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Superconvergence analysis of finite element method for the time-dependent Schrödinger equation
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Superconvergence analysis of finite element method for the time-dependent Schrödinger equation Jianyun Wang ∗ , Yunqing Huang, Zhikun Tian, Jie Zhou School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan, China
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Article history: Received 2 July 2015 Received in revised form 6 March 2016 Accepted 12 March 2016 Available online xxxx
In this paper, we consider the two-dimensional time-dependent Schrödinger equation. Firstly, we use the rectangular Lagrange type finite element of order p to get a semidiscrete scheme of the equation and discuss the superconvergence error estimate in the H 1 norm. Secondly, we use the Crank–Nicolson method in time to get a fully discrete scheme of the equation, and the superconvergence estimate in the H 1 norm can be obtained in this scheme. Finally, a numerical example with the order p = 1 is provided to verify our theoretical results. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Schrödinger equation Finite element method Superconvergence
1. Introduction In this paper, we consider an initial boundary value problem of the two-dimensional time-dependent Schrödinger equation as the following
1 iut = − 1u + Vu + f ,
∀(x, y, t ) ∈ Ω × [0, T ],
u(x, y, t ) = 0, u(x, y, 0) = u0 (x, y),
∀(x, y, t ) ∈ ∂ Ω × [0, T ], ∀(x, y) ∈ Ω ,
2
(1.1)
where Ω ∈ R2 is a rectangular domain, functions u0 (x, y), f (x, y, t ) and unknown function u(x, y, t ) are complex-valued, the potential (real-valued) function V (x, y, t ) is non-negative, functions V (x, y, t ), Vt (x, y, t ) and Vtt (x, y, t ) are bounded for
all (x, y, t ) ∈ Ω × [0, T ], and 1u = ∂∂ x2u + ∂∂ y2u . The Schrödinger equation is widely used in quantum mechanics, optics, seismology, and plasma physics. Numerical methods and analysis for both the linear and nonlinear Schrödinger equation have been investigated extensively. For instance, the finite difference method (see [1,2]), the finite element method (see [3–10]), the spectral method (see [11,12]), the discontinuous Galerkin method (see [13,14]), the meshless local Petrov–Galerkin method (see [15]) and the two-grid method (see [16–18]). In [2], Delfour, Fortin and Payr described a finite-difference method to approximate a Schrödinger equation with a power non-linearity. In [4], Jin and Wu constructed and analyzed a fully discrete finite element scheme for the two-dimensional time-dependent Schrödinger equation on an infinite strip by using artificial boundary conditions. In [6], Kyza proved a posteriori error estimate of optimal order for linear Schrödinger type equations in the L∞ (L2 ) norm and the L∞ (H 1 ) norm. In [11], Feit, Fleck and Steiger described a spectral method to solve the linear Schrödinger equation. In [14], Xu and Shu developed a local discontinuous Galerkin method to solve the generalized nonlinear 2
∗
2
Corresponding author. E-mail addresses: [email protected] (J. Wang), [email protected] (Y. Huang).
http://dx.doi.org/10.1016/j.camwa.2016.03.015 0898-1221/© 2016 Elsevier Ltd. All rights reserved.
