m2an.org

A POSTERIORI ERROR ANALYSIS FOR THE CRANK-NICOLSON METHOD FOR LINEAR SCHR ¨ ODINGER EQUATIONS

Irene Kyza

Abstract. We provea posteriori error estimates of optimal order for linear Schr¨odinger-type equa- tions in the L^∞(L²)- and the L^∞(H¹)-norm. We discretize only in time by the Crank-Nicolson method. The direct use of the reconstruction technique, as it has been proposed by Akrivis et al.

in [Math. Comput. 75(2006) 511–531], leads toa posteriori upper bounds that are of optimal order in theL^∞(L²)-norm, but of suboptimal order in the L^∞(H¹)-norm. The optimality in the case of L^∞(H¹)-norm is recovered by using an auxiliary initial- and boundary-value problem.

Mathematics Subject Classification. 65M15, 35Q41.

Received December 19, 2009. Revised August 16, 2010.

Published online February 21, 2011.

1. Introduction

In this paper we focus on thea posteriori error analysis for time discrete Crank-Nicolson approximations, of linear Schr¨odinger-type equations in theL^∞(L²)- and theL^∞(H¹)-norm.

To fix notation, let Ω ⊂R^d be a bounded domain with boundary ∂Ω and let 0< T < ∞be given. The initial- and boundary-value problem for the general linear Schr¨odinger equation reads as

⎧⎪

⎨

⎪⎩

u_t−iαΔu+ ig(x, t)u=f(x, t) in ¯Ω×[0, T],

u= 0 on∂Ω×[0, T],

u(·,0) =u₀ in ¯Ω,

(1.1)

whereαis a positive constant,g: ¯Ω×[0, T]→Randf : ¯Ω×[0, T]→Care given functions andu₀: ¯Ω→Cis a given initial value. Problem (1.1) can be equivalently written, fort∈[0, T], in variational form as

(u_t(t), υ) + iα(∇u(t),∇υ) + i

g(t)u(t), υ

= f(t), υ

, ∀υ∈H₀¹(Ω),

u(·,0) =u₀ in ¯Ω, (1.2)

Keywords and phrases.Linear Schr¨odinger equation, Crank-Nicolson method, Crank-Nicolson reconstruction,a posteriorierror analysis, energy techniques,L^∞(L²)- andL^∞(H¹)-norm.

∗Work supported in part by the European Union grant No. MEST-CT-2005-021122 (Differential Equations with Applications in Science and Engineering), the “Alexander S. Onassis Public Benefit Foundation” and “The A.G. Leventis Foundation”.

1 Department of Mathematics, Mathematics Building, University of Maryland, College Park, 20742-4015 MD, USA.

kyza@math.umd.edu

Article published by EDP Sciences c EDP Sciences, SMAI 2011

where (·,·) denotes theL²-inner product inΩ. It is well known that, if

⎧⎪

⎨

⎪⎩

g∈C¹

[0, T];C¹( ¯Ω) , f ∈L²

[0, T];L²(Ω)

, f_t∈L²

[0, T];H⁻¹(Ω) , u₀∈H₀¹(Ω),

(1.3)

then problem (1.2) has a unique weak solutionu∈C

[0, T];H₀¹(Ω)

; alsou_t∈C

[0, T];H⁻¹(Ω)

,cf.,e.g., [7], pp. 620–630, [1,10] and references [6,15] therein.

A posteriori error estimates for problem (1.1) (or equivalently for problem (1.2)) in theL^∞(L²)-norm for the Crank-Nicolson method have been proven by D¨orfler in [8]; these estimates are of first order of accuracy in time.

As a continuation of this paper, Katsaounis and Kyza prove in [11] (see also [14], Chap. 7), optimal L^∞(L²) a posteriori error estimates for fully discrete schemes. The derivation of the basic estimates in [11] follows the approach in [17], i.e., an appropriate reconstruction is used which involves the Crank-Nicolson reconstruction that has been proposed by Akriviset al. in [2]. Another Crank-Nicolson reconstruction that has been proposed by Lozinskiet al. in [16], has been used in [14], to obtain similar estimates.

Our aim in this paper is to provide an error control of a posteriori type for Crank-Nicolson time dis- crete approximations of linear Schr¨odinger equation in theL^∞(L²)- and the L^∞(H¹)-norm. Estimates in the L^∞(H¹)-norm are crucial in theL^∞(L²) error analysis of the nonlinear Schr¨odinger equation with cubic non- linearities (cubic NLS). In particular, to complete the arguments in this case, we must have at our disposal L^∞(H¹) error estimates, cf. [12,13] for the a priori error analysis and [14], Chapter 3, for the a posteriori error analysis. More precisely, regarding the a posteriori analysis, standard energy techniques are not enough to lead to estimates in theL^∞(L²)-norm for the two-dimensional cubic NLS. A natural approach then is to take advantage of Strichartz estimates that are valid for the Schr¨odinger operator,cf. [4]. Strichartz estimates are related to the L^∞(H¹)-norm and inevitably the proof ofL^∞(H¹) a posteriori error estimates becomes a ne- cessity. The derivation of a posteriori error estimates in the L^∞(H¹)-norm for the nonlinear case will be the subject of a forthcoming paper. Note, however, that having at hand such estimates in the linear case comprises an important starting point for the proof of the corresponding estimates in the nonlinear case.

