Numerische Mathematik

Transcription

1 umer. Math. (5) 9:55 9 DOI.7/s umerische Mathematik A posteriori error control and adaptivity for Crank icolson finite element approximations for the linear Schrödinger equation Theodoros Katsaounis Irene Kyza Received: April 3 / Revised: 6 February 4 / Published online: 9 May 4 Springer-Verlag Berlin Heidelberg 4 Abstract We derive optimal order a posteriori error estimates for fully discrete approximations of linear Schrödinger-type equations, in the L (L )-norm. For the discretization in time we use the Crank icolson method, while for the space discretization we use finite element spaces that are allowed to change in time. The derivation of the estimators is based on a novel elliptic reconstruction that leads to estimates which reflect the physical properties of Schrödinger equations. The final estimates are obtained using energy techniques and residual-type estimators. Various numerical experiments for the one-dimensional linear Schrödinger equation in the semiclassical regime, verify and complement our theoretical results. The numerical implementations are performed with both uniform partitions and adaptivity in time and space. For adaptivity, we further develop and analyze an existing time-space adaptive algorithm to the cases of Schrödinger equations. The adaptive algorithm reduces the computa- The research of I.K. was supported by the European Social Fund (ESF)-European Union (EU) and ational Resources of the Greek State within the framework of the Action Supporting Postdoctoral Researchers of the Operational Programme Education and Lifelong Learning (EdLL). T.K. was partially supported by European Union FP7 program Capacities (Regpot 9-), through ACMAC ( T. Katsaounis Department of Mathematics and Applied Mathematics, University of Crete, 749 Heraklion-Crete, Greece thodoros@tem.uoc.gr T. Katsaounis I. Kyza Institute of Applied and Computational Mathematics, Foundation of Research and Technology-Hellas, ikolaou Plastira, Vassilika Vouton, 7 3, Heraklion-Crete, Greece I. Kyza (B) Division of Mathematics, University of Dundee, Dundee DD 4H, Scotland, UK ikyza@maths.dundee.ac.uk

2 56 T. Katsaounis, I. Kyza tional cost substantially and provides efficient error control for the solution and the observables of the problem, especially for small values of the Planck constant. Mathematics Subject Classification 65M5 65M6 35Q4 Introduction In this paper we focus on the a posteriori error control and adaptivity for fully discrete Crank icolson finite element (CFE) schemes for the general form of linear Schrödinger equation: t u iαδu + ig(x, t)u = f (x, t) in Ω (, T ], u = on Ω (, T ], (.) u(, ) = u in Ω, where Ω is a convex polygonal domain in R d, d 3, with boundary Ω, and < T <. In (.), α is a positive constant, g : Ω (, T ] R and f : Ω (, T ] Care given functions and u : Ω C is a given initial value. In the sequel we shall use the notation g(t) = g(, t) and f (t) = f (, t). A special case of (.) is the so-called linear Schrödinger equation in the semiclassical regime: t u i ε Δu + i V (x, t)u =, (.) ε with high frequency initial data. It is clear that (.) can be obtained from (.) by setting α := ε, g := ε V and f. In (.), ε ( <ε ) is the scaled Planck constant, V is a time-dependent potential that belongs in L ( (, T ) ), and u is the wave function. The wave function u is used to define primary physical quantities, called observables [,7], such as the position density, and the current density, (x, t) := u(x, t), (.3) J(x, t) := Im ( u(x, t) u(x, t) ). (.4) Problems related to (.) are of great interest in physics and engineering. However, the solution of (.) is complicated from the theoretical as well as the numerical analysis point of view. It is well known that for ε small (close to zero), the solution of (.) oscillates with wavelength O(ε), preventing u to converge strongly as ε. Because of this, standard numerical methods fail to correctly approximate u and the observables, unless very fine mesh sizes and time steps are used. In particular, previous works (cf., e.g., [,7,8]) suggested that for standard finite element (FE) methods there is a very restrictive dispersive relation connecting the mesh sizes (space and time) with parameter ε; cf., e.g., (4.9) below. This restrictive dispersive relation can

3 A posteriori error control and adaptivity for LS 57 be relaxed using the so-called time-splitting spectral methods, introduced earlier by Bao et al. in [], for the approximation of the solution of (.). In this paper, our goal is to show that constructing adaptive algorithms based on rigorous a posteriori error control leads to CFE schemes which are competitive to the best available methods for the approximation of the solution (and the observables) of the semiclassical Schrödinger equation (.), and in general of linear Schrödinger equations of the form (.). It also permits, for the first time, realistic computations for rough potentials for the linear Schrödinger equation in the semiclassical regime. To achieve our goal, in the current work we: Provide rigorous a posteriori error analysis for (.) for CFE approximations using FE spaces that are allowed to change in time; Study the advantages of adaptivity through the obtained estimators for the efficient error control of (.). Optimal order a posteriori error estimates for the heat equation for CFE schemes with FE spaces that are allowed to change in time have been derived very recently by Bänsch et al. [5]. However the extension of those ideas from the simple heat equation to the linear Schrödinger equation (.) is of increased difficulty due to the complexvalue and multiscale nature of the problem. Because of this, novel ideas and techniques are introduced. More precisely, our main contributions are: Derivation of optimal order a posteriori error bounds in the L (L )-norm for CFE schemes for (.). The term optimal order means that the derived a posteriori error estimator converges with the same order as the exact error in some norm. This is a significant property that an a posteriori estimator must enjoy so that when it is used will be possible to capture efficiently qualitative and quantitative characteristics of the exact solution. The (optimal) order of convergence for CFE schemes with finite elements of degree r, given by the a priori error analysis for problems with smooth data and verified numerically, is two in time and r + in space. Our analysis includes time-dependent potentials. This fact makes the problem more challenging since there are no rigorous results for Schrödinger equations for such potentials. In addition the existing literature on a posteriori error analysis for problems with time-dependent operators of the form A(t) := αδ + g(x, t) is quite limited. To the best of our knowledge, only in [8] the authors consider similar operators. Moreover the derived estimates hold for L (L )-type potentials as well, in contrast to the existing literature. In particular, existing results require smooth C (C )-type potentials. However, this regularity requirement on the potential is rather restrictive from applications point of view. Including L (L ) timedependent potentials in the analysis is important for another reason: It can be considered as the first step for the a posteriori error control of nonlinear Schrödinger (LS) equations. More precisely, the relaxation scheme introduced by Besse in [3] suggests that a posteriori error bounds for linear Schrödinger equations with L (L )-type time-dependent potentials is essential for the efficient approximation of the solution of certain LS equations. Introduction of a novel elliptic reconstruction leading to upper bounds that do not involve the global L (L )-norm of g, and thus, to bounds that do reflect the