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Schrödinger equation and the coupled nonlinear Schrödinger equation. In [15], Dehghan and Mirzaei presented the meshless local Petrov–Galerkin (MLPG) method for the numerical solution of the two-dimensional non-linear Schrödinger equation. In [16], Jin, Shu and Xu firstly solved the coupled system of partial differential equations by the two-grid method and analyzed the convergence. Later in [18], Zhang, Jin and Wang extended this approach to the time-dependent Schrödinger equation. The semi-discrete schemes are proved to be convergent with an optimal convergence order and the full-discrete schemes are verified by a numerical example. In [17], Wu developed the two-grid mixed finite element schemes for solving both steady state and unsteady state nonlinear Schrödinger equations, where the schemes were based on the mixed finite element method. Superconvergence analysis is a powerful tool to improve the approximation accuracy and the efficiency of the finite element method. There are numerous and extensive studies on superconvergence analysis (see [8,9,19–28]). In [22], Douglas firstly proposed the research on superconvergence of the rectangular finite element in 1972. In [8], Lin and Liu considered a kind of initial boundary value problem of the Schrödinger equation and presented superconvergence estimates in semi-discrete and fully discrete schemes. In [9], Shi, Wang and Zhao applied the simplest anisotropic linear triangular finite element to solve the nonlinear Schrödinger equation, and provided the error estimate and superconvergence analysis. In [21], Chen and Huang detailed and summarized various theories of the finite element high accuracy research and superconvergence analysis. In [23], Huang, Li and Lin considered the time-dependent Maxwell’s equations modeling wave propagation in metamaterials. One-order higher global superclose results in the L2 norm are proved for several semi-discrete and fully discrete schemes. Furthermore, L∞ superconvergence at element centers is proved for the lowest order rectangular edge element. In [28], Yan gave an essentially self-contained presentation of the mathematical theory underlying the global superconvergence analysis and the recovery type posteriori error estimates. In this paper, we extend the idea in [8] to the case of a more general time-dependent linear Schrödinger equation (1.1) with the potential function V (x, y, t ). Furthermore, we generalize the rectangular linear finite element to the rectangular finite element of order p, and a numerical example with the order p = 1 is also provided. We derive L2 - and H 1 -error estimates in the fully discrete scheme compare with [18]. The superconvergence results are obtained both in the semi-discrete scheme and the fully discrete scheme by the theory of interpolation error estimation, where the main difficulty to us is the analysis of the H 1 -error in the fully discrete scheme. The paper is organized as follows. In Section 2, we present a finite element semi-discrete scheme with the rectangular Lagrange type finite element of order p. Then, we obtain the error estimate in the L2 norm and the superconvergence estimate in the H 1 norm with the order O(hp+1 ), respectively. In Section 3, we present a Crank–Nicolson finite element fully discrete scheme. We obtain the error estimate in the L2 norm with the order O(hp+1 + τ 2 ) and the superconvergence estimate in the 3
H 1 norm with the order O(hp+1 + τ 2 ). In Section 4, a numerical example with the order p = 1 is presented to demonstrate our theoretical analysis. Conclusions are given in the last section. Throughout this paper, the symbol C is used for a positive constant which may vary with the context but is independent of the mesh size h. 2. Superconvergence analysis for the semi-discrete finite element scheme Let ΩT = Ω × [0, T ]. For any complex-valued function u(x, y), v(x, y) ∈ L2 (Ω ), let (u, v) denote the inner product
(u, v) =
Ω
u(x, y)¯v (x, y)dx,
and ∥u∥ denotes the corresponding norm
∥ u∥ =
(u, u),
where v¯ denotes the complex conjugate of v . Then, we introduce the complex-valued function spaces H 1 (ΩT ) = {w(x, y, t )|w, wt , wx , wy ∈ L2 (ΩT )}, H01 (ΩT ) = {w(x, y, t )|w ∈ H 1 (ΩT ), w|∂ Ω = 0}, and for 1 ≤ p, q < ∞, integers k, l ≥ 0, W k,p (0, T ; W l,q (Ω )) = {w(x, y, t )| ∥w(x, y, t )∥W k,p (0,T ;W l,q (Ω )) < +∞}, with the norm
∥w(x, y, t )∥W k,p (0,T ;W l,q (Ω )) =
k s=0
0
T
s ∂ w(x, y, t ) p ∂ts
W l,q (Ω )
1p .
To simplify the notation, we denote the Sobolev space W l,2 by H l . Then the weak solution u(x, y, t ) of problem (1.1) is defined as follows: for all t ∈ [0, T ], find u(x, y, t ) ∈ H01 (ΩT ) such that
i(ut , v) = a(u, v) + (f , v), u(x, y, 0) = u0 (x, y),
where a(u, v) =
1 2
∀v ∈ H01 (ΩT ), t > 0, ∀(x, y) ∈ Ω ,
(∇ u, ∇v) + (Vu, v), and ∇ u = ( ∂∂ux , ∂∂ uy ).
(2.1)
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Assume that Th be a quasi-uniform rectangular partition of Ω with mesh size h > 0, each element K be a rectangle and
Ω=
K.
K ∈Th
Let the Lagrange type finite element space of order p V h,p = {v ∈ C (ΩT ) : v|K ∈ Qp , ∀K ∈ Th }, where Qp = span{xi yj , 0 ≤ i, j ≤ p}. Moreover h,p
V0
= V h,p
H01 (ΩT ).