To obtain the a posteriori error estimates in the linear case, we will use energy techniques and the Crank- Nicolson reconstruction proposed in [2]. The derivation of theL^∞(H¹)-estimates in this case is more involved compared to the L^∞(L²)-estimates. The direct application of the reconstruction technique, as it is proposed in [2], leads to suboptimalL^∞(H¹) upper bounds. As we shall see in Section4, to recover the optimality we need to use technically more involved arguments compared to [2] and new key ideas. In particular, the introduction of an auxiliary initial- and boundary-value problem will play a significant role for the analysis of Section4.

More precisely, the paper is organized as follows: In Section2 we introduce the Crank-Nicolson method and the reconstruction for problem (1.1). In Section3we discuss the regularity of the reconstruction and we address the optimal ordera posteriori error estimate in theL^∞(L²)-norm, Theorem3.1.

The main results of the paper are stated in Section4. In particular, Section4 deals with thea posteriori error analysis in the L^∞(H¹)-norm. To prove the estimates, we first proceed as in the a priori error analysis for Schr¨odinger-type equations (see for example [12,13]), but in the continuous level instead of the discrete.

From this procedure we conclude that to prove an optimala posteriori error estimate in theL^∞(H¹)-norm, we first need to estimatea posteriori the quantity sup_0≤t≤Tˆe_t(t)_L²_(Ω), where ˆerepresents the error between the exact solution and the reconstruction (see Sect. 3). The “obvious way” to do this leads to suboptimal upper bounds, Theorem4.1. In the second part of Section4we describe in detail how we can recover the optimality.

The estimates of optimal order of accuracy for the general case are presented in Theorems4.2 and4.3. In the special casef ≡0 and function g is independent of the time variable, we conclude some simplifications on the main results, which are presented in the last part of Section4. The casef ≡0 andg≡g(x) is very interesting, since the so-called linear Schr¨odinger equation in the semiclassical regime,

u_t−iε 2Δu+ i

εV(x)u= 0, (1.4)

is a special case of it with α := ^ε₂ and g := ¹_εV. In (1.4), ε (0 < ε 1) is the scaled Planck constant and V is a given electrostatic potential. Problems related to equation (1.4) are of great interest in Physics and Engineering, since they describe models of solid state physics, cf., e.g., [5], Chapter 9, and [18], Chapter 5.

Finally, in the last section of the paper, we confirm the theoretical results of Sections3and4by investigating numerically the behaviour of the a posteriori error estimators.

2. Preliminaries 2.1. The Crank-Nicolson method

Let 0 = t⁰ < t¹ < · · · < t^N = T be a partition of [0, T], I_n := (tⁿ⁻¹, tⁿ], k_n := tⁿ−tⁿ⁻¹ and k = max_1≤n≤Nk_n. We discretize problem (1.1) only in time by the Crank-Nicolson method and we end up with approximationsUⁿ∈H₀¹(Ω) to the values u(tⁿ), n= 0,1, . . . , N,defined by

∂U¯ ⁿ−iαΔUⁿ⁻¹² + ig tⁿ⁻¹²

Uⁿ⁻¹² =f tⁿ⁻¹²

, n= 1, . . . , N, (2.1) withU⁰=u₀. Here, we have used the notation

∂U¯ ⁿ:= Uⁿ−Uⁿ⁻¹

k_n , Uⁿ⁻¹² := Uⁿ+Uⁿ⁻¹

2 and tⁿ⁻¹² :=tⁿ+tⁿ⁻¹

2 ·

The Crank-Nicolson method is of second order of accuracy. Thus, it is natural to define the continuous in time approximation U(t) to u(t), for t ∈ [0, T], by linearly interpolating between the nodal values Uⁿ⁻¹ and Uⁿ. I.e.,U : [0, T]→H₀¹(Ω) is defined by

U(t) :=Uⁿ⁻¹² + t−tⁿ⁻¹²

∂U¯ ⁿ, t∈I_n. (2.2)

Then it is clear that, for t ∈ [0, T], u(t)−U(t) = O(k²). However, as it was observed in [2], the use of this continuous approximationU in thea posteriori error analysis yields estimates of first instead of optimal second order of accuracy. Even in the case of g ≡ 0, the error e := u−U satisfies the equation e_t−iαΔe = −r with r(t) := −iα(t−tⁿ⁻¹²)Δ∂U¯ ⁿ +

f(tⁿ⁻¹²)−f(t)

, t ∈ I_n, cf. [2]. Then, r is an a posteriori quantity of first order. Thus, by applying energy techniques to the error equation e_t−iαΔe=−r,as in [8,20], we derive estimates of suboptimal order. That r is not of optimal order is due to the fact that it contains U_t and U_t (U_t(t) = ¯∂Uⁿ, t∈I_n) is a first order approximation tou_t.