4 58 T. Katsaounis, I. Kyza physical properties of the problem. The elliptic reconstruction was developed by Makridakis and ochetto in [5] to derive optimal order L (L ) a posteriori error bounds for FE spatial discrete schemes for the heat equation using energy techniques. A straightforward generalization of this notion of the elliptic reconstruction to Schrödinger equations leads to estimates that involve the L (L )-norm of the potential. Consequently, the obtained estimates are practically useless and adaptivity is inefficient, even in the simplest case of constant potentials. Therefore, proposing a modified elliptic reconstruction based on the physical properties of the problem under consideration is crucial for the efficient error control of (.) (and so (.)). Additionally, the new ideas developed for this purpose might be useful for other problems as well, such as convection-diffusion or reaction-diffusion problems. A detailed numerical study on the reliability and robustness of the a posteriori estimators through a time-space adaptive algorithm. Our starting point is the adaptive algorithm proposed in [9], adapted to the linear Schrödinger equation (.). The a posteriori estimators derived in this work are on the solution u of (.). However, in many applications observables like the position density (.3), or the current density (.4) are far more important than the solution itself. Thus, we introduce an appropriate modification of the a posteriori estimators and the adaptive algorithm. This modification is based on a heuristic idea and the results concerning the observables are very encouraging. Overall, the adaptive algorithm reduces the computational cost substantially and provides efficient error control of u and the observables for small values of the Planck constant ε. It is very difficult to obtain such results via standard techniques and without adaptivity. We point out that our purpose is not to prove convergence and optimality of the considered time-space adaptive algorithm, but rather to show that adaptivity based on rigorous a posteriori error control can be proven beneficial for the approximation of the solution (and the observables) of the linear semiclassical Schrödinger equation (.). In addition, it is to be emphasized that as long as the adaptive algorithm converges, we can guarantee rigorously, based on the a posteriori error analysis, that total error remains below a given tolerance. For parabolic problems, a number of adaptive algorithms exists in the literature; cf., e.g., [,3] and the references therein. However, convergence and optimality of timespace adaptive algorithms are very delicate and difficult issues. In the literature exists only one proven convergent time-space adaptive algorithm for evolution problems and finite element-type methods. This result follows the preliminary work on time-space adaptive algorithms of Chen and Feng, cf. [], it is due to Kreuzer, Möller, Schmidt and Siebert and can be found in [8]. The proposed algorithm is appropriate for the heat equation and backward Euler FE schemes and it is not clear how to generalize it to other problems and higher order in time methods. At this point it is also worth mentioning the paper by Schwab and Stevenson, cf. [34], in which they proposed an optimal, in terms of computationnal complexity, time-space adaptive algorithm for parabolic-type problems using wavelets. It is to be emphasized once more that the introduction of optimal and convergent time-space adaptive algorithms is a very hard and challenging problem, not only because through adaptivity the problem becomes

5 A posteriori error control and adaptivity for LS 59 highly nonlinear. In contrast to elliptic-type problems, the existence of time variable introduces additional challenges in the concept of adaptivity. For example it is not clear how with adaptive time steps it is possible to reach a given final time T and at the same time guarantee convergence (and in particular that the error will remain below a given tolerance in some norm) and optimal computational complexity of the adaptive algorithm. Despite the fact that problem (.) (and thus (.)) is linear, a posteriori error bounds and adaptive algorithms for linear Schrödinger equations are very limited in the literature. In particular, a posteriori error estimates in the L (L )-norm for fully discrete CFE schemes have been proven earlier by Dörfler in [4]; these estimates are first order accurate in time, thus not optimal. Using these estimates, Dörfler also proposes an adaptive algorithm in [4]. In [9] (see also[]), we considered only time-discrete approximations and we managed to prove optimal order a posteriori error estimates for (.)inthel (L ) and L (H )-norms. This was achieved using the Crank-icolson reconstruction proposed by Akrivis et al. []. Similar estimates for (.), using an alternative reconstruction, proposed by Lozinski et al. [4], can be found in []. To the best of our knowledge, optimal order a posteriori error estimates for fully discrete CFE schemes do not exist in the literature. Some preliminary results to that direction can be found in []. However, the a posteriori estimators derived in [] are scaled by the global L (L )-norm of g. Hence, as already mentioned, the derived estimators do not reflect the physical properties of the problem, which makes adaptivity through these estimates not reliable. A posteriori error estimates in the L (L )-norm have been proven earlier in []for uniform partitions and the time-splitting spectral methods for the linear Schrödinger equation in the semiclassical regime (.). In [], only the one-dimensional case in space is studied and the analysis, as in [], permits only time-independent potentials, without being obvious how the theory can be extended to time-dependent potentials. In addition, the time-spectral methods require smooth potentials; the particular analysis is not applicable for L (L )-type potentials. The analysis of the current paper is based on the introduction of appropriate timespace reconstructions. Such reconstructions for CFE methods and FE spaces that are allowed to change in time were introduced, for the first time, very recently, by Bänsch et al. [5], for the proof of optimal order a posteriori estimates in the L (L )-norm for the heat equation. To define those time-space reconstructions, the authors combined the idea of the elliptic reconstruction in [5] with the Crank icolson reconstruction of [,4]. The notion of the elliptic reconstruction has also been used earlier in [] and [6] for the derivation of optimal order a posteriori error estimates for backward Euler FE schemes for the heat and the wave equation, respectively. The reconstruction technique is a useful tool for deriving optimal order a posteriori error bounds; usually, this is not feasible via a direct comparison of the exact and the numerical solution; cf., e.g., [4,35]. In our context, time-space reconstructions can be defined through the novel elliptic reconstruction we introduce and the Crank icolson reconstruction of []. More precisely, the paper is organized as follows. In Sect., we introduce notation, the variational formulation of problem (.) and the fully discrete scheme. We propose the novel elliptic reconstruction and discuss its properties. With the aim of this new