Let uI ∈ V h,p be the standard Lagrange interpolating function of u when p = 1. We define uI ∈ V h,p to be the ‘‘point-edgeelement’’ interpolating function of u such that on the element K ∈ Th when p ≥ 2, uI (zi ) = u(zi ),
i = 1, 2, 3, 4,
(uI − u)v dl = 0,
∀v ∈ Pp−2 (li ), i = 1, 2, 3, 4,
(uI − u)v dxdy = 0,
∀v ∈ Qp−2 (K ),
li
K
where zi and li , i = 1, 2, 3, 4, are nodes and edges of the element K respectively, the symbol Pp−2 is the polynomial of order p − 2, and Qp−2 has been defined on the above. h,p
∈ H01 (ΩT ) be the conforming finite element space of order p on the rectangular mesh. Then we define the h,p semi-discrete finite element solution uh (x, y, t ) of problem (1.1) as follows: for all t ∈ [0, T ], find uh (x, y, t ) ∈ V0 Let V0
such that
i((uh )t , vh ) = a(uh , vh ) + (f , vh ), uh (x, y, 0) = uI0 (x, y), h ,p
where uI0 (x, y) ∈ V0
∀vh ∈ V0h,p , t > 0, ∀(x, y) ∈ Ω ,
(2.2)
is the interpolating function of u0 (x, y). h,p
Lemma 1 ([28]). Let u and uh be the solutions of (2.1) and (2.2), respectively, and let uI ∈ V0
be the interpolating function of u.
h ,p
Let V0 be the conforming finite element space of order p on the rectangular mesh. Assume that u ∈ H p+3 (Ω ) and ut ∈ H p+2 (Ω ) for all t ∈ [0, T ]. Then for all t ∈ [0, T ], there hold
|(∇(u − uI ), ∇v)| ≤ Chp+1 ∥u∥p+3 ∥v∥, |(u − u , v)| ≤ Ch I
p+1
∥u∥p+2 ∥v∥,
|((u − uI )t , v)| ≤ Chp+1 ∥ut ∥p+2 ∥v∥,
∀v ∈ V0h,p ,
(2.3)
∀v ∈
h,p V0
,
(2.4)
∀v ∈
h,p V0
.
(2.5)
Theorem 1. Assume that u and uh be the solutions of (2.1) and (2.2), respectively, and let uI be the corresponding interpolating function of u. If u, ut ∈ H p+3 (Ω ) and utt ∈ H p+2 (Ω ) for all t ∈ [0, T ], then we have the following error estimates
∥uI − uh ∥ ≤ Chp+1 , ∥(u − uh )t ∥ ≤ Ch I
p+1
(2.6)
.
(2.7)
Proof. From (2.1) and (2.2), we can get i((u − uh )t , vh ) = a((u − uh ), vh ),
∀vh ∈ V0h,p .
(2.8)
Let u − uh = ( u − uI ) + θ , with θ = uI − uh , then from (2.8), we have i(θt , vh ) =
1 2
1
(∇θ , ∇vh ) + (V θ , vh ) − i((u − uI )t , vh ) + (∇(u − uI ), ∇vh ) + (V (u − uI ), vh ). 2
(2.9)
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Taking vh = θ in (2.9), we can derive i(θt , θ ) =
1 2
1
(∇θ , ∇θ ) + (V θ , θ ) − i((u − uI )t , θ ) + (∇(u − uI ), ∇θ ) + (V (u − uI ), θ ). 2
(2.10)
Noticing that 1 d 2 dt
∥θ∥2 = Re{(θt , θ )},
comparing the imaginary parts of (2.10), we have
1 ∥θ∥2 = −Re{((u − uI )t , θ )} + Im (∇(u − uI ), ∇θ ) + (V (u − uI ), θ ) 2 dt 2 1 I I ≤ |((u − u )t , θ )| + (∇(u − u ), ∇θ ) + |(V (u − uI ), θ )|, 1 d
2
using (2.3)–(2.5), we can get d dt
∥θ ∥ ≤ Chp+1 (∥ut ∥p+2 + ∥u∥p+3 + ∥u∥p+2 ),
integrating from 0 to t, we obtain
∥θ∥ ≤ ∥θ (·, 0)∥ + Chp+1
t
(∥ut ∥p+2 + ∥u∥p+3 + ∥u∥p+2 )dt .
(2.11)
0
From (2.2), we get
θ (·, 0) = 0.