2.2. The Crank-Nicolson reconstruction

The Crank-Nicolson reconstruction ˆU : [0, T]→H⁻¹(Ω) ofU that we will use in the sequel, was the main tool in the analysis in [2]. This reconstruction is a piecewise quadratic polynomial defined by

Uˆ(t) :=Uⁿ⁻¹+ iαΔ _t

tⁿ⁻¹

U(s) ds−i _t

tⁿ⁻¹

G_U(t) dt+ _t

tⁿ⁻¹

F(t) dt, t∈I_n, (2.3) whereG_U :I_n→L²(Ω) andF :I_n →L²(Ω) are the linear interpolants ofgU andf,respectively, at the nodes tⁿ⁻¹ andtⁿ⁻¹²,i.e.,

G_U(t) :=g tⁿ⁻¹²

Uⁿ⁻¹² + 2

k_n t−tⁿ⁻¹² g tⁿ⁻¹²

Uⁿ⁻¹² −g tⁿ⁻¹

Uⁿ⁻¹

, t∈I_n, (2.4) and

F(t) :=f tⁿ⁻¹² + 2

k_n t−tⁿ⁻¹² f tⁿ⁻¹²

−f tⁿ⁻¹

, t∈I_n. (2.5)

Note that to define the reconstruction ˆU we use at each time intervalI_n the Crank-Nicolson approximations Uⁿ⁻¹andUⁿ,n= 1, . . . , N. This is the reason that this reconstruction is called “two point estimator”. A three point Crank-Nicolson reconstruction is proposed in [16],i.e., to define it at eachI_n, we invoke the valuesUⁿ⁻², Uⁿ⁻¹ and Uⁿ, n = 2, . . . , N. The latter Crank-Nicolson reconstruction can be used alternatively to derive estimates of the same form as those we present in this paper.

From (2.3) we can easily see that ˆU can be written as Uˆ(t) =Uⁿ⁻¹+ iα

2

t−tⁿ⁻¹ Δ

U(t) +Uⁿ⁻¹ +

t−tⁿ⁻¹

f tⁿ⁻¹²

−ig tⁿ⁻¹² Uⁿ⁻¹²

+ 1 k_n

t−tⁿ⁻¹

(tⁿ−t)

× f tⁿ⁻¹²

−f tⁿ⁻¹

−i g tⁿ⁻¹²

Uⁿ⁻¹² −g tⁿ⁻¹

Uⁿ⁻¹

, t∈I_n.

(2.6)

Therefore, ˆU(tⁿ) =U(tⁿ) =Uⁿ, n= 0,1, . . . , N.In particular, ˆU is continuous. From (2.3) we also have that Uˆ satisfies

Uˆ_t(t)−iαΔU(t) =−iG_U(t) +F(t), t∈I_n. (2.7) Thea posteriori quantity ˆr(t) := [ ˆU_t−iαΔUˆ + igUˆ −f](t), t∈I_n,is the residual of ˆU. Using (2.7) we see that the residual can also be expressed as

r(t) =ˆ −iαΔ Uˆ −U

(t) + ig(t) Uˆ −U

(t) + i (gU−G_U) (t) + (F−f) (t), t∈I_n. (2.8) Furthermore, the difference ˆU(t)−U(t), t∈I_n,can be written as in [2],

Uˆ(t)−U(t) = 1 2

t−tⁿ⁻¹

(tⁿ−t)

×

−iαΔ∂U¯ ⁿ+ 2i

k_n g tⁿ⁻¹²

Uⁿ⁻¹² −g tⁿ⁻¹

Uⁿ⁻¹

− 2

k_n f tⁿ⁻¹²

−f

tⁿ⁻¹

. (2.9) Since{Uⁿ}^N_n=0are second order approximations touat the nodestⁿ, n= 0,1, . . . , N,we expect, in view of (2.9), that the difference ˆU−U will be of second order of accuracy in time. Therefore, ˆr(t) =O(k²), t∈[0, T].Thus, the use of the reconstruction ˆU will lead to optimal bounds in theL^∞(L²)-norm,cf. Theorem 3.1in the next section.

3. Estimate in the L

(L

)-norm

In this section we prove an optimal L^∞(L²) a posteriori error estimate. In the analysis below, we would like the reconstruction ˆU(t) to belong to H₀¹(Ω), for t ∈ [0, T], and the residual ˆr(t) to belong to L²(Ω), for t∈[0, T].From the definition of the residual (see also (2.8)), we see that a sufficient condition for ˆr(t) to belong toL²(Ω),fort∈[0, T],is ˆU(t)∈H²(Ω).Therefore, we would like to have ˆU(t)∈H₀¹(Ω)∩H²(Ω),fort∈[0, T].

In general, the reconstruction we have defined in Section2.2does not belong toH₀¹(Ω)∩H²(Ω),but toH⁻¹(Ω) instead. In order to ensure that ˆU(t) ∈ H₀¹(Ω)∩H²(Ω), for t ∈ [0, T], we may have to assume additional regularity and compatibility conditions on the data of the problem. In the following lemma we givesufficient conditions which ensure that ˆU(t)∈H₀¹(Ω)∩H²(Ω), fort∈[0, T].

Lemma 3.1. For the data of problem (1.2)we further assume that g(t)∈C²( ¯Ω), ∀t∈[0, T] and

Δu₀, f(t)∈H₀¹(Ω)∩H²(Ω), ∀t∈[0, T].

Then Uˆ(t)∈H₀¹(Ω)∩H²(Ω).