6 6 T. Katsaounis, I. Kyza elliptic reconstruction, we then define appropriate time-space reconstructions. The main theoretical results are stated in Sect. 3, where the a posteriori analysis is developed and optimal order error bounds are derived using energy techniques, residual-type error estimators and the properties of the reconstructions. The two last sections are devoted to the numerical investigation of the efficiency of the estimators. In particular, in Sect. 4, we validate numerically the optimal order of convergence of the estimators using uniform partitions. For the linear Schrödinger equation in the semiclassical regime, we verify numerically that the estimators have the expected behavior with respect to the scaled parameter ε. Finally, in Sect. 5, we appropriately modify and apply to the one-dimensional semiclassical Schrödinger equation a time-space adaptive algorithm described in [,3] (see also[9]). We further develop the algorithm and we make it applicable for the approximation not only of the exact solution u but also for the observables, and we discuss in detail the benefits of adaptivity for equations of the form (.). Preliminaries. The continuous problem Problem (.) can be rewritten equivalently in variational form as { t u(t), υ +iα u(t), υ +i g(t)u(t), υ = f (t), υ, υ H (Ω), t [, T ], u(, ) = u in Ω, (.) where, denotes the L -inner product, or the H H duality pairing, depending on the context. We also denote by the norm in L (Ω). It is well known that, if g C ( [, T ]; C (Ω) ), f L ( [, T ]; L (Ω) ), f t L ( [, T ]; H (Ω) ), and u H ( ), then problem (.) admits a unique weak solution u C( [, T ]; H (Ω)) with u t C ( [, T ]; H (Ω) ) ; cf., e.g., [,3, pages 6 63]. We thus assume that the data of (.) have the necessary regularity to guarantee the existence of a unique weak solution of (.). We emphasize that the a posteriori error estimates derived in the sequel, remain valid for g L ( ) Ω (, T ) as well, provided that (.) is well-posed. In other words, in contrast to the existing analyses, ours includes rough potentials as well, under the knowledge of the well-posedness of (.). To avoid making the forthcoming analysis more technical, we further assume that g satisfies sup g(x, t) inf g(x, t), t [, T ]. (.) x Ω x Ω Condition (.) is not restrictive from applications point of view, as, in most applications, g denotes a nonnegative potential and thus (.) is automatically satisfied.

7 A posteriori error control and adaptivity for LS 6. The method We consider a partition =: t < t < < t := T of [, T ], and let I n := (t n, t n ] and k n := t n t n, n, denote the subintervals of [, T ] and the time steps, respectively. Let also k := max n k n. We discretize (.) by a Galerkin finite element method. To this end, we introduce a family {T n } n= of conforming shape-regular triangulations of Ω. We further assume that each triangulation T n, n, is a refinement of a macro-triangulation of Ω and that T n and T n are compatible. Two triangulations are said to be compatible if they are derived from the same macro-triangulation by an admissible refinement procedure. For precise definitions of these properties of the family {T n } n=, we refer to [3,]. ote that the triangulations are allowed to change arbitrarily from one step to another, provided they satisfy the aforementioned compatibility conditions. These conditions are minimal and allow for heavily graded meshes and adaptivity. Additionally, the forthcoming analysis is applicable without any quasiuniformity type assumptions on the mesh and without any restrictions on the sizes of neighboring elements of the triangulation. For an element K T n, we denote its boundary by K.Leth K be the diameter of K T n and h := max n max K Tn h K. Let also Σ n (K ) be the set of internal sides of K T n (points in d =, edges in d = and faces in d = 3) and define Σ n := K T n Σ n (K ). To any side e Σ n, we associate a unit vector n e on e and for x e and a function υ, we define [ ] J[ υ](x) := lim υ(x + δn e ) υ(x δn e ) n e. δ To each triangulation T n, we associate the finite element space V n, V n := {Φ n H (Ω) : K T n,φ n K P r }, where P r denotes the space of polynomials in d variables of degree at most r. With T n := T n T n we denote the finest common coarsening triangulation of T n and T n and by V n := V n V n its corresponding finite element space. Finally, let ˇΣ n := Σ n Σn, and for K T n, let ˇΣ n K := ˇΣ n K, where the element K T n is taken to be closed. Definition. (Discrete Laplacian) For n, the discrete version Δ n : V n V n of the Laplace operator Δ onto V n is defined as Δ n υ, Φ n = υ, Φ n, Φ n V n. (.3) We now discretize problem (.)by a modified Crank icolson Galerkin scheme, introduced earlier for the heat equation in [5]. Given an approximation U n V n to the exact solution at t n we define approximation U n V n to the exact solution u at the nodes t n, n, by the numerical method:

8 6 T. Katsaounis, I. Kyza U n Π n U n k n iα Π n Δ n U n + Δ n U n +ip n( ) g(t n )U n =P n f (t n ), (.4) for n, with U := P u in Ω.In(.4), t n := t n +t n, U n := U n +U n, and P n : L (Ω) V n, Π n : V n V n are appropriate projections or interpolants. In Sects. 4 and 5, where we discuss the numerical experiments, P n and Π n are taken to be the L -projection. However, the theory is still valid for other choices of P n and Π n (cf. [5,6]), and therefore we consider the method in this general setting. Another non-standard term appearing in (.4) isπ n Δ n U n instead of Δ n U n. As it was observed in [5,6], considering Δ n U n instead of Π n Δ n U n may lead to oscillatory behavior of the obtained a posteriori estimators. For this reason, we consider the modified scheme (.4) instead of the standard one..3 ovel elliptic reconstruction Residual-type estimators The elliptic reconstruction was originally introduced by Makridakis and ochetto in [5] for the proof of optimal order a posteriori error estimates in space in the L (L )- norm for evolution problems, using energy techniques. It was also one of the main tools in the a posteriori error analysis of the heat equation for Crank icolson fully discrete schemes; cf. [5]. For the linear Schrödinger equation (.), we introduce a new type of elliptic reconstruction which reflects the physical properties of the problem, and in particular the physical properties of the semilcassical Schrödinger equation (.). To this end, for n, we introduce the constant ḡ n := [ ] sup g(x, t n ) + inf g(x, t x Ω x Ω n ). (.5) with t :=. The main reason for the choice of (.5) is that the knowledge on how far from ḡ n is g in Ω gives qualitative information on the behavior of the exact solution, especially in the case of linear Schrödinger equation in the semiclassical regime. In order for the elliptic reconstruction we introduce below to be well defined, we need ḡ n, which is automatically satisfied due to (.). Definition. (ovel elliptic reconstruction) For fixed V n V n we define the elliptic reconstruction R n V n H (Ω) of V n to be the weak solution of the elliptic problem α R n V n, φ +ḡ n R n V n,φ = ( αδ n +ḡ n )V n,φ, φ H (Ω). (.6) As we shall see in the sequel, the above modified elliptic reconstruction will allow us to obtain qualitatively better a posteriori error estimators compared to those obtained using the standard elliptic reconstruction; cf., []. In fact, the sup x Ω g(x, t) that appears in the a posteriori error analysis using the standard elliptic reconstruction, can now be replaced, due to (.6), by quantities of the form sup x Ω g(x, t) ḡ n, t I n, leading to better constants. A very interesting question here, that needs further