(2.12)
Therefore, (2.6) follows from (2.11) and (2.12). Next we prove (2.7). Taking vh = θt (·, 0) in (2.9) with t = 0, we have i(θt (·, 0), θt (·, 0)) = −i((u(·, 0) − uI (·, 0))t , θt (·, 0)) +
1
(∇(u(·, 0) − uI (·, 0)), ∇θt (·, 0))
2 + (V (·, 0)(u(·, 0) − uI (·, 0)), θt (·, 0)),
then using (2.3)–(2.5), we can get
∥θt (·, 0)∥2 ≤ Chp+1 (∥ut (·, 0)∥p+2 + ∥u(·, 0)∥p+3 + ∥u(·, 0)∥p+2 )∥θt (·, 0)∥, thus
∥θt (·, 0)∥ ≤ Chp+1 .
(2.13)
Differentiating (2.9) with respect to t, we have i(θtt , vh ) =
1 2
(∇θt , ∇vh ) + (Vt θ , vh ) + (V θt , vh ) − i((u − uI )tt , vh ) 1
+ (∇((u − uI )t ), ∇vh ) + (Vt (u − uI ), vh ) + (V (u − uI )t , vh ), 2
(2.14)
taking vh = θt in (2.14), we can derive i(θtt , θt ) =
1 2
(∇θt , ∇θt ) + (Vt θ , θt ) + (V θt , θt ) − i((u − uI )tt , θt ) 1
+ (∇((u − uI )t ), ∇θt ) + (Vt (u − uI ), θt ) + (V (u − uI )t , θt ). 2
Noticing that 1 d 2 dt
∥θt ∥2 = Re{(θtt , θt )},
comparing the imaginary parts of (2.15), we obtain 1 d 2 dt
1
∥θt ∥2 = Im{(Vt θ , θt )} − Re{((u − uI )tt , θt )} + Im{(∇((u − uI )t ), ∇θt )} 2
+ Im{(Vt (u − uI ), θt ) + (V (u − uI )t , θt )},
(2.15)
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using (2.3)–(2.5), we can get d dt
∥θt ∥2 ≤ C (∥θ∥ + hp+1 ∥utt ∥p+2 + hp+1 ∥ut ∥p+3 + hp+1 ∥u∥p+2 + hp+1 ∥ut ∥p+2 )∥θt ∥,
(2.16)
from (2.6) and (2.16), we have d dt
∥θt ∥ ≤ Chp+1 ,
integrating from 0 to t, we have
∥θt ∥ ≤ ∥θt (·, 0)∥ + Chp+1 .
(2.17)
Therefore, (2.7) follows from (2.13) and (2.17). Theorem 2. Assume that u and uh be the solutions of (2.1) and (2.2), respectively, and let uI be the corresponding interpolating function of u. If u ∈ H p+3 (Ω ) and ut ∈ H p+2 (Ω ) for all t ∈ [0, T ], then we have the following superconvergence estimate
∥uI − uh ∥1 ≤ Chp+1 .
(2.18)
Proof. Taking vh = θt in (2.9), we can get i(θt , θt ) =
1 2
1
(∇θ , ∇θt ) + (V θ , θt ) − i((u − uI )t , θt ) + (∇(u − uI ), ∇θt ) + (V (u − uI ), θt ).
(2.19)
2
Noticing that 1 d 2 dt
∥∇θ∥2 = Re{(∇θ , θt )},
comparing the real parts of (2.19), we have 1 d 2 dt
∥∇θ∥2 = −Re{2(V θ , θt ) + (∇(u − uI ), ∇θt ) + 2(V (u − uI ), θt )} − Im{2((u − uI )t , θt )},
using (2.3)–(2.5), we get d dt
∥∇θ∥2 ≤ C (∥θ ∥ + hp+1 ∥u∥p+3 + hp+1 ∥u∥p+2 + hp+1 ∥ut ∥p+2 )∥θt ∥,
(2.20)
from (2.6), (2.7) and (2.20), we get d dt
∥∇θ∥2 ≤ Ch2p+2 ,
integrating from 0 to t, we have
∥∇θ ∥2 ≤ ∥∇θ (·, 0)∥2 + Ch2p+2 ,
(2.21)
from (2.12) and (2.21), we can get
∥∇θ ∥2 ≤ Ch2p+2 , that is
∥∇θ ∥ ≤ Chp+1 , which completes the proof. 3. Superconvergence analysis for the fully discrete scheme In this section, we focus on the fully discrete Crank–Nicolson scheme and discuss the superconvergence error estimate. Let τ = NT be the mesh size of the interval [0, T ], tn = nτ , n = 0, 1, . . . , N be the time nodes, and tn− 1 = 21 (t n + t n−1 ). 2
We introduce some notations. 1
An− 2 (u, v) = An (u, v) =
1 2
1 2
(∇ u, ∇v) + V x, y, tn− 1 u, v , 2
(∇ u, ∇v) + (V (x, y, tn )u, v).