Proof. The regularity and compatibility assumptions on the data of problem (1.2) ensure that Uⁿ ∈H₀¹(Ω)∩ H²(Ω), n = 0,1, . . . , N, in light of the method (2.1). Therefore, g(tⁿ⁻¹²)Uⁿ⁻¹² ∈ H₀¹(Ω)∩H²(Ω) because g(t)∈C²( ¯Ω), fort∈[0, T]. Using once more (2.1) and the fact that f(t)∈H₀¹(Ω)∩H²(Ω), fort∈[0, T],we immediately obtain that ΔUⁿ⁻¹² ∈H₀¹(Ω)∩H²(Ω), n = 1, . . . , N. Since ΔU⁰ =Δu₀ ∈ H₀¹(Ω)∩H²(Ω), an inductive argument ensures thatΔUⁿ∈H₀¹(Ω)∩H²(Ω), n= 0,1, . . . , N.This yieldsΔU(t)∈H₀¹(Ω)∩H²(Ω), for t ∈ [0, T]. On the other hand it is easily seen that G_U(t), F(t) ∈ H₀¹(Ω)∩H²(Ω), for t ∈ [0, T], and the

proof is complete, in view of (2.3).

Remark 3.1. ConditionΔu₀∈H₀¹(Ω) is actually needed in order to ensure thatUⁿ, n= 1, . . . , N,are indeed second order approximations touat the nodestⁿ, n= 1, . . . , N (see for example [19], Chap. 7).

Remark 3.2. A more detailed analysis about when the reconstruction ˆU belongs to the correct space can be found in [3]. However, we would like to point out that in cases of fully discrete schemes, the reconstruction and the residual belong to the spaces H₀¹(Ω) and L²(Ω), respectively, without further assumptions on the data of the problem (see for example [11], or [14], Second Part). Therefore, from now on we will assume that Uˆ : [0, T]→H₀¹(Ω) and ˆr: [0, T]→L²(Ω).

We are now ready to derive the estimate in the L^∞(L²)-norm. Let the error ˆe: [0, T]→H₀¹(Ω) be defined by ˆe:=u−Uˆ. By the residual’s definition and problem (1.1) we conclude, fort ∈I_n, n = 1, . . . , N,that the error satisfies, in weak form, the problem

(ˆe_t(t), υ) + iα(∇ˆe(t),∇υ) + i

g(t)ˆe(t), υ

=− r(t), υˆ

, ∀υ∈H₀¹(Ω),

ˆe(·,0) = 0, in ¯Ω. (3.1)

Setting in (3.1)υ= ˆe, then taking real parts and using the Cauchy-Schwarz inequality we arrive at 1

2 d

dtˆe(t)²_L2(Ω)≤ ˆr(t)_L²_(Ω)ˆe(t)_L²_(Ω), or,

d

dtˆe(t)_L²_(Ω)≤ ˆr(t)_L²_(Ω).

Integrating the above relation from 0 totⁿ, n= 1, . . . , N, we immediately obtain the following:

Theorem 3.1 (a posteriori error estimate in the L^∞(L²)-norm). Let ube the weak solution of problem (1.1), Uˆ the Crank-Nicolson reconstruction (2.3)andeˆ=u−U. Then the following optimal orderˆ a posteriorierror estimate holds for n= 1, . . . , N,

0≤t≤tmaxⁿe(t)ˆ L²(Ω)≤ _tⁿ

0 r(t)ˆ L²(Ω)dt, (3.2)

where thea posterioriquantityrˆis given by (2.8).

In the next section we will see that the error equation (3.1) is not sufficient to control the error in the L^∞(H¹)-norm in an optimal way.

4. A

error estimates in the L

(H

)-norm

To obtain estimates in the L^∞(H¹)-norm we need further regularity for the solutionuof problem (1.2). In particular, in this section, in addition to the conditions (1.3) we assume that

⎧⎪

⎨

⎪⎩

g_tt∈L²

[0, T];L²(Ω) , f ∈L²

[0, T];H¹(Ω)

, f_t∈L²

[0, T];L²(Ω)

, f_tt∈L²

[0, T];H⁻¹(Ω) , iαΔu₀+f(0)∈H₀¹(Ω).

(4.1)

Then it can be proven thatu∈C

[0, T];H₀¹(Ω)∩H²(Ω)

, u_t∈C

[0, T];H₀¹(Ω)

andu_tt∈C

[0, T];H⁻¹(Ω) . The proof of this statement follows similar arguments to those in the proof of Theorems 5 and 6, pp. 389–393, in [9], it is beyond the scope of the paper and thus it is omitted.

The starting point to derive estimates in theL^∞(H¹)-norm is equation (3.1). Indeed, settingυ= ˆe_tin (3.1) and then taking imaginary parts, we get

α 2

d

dt∇ˆe(t)²_L2(Ω)≤

x∈Ωsup|g(x, t)| ˆe(t)_L²_(Ω)+ˆr(t)_L²_(Ω)

ˆe_t(t)_L²_(Ω), t∈I_n. (4.2)

From (4.2), it is clear that to estimate ∇ˆe(t)L²(Ω), we must control the quantityˆe_t(t)L²(Ω).