9 A posteriori error control and adaptivity for LS 63 investigation, is whether the global constant sup x Ω g(x, t) ḡ n can be localized in each element. This will not only lead to better constants in the final a posteriori error estimators, but also might give the inspiration of proposing appropriate adaptive strategies. Using (.3), we see that R n satisfies the orthogonality property α (R n I)V n, Φ n +ḡn (R n I)V n,φ n =, Φn V n. (.7) Let now z H (Ω) be the weak solution of the following elliptic problem z, φ = (R n I)V n,φ, φ H (Ω), (.8) and let I n z be its Clément-type interpolant in V n (for the definition of the Clémenttype interpolant and its properties we refer to [4,,33]). Then we can prove the next auxiliary lemma. Lemma. Let z be the solution of (.8) and I n z its Clément-type interpolant. Then, for all V n V n, we have the following estimate for R n V n (R n I)V n Δ n V n, z I n z V n, (z I n z). (.9) Proof Using (.8), we obtain (R n I)V n = (R n I)V n, z, and thus, invoking the definitions of the modified elliptic reconstruction (.6) and of the discrete Laplacian (.3) wearriveat (R n I)V n = Δ n V n, z I n z V n, (z I n z) α ḡn (R n I)V n, z. Since both α and ḡ n are positive, (.9) follows by (R n I)V n, z = z ; cf. (.8). Since we use finite element spaces that are allowed to change from t n to t n, we will need to work with quantities of the form (R n I)V n (R n I)V n for V n V n and V n V n. To estimate such a quantity, we consider the elliptic problem ẑ, φ = (R n I)V n (R n I)V n,φ, φ H (Ω) with solution ẑ and we denote by Î n ẑ its Clément-type interpolant onto V n. Lemma. For V n V n and V n V n we have that (R n I)V n (R n I)V n Δ n V n, ẑ Î n ẑ V n, (ẑ Î n ẑ) + Δ n V n, ẑ Î n ẑ + V n, (ẑ Î n ẑ). (.)

10 64 T. Katsaounis, I. Kyza Proof The proof is similar to the proof of Lemma.. To estimate a posteriori the errors (R n I)V n and (R n I)V n (R n I)V n, we use residual-type error estimators. To this end, for a given V n V n, n, we define the following L elliptic estimator: ( η V n (V n ) := h K (Δ Δn )V n L (K ) + h 3 ) K J[ V n ] L. (.) ( K ) K T n In case d =, the term with the discontinuities in (.) vanishes. For V n V n and V n V n, n, we also define ( η V n (V n, V n ) := h [ K (Δ Δ n )V n (Δ Δ n ] )V n L (K ) K T n + h 3 ) K J[ V n V n ] L ( ˇΣ K n ). (.) In view of the definition of η V n and of (.9), the Lemma below is standard. Its proof is based on duality arguments and the elliptic regularity estimate for the Laplace operator. For details on the proof we refer, for example, to [,5]. Lemma.3 For all V n V n, n, it holds (R n I)V n Cη V n (V n ), (.3) where the constant C depends only on the domain Ω and the shape regularity of the family of triangulations. Similarly, by (.) the estimate (.4) in the next lemma holds. For a detailed proof, we refer to [5,]. Lemma.4 For V n V n and V n V n, n, we have (R n I)V n (R n I)V n Ĉη V n (V n, V n ), (.4) where the constant Ĉ depends only on the domain Ω, the shape regularity of the triangulations, and the number of refinement steps necessary to pass from T n to T n..4 Space and time-space reconstructions We first define the continuous, piecewise linear interpolant U :[, T ] H (Ω) between the nodal values U n V n and U n V n, i.e., U(t) := l n (t)u n + l n (t)u n, t I n, (.5)

11 A posteriori error control and adaptivity for LS 65 with l n (t) := t n t and l n k (t) := t t n, t I n. The space reconstruction of n k n U, thatwasusedin[] to obtain optimal order a posteriori error estimates for the backward Euler Galerkin fully discrete scheme is given via ω(t) := l n (t)rn U n + l n (t)rn U n, t I n. However, as the authors note in [,4] to obtain optimal order in time a posteriori error estimates for the Crank icolson method, a reconstruction in time is also needed. Here, with the aid of the new elliptic reconstruction (.6), we propose a two-point time-space reconstruction, appropriate for the linear Schrödinger equation (.) and the corresponding method (.4). Definition.3 (Time-space reconstruction) For n, we define the two-point time-space reconstruction Û : I n H (Ω) of the CFE scheme (.4) as Û(t) := R n U n + t t n ( R n Π n U n R n U n ) t iα R n (s) ds k n t n i t t n R n P n G U (s) ds + t t n R n P n F(s) ds, t I n, (.6) where G U (t) := g(t n )U n [ + (t t k n ) g(t n n )U n g(tn )U n ] (.7) and F(t) := f (t n ) + [ ] (t t k n ) f (t n n ) f (t n ), (.8) denote the linear interpolants of gu and f, respectively, at the nodes t n and t n, and (t) := l n (t)π n ( Δ n )U n + l n (t)( Δn )U n. (.9) In order to write compactly method (.4) and the reconstruction Û, we introduce the notation ( ) W (t) := iα + ip n G U P n F (t), t I n. (.) With this notation, the reconstruction Û is rewritten as Û(t) = R n U n + t t n ( R n Π n U n R n U n ) t R n W (s) ds, t I n, k n t n (.)