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For function w(x, y, t ) and function series un (x, y), n = 0, 1, . . . , let 1
1
∂t un− 2 (x, y) = 1
un− 2 (x, y) =
1 2
τ
[un (x, y) + un−1 (x, y)], 1
1
∂t w n− 2 (x, y) = 1
wn− 2 (x, y) =
[un (x, y) − un−1 (x, y)],
τ
1 2
[w(x, y, tn ) − w(x, y, tn−1 )],
[w(x, y, tn ) + w(x, y, tn−1 )]. h,p
Then we define the fully discrete finite element solution U n (x, y) ∈ V0 , n = 0, 1, . . . , N of problem (1.1) satisfying the Crank–Nicolson scheme
1 1 1 1 i ∂t U n− 2 , vh = An− 2 U n− 2 , vh + (f n− 2 , vh ),
∀vh ∈ V0h,p , n = 1, 2, . . . , N ,
U 0 (x, y) = uI0 (x, y).
(3.1)
Theorem 3. Let u(x, y, t ) be the solution defined in (2.1), function series U n (x, y) be the fully discrete finite element solution defined in (3.1), and let uI be the corresponding interpolating function of u, then we have the following error estimate
∥uI (·, tn ) − U n ∥ ≤ Chp+1 + C τ 2 .
(3.2)
Proof. Let u(x, y, tn ) − U n (x, y) = ρ n (x, y) + θ n (x, y) with
ρ n (x, y) = u(x, y, tn ) − uI (x, y, tn ), θ n (x, y) = uI (x, y, tn ) − U n (x, y). From (2.1) and (3.1), we can get
1
1
i ∂t θ n− 2 , vh + i ∂t ρ n− 2 , vh + i ut ·, tn− 1 2
1 1 1 1 − ∂t u ·, tn− 1 , vh = An− 2 θ n− 2 , vh + An− 2 ρ n− 2 , vh , 2
(3.3) taking vh = θ
i ∂t θ
n− 12
n− 12
in (3.3), we get 1
, θ n− 2
1 1 1 = −i ∂t ρ n− 2 , θ n− 2 − i ut ·, tn− 1 − ∂t u ·, tn− 1 , θ n− 2 2 2 1 1 1 1 1 1 + An− 2 θ n− 2 , θ n− 2 + An− 2 ρ n− 2 , θ n− 2 .
(3.4)
Noticing that 1 2τ
(∥θ n ∥2 − ∥θ n−1 ∥2 ) = Re
1 1 ∂t θ n− 2 , θ n− 2 ,
comparing the imaginary parts of (3.4), we get 1 2τ
1 1 1 ∂t ρ n− 2 , θ n− 2 + ut ·, tn− 1 − ∂t u ·, tn− 1 , θ n− 2 2 2 1 1 1 1 n− 2 n− 2 n− 2 n− 12 + Im ∇ρ , ∇θ + V ·, tn− 1 ρ ,θ ,
(∥θ n ∥2 − ∥θ n−1 ∥2 ) = −Re
2
2
thus 1 2τ
1 1 1 1 1 (∥θ n ∥2 − ∥θ n−1 ∥2 ) ≤ ξ1n , θ n− 2 + ∥ξ2n ∥ · θ n− 2 + ∇ξ3n , ∇θ n− 2 + ξ4n , θ n− 2 , 2
where 1
ξ1n = ∂t ρ n− 2 , 1
ξ3n = ρ n− 2 ,
ξ2n = ut ·, tn− 1 − ∂t u ·, tn− 1 , 2 2 n− 21 n ξ4 = V ·, tn− 1 ρ . 2
(3.5)
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Noticing that
|(ξ1n , v)| = ∂t u ·, tn− 1 − ∂t uI ·, tn− 1 , v 2
2
= |(τ −1 (u(·, tn ) − u(·, tn−1 )) − τ −1 (uI (·, tn ) − uI (·, tn−1 )), v)| = τ −1 |(u(·, tn ) − u(·, tn−1 ) − (uI (·, tn ) − uI (·, tn−1 )), v)|, using (2.5), we get
|(ξ1n , v)| ≤ C τ −1 hp+1 ∥u(·, tn ) − u(·, tn−1 )∥p+2 ∥v∥.