4.1. First order estimates

To controlˆe_t(t)_L²_(Ω), we differentiate (2.6) to obtain Uˆ_t(t) = iαΔUⁿ⁻¹+ iα

t−tⁿ⁻¹ Δ∂U¯ ⁿ

−i

g tⁿ⁻¹²

Uⁿ⁻¹² + 2

k_n t−tⁿ⁻¹²

g tⁿ⁻¹²

Uⁿ⁻¹² −g tⁿ⁻¹

Uⁿ⁻¹ +

f tⁿ⁻¹²

− 2

k_n t−tⁿ⁻¹²

f tⁿ⁻¹²

−f

tⁿ⁻¹

, t∈I_n.

(4.3)

At this point, a careful analysis is needed, since ˆU_tis discontinuous at the nodestⁿ, n= 1, . . . , N−1.Indeed, it is easily seen that

Uˆ_t(tⁿ⁻) = iαΔUⁿ−i 2g tⁿ⁻¹²

Uⁿ⁻¹² +g tⁿ⁻¹

Uⁿ⁻¹

+ 2f tⁿ⁻¹²

−f tⁿ⁻¹ and

Uˆ_t(tⁿ⁺) = iαΔUⁿ−ig(tⁿ)Uⁿ+f(tⁿ).

Therefore, in general, ˆU_t(tⁿ⁻)= ˆU_t(tⁿ⁺). However, we can conclude that Uˆ_t(tⁿ⁺)−Uˆ_t(tⁿ⁻) =O

k²

, n= 1, . . . , N−1. (4.4)

It is to be emphasized that (4.4) is crucial to deduce the order of the upper bound in the estimate (4.9) below.

By differentiation of (3.1), we get, fort∈I_n, n= 1, . . . , N, eˆ_tt(t), υ

+ iα

∇ˆe_t(t),∇υ + i

g(t)ˆe_t(t), υ

=−i

g_t(t)ˆe(t), υ

−

ˆr_t(t), υ

, ∀υ∈H₀¹(Ω). (4.5) Thus, by taking in (4.5)υ= ˆe_t, then real parts and integrating fromtⁿ⁻¹ tot, we obtain

eˆ_t(t)_L²_(Ω)≤ˆe_t t⁽ⁿ⁻¹⁾⁺

L²(Ω)+ max

tⁿ⁻¹≤t≤tⁿe(t)ˆ _L²_{(Ω) t}ⁿ

tⁿ⁻¹

x∈Ωsup|g_t(x, t)|dt +

_tⁿ

tⁿ⁻¹

rˆ_t(t)_L²_(Ω)dt, t∈I_n.

(4.6)

Notice now that the following inequality holds forn= 1, . . . , N, ˆe_t t⁽ⁿ⁻¹⁾⁺

L²(Ω)≤eˆ_t t⁽ⁿ⁻¹⁾⁻

L²(Ω)+Uˆ_t t⁽ⁿ⁻¹⁾⁺

−Uˆ_t t⁽ⁿ⁻¹⁾⁻

L²(Ω), (4.7)

becauseu_tis a time-continuous function,cf. Introduction. In view of (4.7), (4.6) is rewritten as ˆe_t(t)L²(Ω)≤eˆ_t t⁽ⁿ⁻¹⁾⁻

L²(Ω)+Uˆ_t t⁽ⁿ⁻¹⁾⁺

−Uˆ_t t⁽ⁿ⁻¹⁾⁻

L²(Ω)

+ max

tⁿ⁻¹≤t≤tⁿˆe(t)_L²_{(Ω) t}ⁿ

tⁿ⁻¹

x∈Ωsup|g_t(x, t)|dt+ _tⁿ

tⁿ⁻¹

ˆr_t(t)_L²_(Ω)dt, t∈I_n,

and therefore, using induction we conclude the estimate (we implicitly set₀

m=1:= 0) ˆe_t(t)_L²_(Ω)≤ ˆe_t(0)_L²_(Ω)+

n−1

m=1

Uˆ_t t^m+

−Uˆ_t t^m−

L²(Ω)

+ max

0≤t≤tⁿˆe(t)_L²_{(Ω) t}ⁿ

0 sup

x∈Ω|g_t(x, t)|dt+ _tⁿ

0 rˆ_t(t)_L²_(Ω)dt, t∈I_n.

(4.8)

Since ˆe(0) = 0 and ˆr(0) = 0 we have, in view of (3.1), that ˆe_t(0) = 0 as well. Hence (4.8) is now written, in light of (3.2), as

0≤t≤tsupⁿˆe_t(t)_L²_(Ω)≤ⁿ⁻¹

m=1

Uˆ_t(t^m+)−Uˆ_t(t^m−)_L²_(Ω)

+ _tⁿ

0 sup

x∈Ω|g_t(x, t)|dt _tⁿ

0 ˆr(t)_L²_(Ω)dt+ _tⁿ

0 rˆ_t(t)_L²_(Ω)dt.

(4.9)

Remark 4.1. Relation (4.9) gives an a posteriori estimate for ˆe_t in the L^∞(L²)-norm of first order. This is because both terms _n−1

m=1Uˆ_t(t^m+)−Uˆ_t(t^m−)_L²_(Ω) and _tⁿ

0 ˆr_t(t)_L²_(Ω) dt are in general of first instead of second order of accuracy. This immediately follows from (4.4) for the first term while for the second, this can be verified by differentiation in time of (2.8). Even in the simplest case wheng ≡0 and f ≡0 we have rˆ_t(t) =O(k).Indeed, in this case, it is easily seen that

rˆ_t(t) = iα 2

2t−tⁿ⁻¹−tⁿ

Δ∂U¯ ⁿ, t∈I_n, and therefore, in general, ˆr_t(t) =O(k).