12 66 T. Katsaounis, I. Kyza and method (.4) as U n Π n U n k n + W (t n ) =, n. (.) ote that in each [t n, t n ], W is a linear polynomial between the values ( t n, W (t n ) ) and ( t n, W (t n ) ). Thus, it is straightforward to see that W (t) W (t n ) = (t t n ) t W (t), t I n. (.3) Proposition. For n, we denote by Û(t n + ) the limit of Û from above at t n. Then there holds Û(t + n ) = Rn U n and Û(t n ) = R n U n. In particular, Û is continuous in time on [, T ]. Furthermore, it satisfies t Û + iαr n + ir n P n G U = R n P n F + Rn Π n U n R n U n in I n. k n (.4) Proof That Û(t + n ) = Rn U n is obvious from the definition of Û. Moreover, Û(t n ) = R n Π n U n R n W (t) dt. I n Since W is a linear polynomial in time in I n, we have that I n W (t) dt = k n W (t n ) and that Û(t n ) = R n U n follows invoking (.). Finally, (.4) is an immediate consequence of differentiation in time of (.6). We conclude the section by computing the difference Û ω. For this, we introduce, for n, the notation W n [ ] := W (t k n ) W (t n ). (.5) n Lemma.5 (The difference Û ω) The difference Û ω satisfies (Û ω)(t) = (t n t)(t t n )R n W n, t In. (.6) Proof Using the definitions of Û and ω and the method in the form (.) we obtain

13 A posteriori error control and adaptivity for LS 67 t (Û ω)(t) = R n( ) W (t) W (t n ). Thus, using (.3) and the fact that t t n (s t n ) ds = (t t n )(t t n ), we obtain (Û ω)(t) = (t n t)(t t n )R n t W (t), t I n. (.7) Equality (.6) follows now from (.7), by noting that t W (t) = W n, t I n ; cf. (.5) and the definition (.) ofw (t). 3 A posteriori error estimates in the L (L )-norm 3. Main ideas In this section, we establish a posteriori error estimates in the L (L )-norm for problem (.), using the tools developed in the previous section. To this end, we denote by e := u U the error, where recall that U is the piecewise linear interpolant between the nodal values U n and U n ;cf.(.5). To achieve proving optimal order a posteriori error estimates in the L (L )-norm for (.) we split the error as e := ˆρ + σ + ɛ, with ˆρ := u Û, σ := Û ω and ɛ := ω U. We refer to ˆρ as the main error, to σ as the time-reconstruction error and to ɛ as the elliptic-reconstruction error. The term σ measures the error due to the reconstruction in time. This term is of optimal order in time, cf. (.6), but not yet an a posteriori quantity. It can be estimated a posteriori using the residual-type error estimators. The residual estimators will also be used for the direct estimation of the elliptic-reconstruction error. Finally, as we shall see, the main error ˆρ satisfies a perturbation of the original PDE and it will be bounded by the perturbed terms using energy techniques. The perturbed terms are either a posteriori quantities of optimal order, or can be estimated a posteriori by estimators of optimal order. These terms will include quantities that measure the time and space errors, the effect of mesh changes and the variation of the data f and g. We now proceed with the estimation of σ and ɛ in Propositions 3. and 3., respectively. Proposition 3. (Estimation of the time-reconstruction error) For m, the following estimate is valid for the time reconstruction error σ = Û ω: max σ(t) Em T, with Em T, t t m := max n m kn ] [ W n +CηV n ( W n ). 8 (3.) Proof We write R n W n = W n + (R n I) W n and the desirable result now follows using (.3) and (.6).

14 68 T. Katsaounis, I. Kyza Proposition 3. (Estimation of the elliptic error) For the elliptic error ɛ = ω U we have, for m : max ɛ(t) CEm S, with Em S, := max η V t t m n m n (U n ). (3.) Proof For t I n,ɛ(t) = l n (t)(rn I)U n + l n (t)(rn I)U n. Hence, { } ɛ(t) max (R n I)U n, (R n I)U n, t I n, from where we immediately conclude (3.), in view of (.3). 3. Estimation of the main error In view of (.4) we see that the reconstruction Û satisfies, for t I n, the equation t Û(t), φ +iα Û(t), φ +i g(t)û(t), φ = R(t), φ, φ H (Ω), (3.3) with R(t) := R n W (t)+ Rn Π n U n R n U n +i ( αδ+g(t) ) (ω+σ)(t), t I n. k n (3.4) Proposition 3.3 (Error equation for ˆρ) The main error ˆρ = u Û satisfies, for t I n, the equation t ˆρ(t), φ +iα ˆρ(t), φ +i g(t) ˆρ(t), φ = where the residuals R j, j 4, are given by 4 R j (t), φ, φ H (Ω), j= (3.5) R (t) := (R n I)W (t) Rn Π n U n R n U n k n +iαl n (t)(i Π n )Δ n U n, (3.6) R (t) := i ( (t n t)(t t n )[ αδ n + g(t) ) W n + ( ) ] g(t) ḡ n (R n I) W n, (3.7) [ R 3 (t) := i l n (t)( ) g(t) ḡ n (I R n )U n + l n (t)( ) g(t) ḡ n (I R n )U n], (3.8) and R 4 (t) := i ( P n G U (t) (gu)(t) ) + ( f (t) P n F(t) ). (3.9)

15 A posteriori error control and adaptivity for LS 69 Proof Subtracting (3.3) from (.) we obtain, for t I n, t ˆρ(t), φ +iα ˆρ(t), φ +i g(t) ˆρ(t), φ = f (t), φ R(t), φ, φ H (Ω). (3.) We further write ( αδ + g(t) ) ω(t) = l n (t)( αδ +ḡ n )R n U n + l n (t)( αδ +ḡ n)r n U n + l n (t)( g(t) ḡ n ) R n U n + l n (t)( g(t) ḡ n ) R n U n, where we recall that ω(t) = l n (t)rn U n +l n (t)rn U n, t I n. Thus (.6), (.9) yield ( ) αδ + g(t) ω(t), φ = α (t), φ +αl n (t) (Π n I)Δ n U n,φ + (gu)(t), φ +l n (t) ( ) g(t) ḡ n (R n I)U n,φ +l n (t) ( ) g(t) ḡ n (R n I)U n,φ. (3.) Similarly, in view of (.6), we obtain ( αδ + g(t) ) σ(t), φ = (t n t)(t t n ) ( αδ n + g(t) ) W n + ( g(t) ḡ n ) (R n I) W n,φ. (3.) Combining (3.), (3.4) with (3.), (3.) and using (.) wearriveat(3.5). ext, we prove the following auxiliary lemma. Lemma 3. The residual R in (3.6) can be rewritten as R (t) = (t t n )(R n I) W n (R n I)U n (R n I)U n + (I Π n ) (iαl n (t)δn U n + U n ), t I n. k n k n (3.3) Proof We just note, using the method in the form (.), that (R n I)W (t n ) Rn Π n U n R n U n k n = (I Rn )U n (I R n )U n + (I Π n ) U n. k n k n The result follows in light of (.3), because t W (t) = W n for t I n. Proposition 3.3 and Lemma 3. together with energy methods, lead to the following a posteriori estimation in the L (L )-norm for the main error ˆρ.

16 7 T. Katsaounis, I. Kyza Proposition 3.4 (Estimation of the main error) Let p n := sup Ω In g(x, t) ḡ n and p n,n := sup Ω In g(x, t) ḡ n, n. Then, for the main error ˆρ = u Û and m, it holds that max ˆρ(t) u R U +Em T, t t m where the time estimator E T, m is given by + C(ES, m + ES, m ) + ĈES,3 m + EC m + ED m, (3.4) E T, m := m t n n=t n + C m n= (t n t)(t t n ) ( αδ n + g(t) ) W n dt the space estimators Em S, j, j 3, are given by m Em S, := kn 4 η V n ( W n ), E S, m := m n= n= m and Em S,3 := n= k 3 n 4 p nη V n ( W n ), (3.5) k n k n η V n ( U n k n, U n k n and the coarsening and data estimators E C m and ED m are and m Em C := t n ( U (I Π n n ) k n= n t n E D m := m respectively. t n n=t n ( pn,n η V n (U n )+ p n η V n (U n ) ), ), (3.6) ) + iαl n (t)δn U n dt, (3.7) [ P n G U (t) (gu)(t) + f (t) P n F(t) ] dt, (3.8) Proof Setting φ =ˆρ in (3.5) and taking real parts yields 4 d dt ˆρ(t) = Re R j (t), ˆρ(t) j= 4 R j (t) ˆρ(t), t I n, j=

17 A posteriori error control and adaptivity for LS 7 or, Then, it is easily seen that max ˆρ(t) ˆρ() + t t m 4 t m j= R j (t) dt. (3.9) cf. (3.3), and t m R (t) dt E T, t m m, tm R (t) dt E S, m R 3 (t) dt E S, + ES,3 m + EC m ; (3.) m, tm R 4 (t) dt E D m ; (3.) cf. (3.7) (3.9). Going back to (3.9) and plugging in (3.) (3.) we readily obtain (3.4). Remark 3. (Optimal order of the estimators in (3.4)) It is clear that the space estimators Em S, j, j 3, are expected to be of optimal order of accuracy in space. In fact, estimator Em S, is expected to be of optimal order in space and of order one in time, i.e., it is a superconvergent term. As far as the first part of the time estimator is concerned, we note that E T, m t n t n (t n t)(t t n ) ( αδ n + g(t) ) W n dt k3 n sup ( αδ n n + g(t) ) W t I n So, it is expected to be of optimal order of accuracy in time. umerically, this term can be computed by invoking a quadrature in time, which is at least second order accurate (i.e., at least as accurate as the accuracy of the discretization method in time). The second part of Em T, is expected to be of optimal order in both time and space. On the other hand, note that estimator Em C is not identically zero, only during the coarsening procedure. Finally, for the estimators related to the data of the problem we have u R U u U +Cη V (U ) and P n G U (t) (gu)(t) (I P n )G U (t) + (G U gu)(t). The term f (t) P n F(t) is handled similarly. Thus, it is straightforward to see that Em D can be split into optimal order estimators in time and space, while u R U is easily estimated a posteriori via optimal order estimators in space. Remark 3. (The constants p n and p n,n ) For the constants p n we note that p n p n, + p n, with p n, := sup Ω In g(x, t) g(x, t n ) and p n, =.

18 7 T. Katsaounis, I. Kyza [ supx Ω g(x, t n ) inf x Ω g(x, t n ) ]. Therefore, p n, = O(k n ), while p n, is relatively small, provided that g does not change much, with respect to the spatial variable. More precisely, p n, when g is constant in space, while the estimators that are multiplied by p n in (3.4) vanish for constant potentials. This particular behavior of the estimators is natural from physical point of view. Similar comments can be made for constants p n,n. We conclude with the main theorem of the paper. Theorem 3. (A posteriori error estimate in the L (L )-norm) Let u be the exact solution of (.) and let U be the continuous approximation (.5) of u related to the modified Crank icolson Galerkin method (.4). Then, the following estimate is valid for m : max (u U)(t) u R U +Em T, + t t ET, m where E T, m and Em T,, Em S, m +C j= E S, j m +ĈE S,3 m +EC m + ED m, (3.), Em S, j, j 3, Em C, ED m are given by (3.5), (3.6), (3.7) and (3.8) are as in (3.) and (3.), respectively. Proof We write u U =ˆρ + σ + ɛ, whence, for m, max t t m (u U)(t) max t t m ˆρ(t) + max t t m σ(t) + max t t m ɛ(t). Estimate (3.) is now an immediate consequence of Propositions 3., 3. and umerical experiments: uniform partition In this section, we perform various numerical experiments for the one-dimensional linear semiclassical Schrödinger equation: t u i ε xxu + i V (x, t)u = in(a, b) (, T ], (4.) ε using uniform partitions. Our experiments, not only illustrate and complement our theoretical results, but also give important information in several other interesting aspects, like the behavior of the estimators with respect to the parameter ε. Atthe moment, the particular behavior can only be proven formally; cf. Sect. 4.. Inallof the numerical experiments, the initial data is of the well known semiclassical WKB form: u (x) = n (x)e i S (x) ε. (4.) In (4.), n and S are real and smooth functions on [a, b]. In addition, n is positive on (a, b) and decays to zero exponentially at the endpoints a and b. The modified Galerkin Crank icolson method (.4) and the corresponding a posteriori error estimators for problem (4.) (4.) with homogeneous Dirichlet boundary