Invoking in (4.2) the estimates (3.2), (4.9) and integrating from 0 totⁿ we conclude to:

Theorem 4.1 (suboptimala posteriorierror estimate in L^∞(H¹)- norm). With the notation of Theorem 3.1 the following a posteriorierror estimate is valid, forn= 1, . . . , N,

0≤t≤tmaxⁿ∇ˆe(t)²_L2(Ω)≤ 2 α

1 +

_tⁿ

0 sup

x∈Ω|g(x, t)|dt

× _tⁿ

0 r(t)ˆ L²(Ω)dt sup

0≤t≤tⁿˆe_t(t)L²(Ω),

(4.10)

where the quantitysup_0≤t≤tnˆe_t(t)L²(Ω)is estimated a posteriori via (4.9).

Theorem 4.1 yields ana posteriori error estimate in theL^∞(H¹)-norm of order ³₂ in time, instead of two, which is the optimal order of accuracy. To recover the optimal order we have to proceed in a different way and to introduce new ideas. This will be the topic of the next subsection.

4.2. Recovery of optimality

The main idea of the proof of optimal ordera posterioriestimates in theL^∞(H¹)-norm is based on considering the auxiliary initial- and boundary-value problem:

⎧⎪

⎨

⎪⎩

w_t−iαΔw+ ig(x, t)w=f_t(x, t)−ig_t(x, t) ˆU in ¯Ω×[0, T],

w= 0 on∂Ω×[0, T],

w(·,0) = iαΔu₀−ig(·,0)u₀+f(·,0) in ¯Ω.

(4.11)

Since ˆU is a piecewise quadratic polynomial with ˆU(t),Uˆ_t(t) ∈ L²(Ω), for almost every t ∈ [0, T], it can be proven (cf. [7], pp. 620–630) that, under conditions (1.3) and (4.1), problem (4.11) has a unique solution w∈C

[0, T];H₀¹(Ω)

withw_t∈C

[0, T];H⁻¹(Ω) .

Note that since ˆU is a second order approximation tou, and u_tsatisfies the following problem:

⎧⎪

⎨

⎪⎩

(u_t)_t−iαΔu_t+ ig(x, t)u_t=f_t(x, t)−ig_t(x, t)u in ¯Ω×[0, T],

u_t= 0 on∂Ω×[0, T],

u_t(·,0) = iαΔu₀−ig(·,0)u₀+f(·,0) in ¯Ω,

(4.12)

we expect thatwwill be a second order approximation tou_t, see Lemma4.1below.

We discretize problem (4.11) by the Crank-Nicolson method, i.e., we derive approximations Wⁿ, n = 0,1, . . . , N,defined by

⎧⎨

⎩

∂W¯ ⁿ−iαΔWⁿ⁻¹² + ig tⁿ⁻¹²

Wⁿ⁻¹² =f_t tⁿ⁻¹²

−ig_t tⁿ⁻¹²

Uˆ tⁿ⁻¹² ,

W⁰= iαΔu₀−ig(0)u₀+f(0) in ¯Ω. (4.13)

LetW : [0, T]→H₀¹(Ω) be the Crank-Nicolson approximation to w,i.e., the linear interpolate between the nodal valuesWⁿ⁻¹andWⁿ,

W(t) :=Wⁿ⁻¹² + t−tⁿ⁻¹²

∂W¯ ⁿ, t∈I_n. Also let ˆW : [0, T]→H₀¹(Ω) be the Crank-Nicolson reconstruction ofW,

Wˆ(t) =Wⁿ⁻¹+ iα

t−tⁿ⁻¹

ΔWⁿ⁻¹+ iα 2

t−tⁿ⁻¹₂ Δ∂W¯ ⁿ

−i

t−tⁿ⁻¹

g tⁿ⁻¹² Wⁿ⁻¹²

− i k_n

t−tⁿ⁻¹

(tⁿ−t) g tⁿ⁻¹²

Wⁿ⁻¹² −g tⁿ⁻¹

Wⁿ⁻¹

−i

t−tⁿ⁻¹

g_t tⁿ⁻¹²

Uˆ tⁿ⁻¹²

− i k_n

t−tⁿ⁻¹

(tⁿ−t) g_t tⁿ⁻¹²

Uˆ tⁿ⁻¹²

−g_t tⁿ⁻¹

Uⁿ⁻¹ +

t−tⁿ⁻¹

f_t tⁿ⁻¹² + 1

k_n

t−tⁿ⁻¹

(tⁿ−t) f_t tⁿ⁻¹²

−f_t tⁿ⁻¹

, t∈I_n.

(4.14)

Note that the new reconstruction ˆW we have just defined depends on the reconstruction ˆU.