19 A posteriori error control and adaptivity for LS 73 conditions, were implemented in a double precision C-code, using B-splines of degree r, r, as a basis for the finite element space V n, n, [7]. The involved projections Π n and P n in (.4) aretakentobethel -projection onto V n. In what follows, we present some characteristic examples that allow us to verify the correct order of convergence of the estimators in time and space, and their dependence on the Planck constant ε. We also report on the relation between the time and space mesh sizes with respect to ε in order to have convergence. 4. EOC of the estimators We proceed by studying two different cases. The first one concerns time-independent potentials, while in the second one we consider a time-dependent potential. Experiment (Time-independent potentials). Here, we consider three well-known types of potential: a constant potential, a harmonic oscillator and a double-well potential [5,6,3]. In all three examples, the Planck constant is taken to be of order. More precisely, we study the following cases: a. V (x) =, n (x) = e 5 x, S (x) = x, and ε = ; b. V (x) = x, n (x) = e 5(x.5), S (x) = + x, and ε =.5; c. V (x) = (x.5) = x 4 x + 6, n (x) = e 5 x, S (x) = 5 (e ln 5(x.5) + e 5(x.5)), and ε =.5. All computations are performed in [a, b] [, T ]=[, ] [, ]. Our purpose is to compute the experimental order of convergence (EOC) of the a posteriori error estimators at the final time T =. For this, we consider uniform partitions in both time and space. Let us denote by r the degree of B-splines used for the discretization in space and recall that the order of convergence for the CFE scheme is in time and r + in space. This motivates the relation between the mesh size h and the time step k. In particular, in each implementation, the relation between h and k is taken to be h k r+. (4.3) Let l count the different realizations (runs), h(l) the corresponding meshsize and M(l) = + [ ] b a h(l) where [ ] denotes the integral part of a real number. Then, for each space estimator E S, j, j 3, the EOC is computed as EOC := ( log log E S, j ) (l)/es, j (l + ) ( ), (4.4) M(l + )/M(l) where E S, j j (l) and ES, (l + ) denote the value of the estimators in two consecutive implementations with mesh sizes h(l) and h(l + ), respectively (recall that M(l) = + [ ] [ b a h(l) and M(l + ) = + b a ] h(l+) ). ote that E S, is expected to be of optimal order in space and of order in time, i.e., it is a superconvergent term. Therefore, the EOC we expect to observe is h r+ h r+ = h 3 (r+), due to (4.3) and (4.4). Similarly,

20 74 T. Katsaounis, I. Kyza for the time estimators E T, j, j, the EOC is computed as EOC := ( log E T, j log ) (l)/et, j (l + ) ( ). (4.5) k(l)/k(l + ) We are also interested in computing the effectivity index, defined as the ratio between the total a posteriori error estimator and the corresponding norm of the exact error. Since we do not have at our disposal the exact solution for the three examples, we compute a reference solution u ref instead, by taking very fine mesh and time steps. In particular, we take as kref = 4,96, while in space we discretize by B-splines of degree 5 and take as h ref =. Then, the reference error is defined as Eref := max u ref(t n ) U n. In addition, we define n E total := u U +η V (U ) + E T, + ET, + 3 j= E S, j + ED, and we compute the effectivity index ei as ei := E total /Eref. ote that for uniform partitions, the coarsening estimator E C is identically zero. Our findings are reported in Tables,, 3, 4, 5, 6: the EOC is presented with 5 digits of accuracy which was sufficient to observe the trend in the order of convergence. In the case of constant potential V (x) =, we discretize in space by linear B-splines. We recall that in this case E S, is identically zero and does not appear in Table. As we see in Tables and, all estimators decrease with the correct order. Table Space estimators E S, j, j =,, 3, and EOC for Experiment a M E S, EOC E S, EOC E S,3 EOC e 4.389e 6.493e,8.69e e e.9,56 3.5e e e , 7.884e e e ,4.97e e e Table Time estimators E T, j, j =,, and EOC, total estimator Etotal, reference error Eref, and effectivity index ei for Experiment a k E T, EOC E T, EOC Eref E total ei 6 3.8e e e e 6.44e e e e.678e.767,8 5.48e e e 4.54e.434,56.39e e e 3.4e.456

21 A posteriori error control and adaptivity for LS 75 Table 3 Space estimators E S, j, j 3, and EOC for Experiment b M E S, EOC E S, EOC E S, EOC E S,3 EOC 75.34e.569e.4735e e 3.87e e e e e e e e e e e e e e e e e e e e Table 4 Time estimators E T, j, j =,, and EOC, total estimator Etotal, reference error Eref, and effectivity index ei for Experiment b k E T, EOC E T, EOC Eref E total ei e 3.83e.4e e e e.64e e e e e e e e e 4.757,8.93e e e e ,56 5.e e e 4.53e Table 5 Space estimators E S, j, j 3, and EOC for Experiment c M E S, EOC E S, EOC E S, EOC E S,3 EOC 35.9e 3.4e e 3.978e e e e e e e e e e e e e e e e e e e e e Table 6 Time estimators E T, j, j =,, and EOC, total estimator Etotal, reference error Eref, and effectivity index ei for Experiment c k E T, EOC E T, EOC Eref E total ei 8.555e 3.6e.44e 7.85e e e e e e e e e e e e 4.49e , e e e 5.36e , e e e e

22 76 T. Katsaounis, I. Kyza We observe that the total error is mainly due to the time estimator E T,, while the effectivity index is around.4, i.e., the total estimator E total is very close to the reference error. However constant potentials are the simplest; actually, from physical point of view, having a constant potential is like having no potential at all. In Tables 3, 4 the results for the harmonic oscillator (b) are presented. We use quadratic B-splines for the discretization in space. The correct order of convergence is observed for all estimators. The dominant estimator for the harmonic oscillator is E S,3, while the effectivity index tends asymptotically to the constant value 4.5. Finally, for the double-well potential (c), we discretize in space by cubic B-splines. The results are listed in Tables 5, 6. For this example, the effectivity index seems to be asymptotically constant (around 47.8), but it is certainly larger compared to the previous two examples. This is maybe an indicator that the presented analysis can be improved, in order to end-up with better effectivity indices. Effectivity indices of this size were also observed in experiments for the two-dimensional heat equation, for backward Euler finite element schemes [] and for the corresponding to (.4) method [6]. Experiment (A time-dependent potential). In the second experiment we consider the time-dependent potential V (x, t) = ( + t) x. Such potentials were studied for example in [9,3]. In order to have an example where we can evaluate the exact error, instead of solving numerically problem (4.) (4.) with zero Dirichlet boundary conditions, we replace (4.) by t u i xxu + iv (x, t)u = f (x, t) (4.6) (for this experiment, ε = ). We consider as exact solution u(x, t) = e 5(x t) e i(+t)(+x) and we calculate f through (4.6). We take [a, b] [, T ]=[, ] [, ] and we perform the same computations as in Experiment. In space, we discretize by quadratic B-splines. The numerical results are reported in Tables 7 and 8. The correct order of convergence is observed for the estimators. The effectivity index tends asymptotically to a constant value, which is around 368, which is a strong indication that there might be room for improvement of the analysis. We point out Table 7 Space estimators E S, j, j 3, and EOC for Experiment M E S, EOC E S, EOC E S, EOC E S,3 EOC 75.39e 7.38e e.448e 3.864e e e e e e e e e e e e e e e e e e e e 4 3.,9 3.6e e e e 5 3., e e e e 6 3.

23 A posteriori error control and adaptivity for LS 77 Table 8 Time estimators E T, j, j =,, and EOC, total estimator Etotal, exact error Eex, and effectivity index ei for Experiment k E T, EOC E T, EOC Eex E total ei e e 6.655e e e e e 4 6.5e e e e 5.664e e e e e ,8.553e e e 6.73e , e e e 7.487e ,.5988e e e e , e e e 8.56e though, that no a posteriori error bounds of optimal order exist in the literature for time-dependent potentials and any numerical method. It is the first time that a complete a posteriori error analysis is provided and numerically verified for operators of the form i( Δ + V (x, t)). 4. ε-sensitivity of the estimators In the case of WKB initial data for the problem (4.) (4.) one can show that sup t T m u (t) =O( tm ε m ) and sup t T m u (t) =O( xm ε m ), m, provided n, S and V are regular enough; []. In that respect, and assuming that U n, n, are reasonably good approximations to u at the nodes t n, we expect the following behavior of the a posteriori error estimators with respect to the parameter ε: ( E S, h r+ ) ( = O ε r+, E S, h r+ = O k ε r+ ε E S,3 = O ( h r+ E T, = O ( k ε ε r+ ( + hr+ ) (, E S, h r+ ) = O ε r+, ), (4.7) ε r+ )), E T, = O ( k ε 3 ( + hr+ ε r+ )). (4.8) Relations (4.7) (4.8) give us an idea on how we have to choose the time and space steps so that the estimators converge. The suggested choice seems to be restrictive; however it is the expected one. Indeed the a priori error analysis for CFE schemes gives that ( h max u(t n) U n r+ ) k =O + n εr+ ε 3, (4.9)

24 78 T. Katsaounis, I. Kyza Table 9 Space estimators E S, j, j =,, 3, and time estimators ET, j, j =,, for ε =.5 k = h E S, E S, E S,3 E T, E T, e 3.36e e e e e.39e e e e 4.35e e 5.884e e.535e e e 4.48e 6.365e e e e e e 5.85e cf. [], and naturally, conditions (4.7) (4.8) were not expected to be more relaxed. ext, we verify numerically (4.7) (4.8). To this end, we consider n (x) = e 5(x.5), S (x) = 5 (e ln 5(x.5) + e 5(x.5)), and the constant potential V (x) =. We solve numerically problem (4.) (4.) in(a, b) (, T ]=(, ) (,.54], for ε =.5 and ε =., using B-splines of degree or 3. Since the potential is taken to be constant, estimator E S, is identically zero. The particular example has been considered earlier in [](seealso[8]) and it is interesting because caustics are formed before the final time T =.54. First, we consider the case ε =.5. We discretize by B-splines of degree and we consider uniform partitions in both time and space with k = h. The behavior of the space and time a posteriori error estimators are reported in Table 9. As (4.7) suggests, estimator E S, has the expected behavior for k = h 5 4, while E S,3 for h 4. Similar results, verifying (4.8), are observed for the time estimators E T, and ET,. In particular, note that for k 4, E T, is not reasonable, something we expect, provided that (4.8) is true and ε =.5. ote however, that estimator E S, behaves better than expected, since for h = it already decays with optimal order. ext, we consider the case ε =.. We discretize in space by cubic B-splines. To verify numerically (4.7), we take constant time step, k = 5 3. Then k ε = 5, k i.e., is neither significantly smaller nor significantly larger than one. With this ε choice of time step we will be able to see only the effect of the space discretization with respect to ε for E S,. As before, in Table, the stated relation (4.7) between h and ε is observed for E S, and E S,3. We also verify the corresponding relation between h and ε in (4.7) fore S,. Indeed, despite the fact that ES, is small, even for M = 6 (h = 5 3 ), it does not decay with optimal order. The correct behavior is initiated for M =,5 (h = 3 ), and verified for M = 3, (h = 3 ). For the time estimators, (4.8) is verified with constant mesh size h = 5 4 (M = 6,). Our choice of h is so that h4 is controlled, and ε allow us to exploit the behavior of k with respect to ε. Our findings 5 are shown in Table.