As in the definition of the residual ˆr for the reconstruction ˆU, we define the residual ˆR : [0, T] → L²(Ω) for ˆW as

R(t) := ( ˆˆ W_t−iαΔWˆ + igWˆ −f_t+ ig_tUˆ)(t), t∈I_n, n= 1, . . . , N. (4.15) Remark 4.2. Here we assume again that ˆW(t) ∈ H₀¹(Ω) and ˆR(t)∈ L²(Ω), for t ∈[0, T]. As in Section3, to ensure this we might need to assume additional regularity and compatibility conditions on the data of problem (4.11), in the spirit of Lemma3.1. However, we point out once more that in the cases of fully discrete schemes, no further conditions are required to ensure that ˆW(t)∈H₀¹(Ω) and ˆR(t)∈L²(Ω),fort∈[0, T].

Since problem (4.11) is of the same form as problem (1.1) with the function f replaced by the function f_t−ig_tUˆ and the initial valueu₀ replaced by iαΔu₀−ig(0)u₀+f(0), we have, according to (2.3)–(2.5), that the residual ˆRis written fort∈I_n, n= 1, . . . , N,as

R(t) =ˆ −iαΔ Wˆ −W

(t) + ig(t) Wˆ −W (t) + i

g(t)W(t)−G_W(t) + F(t)˜ −f_t(t)

+ i g_t(t) ˆU(t)−G˜_Uˆ(t)

, (4.16)

with

G_W(t) :=g tⁿ⁻¹²

Wⁿ⁻¹² + 2

k_n t−tⁿ⁻¹² g tⁿ⁻¹²

Wⁿ⁻¹² −g tⁿ⁻¹

Wⁿ⁻¹

, (4.17)

F(t) :=˜ f_t tⁿ⁻¹² + 2

k_n t−tⁿ⁻¹² f_t tⁿ⁻¹²

−f_t tⁿ⁻¹

(4.18) and

G˜_Uˆ(t) := ig_t tⁿ⁻¹²

Uˆ tⁿ⁻¹² + 2i

k_n t−tⁿ⁻¹² g_t tⁿ⁻¹²

Uˆ tⁿ⁻¹²

−g_t tⁿ⁻¹

Uⁿ⁻¹

, (4.19)

Also recall that fort∈I_n, n= 1, . . . , N, Wˆ(t)−W(t) =1

2

t−tⁿ⁻¹

(tⁿ−t)

−iαΔ∂W¯ ⁿ+ 2i

k_n g tⁿ⁻¹²

Wⁿ⁻¹² −g tⁿ⁻¹

Wⁿ⁻¹

− 2

k_n f_t tⁿ⁻¹²

−f_t tⁿ⁻¹

+ 2i

k_n g_t tⁿ⁻¹²

Uˆ tⁿ⁻¹²

−g_t tⁿ⁻¹

Uⁿ⁻¹ .

(4.20)

Since the first and the second time-derivative of ˆU are discontinuous functions at the nodestⁿ, n= 1, . . . , N −1, it is not obvious that the Crank-Nicolson approximations Wⁿ are second order approximations to the values w(tⁿ), n = 1, . . . , N, even if the data of the problem are compatible and smooth enough. So, for completeness, we will prove that the Crank-Nicolson approximationsWⁿare indeed second order approximations to the values w(tⁿ), n= 1, . . . , N.

Lemma 4.1. Let u and w be the weak solutions of problems (1.1) and (4.11), respectively. Then, for n = 1, . . . , N,

0≤t≤tmaxⁿ(ut−w)(t)_L²_(Ω)≤ _tⁿ

0 sup

x∈Ω|g_t(x, t)|dt _tⁿ

0 ˆr(t)_L²_(Ω)dt, (4.21) whererˆis the residual given by (2.8).

Proof. It is easily seen that the differenceu_t−wsatisfies the problem

⎧⎪

⎨

⎪⎩

(u_t−w)_t(t), υ + iα

∇(ut−w)(t),∇υ + i

g(t)(u_t−w)(t), υ

=−i

g_t(t)ˆe(t), υ

, ∀υ∈H₀¹(Ω), t∈I_n, (u_t−w)(0) = 0, in ¯Ω.

(4.22)

Choosing in (4.22)υ=u_t−w, then taking real parts, and integrating from 0 totⁿ,we obtain

0≤t≤tmaxⁿ(ut−w)(t)_L²_(Ω)≤ _tⁿ

0 sup

x∈Ω|g_t(x, t)|dt max

0≤t≤tⁿˆe(t)_L²_(Ω). (4.23)

Estimate (4.23) yields estimate (4.21) in view of (3.2).

In the proposition below, we prove that theWⁿ are second order approximations tow at the nodestⁿ, n= 1, . . . , N. The idea of the proof is based on splitting the errorWⁿ−w(tⁿ) as

Wⁿ−w(tⁿ) = Wⁿ−W˜ⁿ

+ W˜ⁿ−u_t(tⁿ)

+ (u_t−w) (tⁿ) (4.24)

where the{W˜ⁿ}^N_n=0 denote the Crank-Nicolson approximations which correspond to problem (4.12),

⎧⎪

⎪⎨

⎪⎪

⎩

∂¯W˜ⁿ−iαΔW˜ⁿ⁻¹² + ig(tⁿ⁻¹²) ˜Wⁿ⁻¹²

=f_t(tⁿ⁻¹²)−ig_t(tⁿ⁻¹²)u(tⁿ⁻¹²), n= 1, . . . , N, W˜⁰=W⁰ in ¯Ω.

(4.25)

Proposition 4.1. Letwbe the (weak) solution of problem(4.11)andWⁿbe the Crank-Nicolson approximations of wat the nodestⁿ, n= 0,1, . . . , N,defined by the numerical scheme (4.13). Then, if

0≤n≤Nmax

u_t(tⁿ)−W˜ⁿ

L²(Ω)=O(k²), (4.26)

we have that

0≤n≤Nmax Wⁿ−w(tⁿ)_L²_(Ω)=O(k²).

Proof. We setZⁿ:=Wⁿ−W˜ⁿ, n= 0,1, . . . , N.Using the schemes (4.13) and (4.25) we have that theZⁿ, n= 0,1, . . . , N,satisfy the following numerical scheme

⎧⎨

⎩

∂Z¯ ⁿ−iαΔZⁿ⁻¹² + ig tⁿ⁻¹²

Zⁿ⁻¹² = ig_t tⁿ⁻¹²

eˆ tⁿ⁻¹²

, n= 1, . . . , N,

Z⁰= 0 in ¯Ω. (4.27)

Applying standard stability arguments in (4.27) we can easily conclude that

0≤n≤Nmax Zⁿ_L²_(Ω)≤C(Z⁰_L²_(Ω)+ max

0≤t≤Tˆe(t)_L²_(Ω)), (4.28)

where the constantC depends only on the final timeT andg_t. Accordingly,

0≤n≤Nmax

Wⁿ−W˜ⁿ

L²(Ω)=O(k²)

and the proof is complete in light of (4.21), (4.24) and (4.26).

Corollary 4.1. The residual Rˆ : [0, T]→L²(Ω) defined in (4.15)and corresponding to the reconstruction Wˆ is of second order of accuracy in time.

We recall that our goal is to provea posteriori error estimates of optimal (second) order of accuracy for the quantity sup_0≤t≤tnˆe_t(t)L²(Ω), n= 1, . . . , N.To achieve this, we split the error ˆe_tas

ˆe_t= (u_t−w) + w−Wˆ

+ Wˆ −Uˆ_t .

The quantityu_t−wis estimateda posteriori in theL^∞(L²)-normvia the optimal order estimate (4.21), while Wˆ −Uˆ_tis already ana posteriori quantity. For the quantityw−Wˆ we use the following:

Lemma 4.2. Let w be the (weak) solution of problem (4.11)andWˆ the corresponding Crank-Nicolson recon- struction. Then, for n = 1, . . . , N, the following a posteriori estimate is valid for the error w−Wˆ in the L^∞(L²)-norm

0≤t≤tmaxⁿ

w−Wˆ (t)

L²(Ω)≤ _tⁿ

0

R(t)ˆ

L²(Ω)dt, (4.29)

whereRˆ: [0, T]→L²(Ω)is the residual given by (4.15).

Using Lemmata4.1and4.2we can prove the following:

Theorem 4.2 (a posteriori estimate of optimal order for sup_0≤t≤Tˆe_t(t)_L²_(Ω)). With the notation of Theo- rem 3.1, the following a posteriorierror estimate holds, for n= 1, . . . , N,

0≤t≤tsupⁿˆe_t(t)_L²_(Ω)≤ _tⁿ

0 sup

x∈Ω|g_t(x, t)|dt _tⁿ

0 r(t)ˆ _L²_(Ω)dt +

_tⁿ

0 R(t)ˆ L²(Ω)dt+ sup

0≤t≤tⁿ

Wˆ(t)−Uˆ_t(t)

L²(Ω),

(4.30)

where the residuals ˆr and Rˆ are given by (2.8) and (4.16), respectively, andWˆ is the Crank-Nicolson recon- struction corresponding to problem (4.11).

We are now ready to formulate the basic theorem of this section.

Theorem 4.3 (a posteriori error estimate of optimal order in theL^∞(H¹)-norm). With the notation of the previous theorem, the following a posteriori error estimate holds, forn= 1, . . . , N,

0≤t≤tmaxⁿ∇ˆe(t)²_L2(Ω)≤ 2 α

1 +

_tⁿ

0 sup

x∈Ω|g(x, t)|dt

× _tⁿ

0 ˆr(t)_L²_(Ω)dt sup

0≤t≤tⁿˆe_t(t)_L²_(Ω),

(4.31)

where the quantitysup_0≤t≤tnˆe_t(t) is estimated by the optimal order estimate (4.30).

In Theorems4.2and4.3we claim that estimates (4.30) and (4.31) are of optimal second order. This is true if ˆW is a second order approximation in time to ˆU_t. We prove this in Lemma4.3 below, using a priori error analysis. However, we emphasize that estimates (4.30) and (4.31) are validindependently of Lemma4.3. Since Wˆ −Uˆ_tis ana posteriori quantity, the order of sup_0≤t≤T( ˆW−Uˆ_t)(t)_L²_(Ω)can always be checked numerically.

Lemma 4.3. Let Uˆ be the Crank-Nicolson reconstruction given in (2.6) and corresponding to problem (1.1) and letWˆ be the Crank-Nicolson reconstruction given in (4.14)and corresponding to problem (4.11). Then

0≤t≤Tsup

Wˆ −Uˆ_t (t)

L²(Ω)=O k²

.