(3.6)
Further
∥ξ2n ∥ = ut ·, tn− 1 − ∂t u ·, tn− 1 2 2 tn 1 tn− 12 2 2 ( t − t ) u (·, t ) dt = ( t − t ) u (·, t ) dt + n ttt n−1 ttt 2τ tn−1 t 1 n− 2 tn ≤ Cτ ∥uttt (·, t )∥dt .
(3.7)
tn−1
And using (2.3), we have
|(∇ξ3n , ∇v)| = ∇ u ·, tn− 1 − uI ·, tn− 1 , ∇v 2
1
=
2
2
|(∇(u(·, tn ) + u(·, tn−1 ) − (uI (·, tn ) + uI (·, tn−1 ))), ∇v)|
≤ Chp+1 ∥u(·, tn ) + u(·, tn−1 )∥p+3 ∥v∥,
(3.8)
similarly, using (2.4), we obtain
|(ξ4n , v)| = V ·, tn− 1 u ·, tn− 1 − uI ·, tn− 1 , v 2
2
2
≤ C |(u(·, tn ) + u(·, tn−1 ) − (uI (·, tn ) + uI (·, tn−1 )), v)| ≤ Chp+1 ∥u(·, tn ) + u(·, tn−1 )∥p+2 ∥v∥.
(3.9)
1 n− 21 n n−1 = (θ + θ ) ≤ ∥θ n ∥ + ∥θ n−1 ∥, θ
(3.10)
Noticing that
2
from (3.5)–(3.10), we can get n
∥θ ∥ − ∥θ
n−1
p+1
∥ ≤ Ch
∥u(·, tn ) − u(·, tn−1 )∥p+2 + C τ
2
tn
∥uttt (·, t )∥dt
tn−1
+ C τ hp+1 ∥u(·, tn ) + u(·, tn−1 )∥p+3 + C τ hp+1 ∥u(·, tn ) + u(·, tn−1 )∥p+2 ,
(3.11)
summing up for n in (3.11), we have
∥θ n ∥ − ∥θ 0 ∥ ≤ Chp+1
n
j
T1 + C τ 2
j =1
n j =1
j
T2 + C τ hp+1
n j =1
j
T3 + C τ hp+1
n
j
T4 ,
(3.12)
j=1
where n
j
T1 =
j =1
n j =1 n
≤
∥u(·, tn ) − u(·, tn−1 )∥p+2
j =1 tn
=
0
tj
∥ut (·, t )∥p+2 dt
tj−1
∥ut (·, t )∥p+2 dt ,
(3.13)
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n
j
T2 =
j =1
j =1 tn
=
tj
)
–
∥uttt (·, t )∥dt
tj−1
∥uttt (·, t )∥dt ,
(3.14)
0 n
n
j
T3 =
j =1
∥u(·, tn ) + u(·, tn−1 )∥p+3
j =1 n
≤ 2
∥u(·, tj )∥p+3
j=0
≤ ∥u(·, 0)∥p+3 +
tn
(∥ut (·, t )∥p+3 + τ −1 ∥u(·, t )∥p+3 )dt ,
(3.15)
0 n
n
j
T4 =
j =1
∥u(·, tn ) + u(·, tn−1 )∥p+2
j =1 n
≤ 2
∥u(·, tj )∥p+2
j=0
≤ ∥u(·, 0)∥p+2 +
tn
(∥ut (·, t )∥p+2 + τ −1 ∥u(·, t )∥p+2 )dt .
(3.16)
0
From (3.1), we can see that
∥θ 0 ∥ = 0,
(3.17)
from (3.12)–(3.17), we get
∥θ n ∥ ≤ C (hp+1 + τ 2 + τ hp+1 ),
(3.18)
noticing that
τ hp+1 ≤ τ 2 + h2p+2 ,
(3.19)
from (3.18), (3.19) and noticing that 0 ≤ h ≤ 1, we have
∥θ n ∥ ≤ Chp+1 + C τ 2 , which completes the proof. Theorem 4. Let u(x, y, t ) be the solution defined in (2.1), function series U n (x, y) be the fully discrete finite element solution defined in (3.1), and let uI be the corresponding interpolating function of u, then we have the following superconvergence estimate 3
∥uI (·, tn ) − U n ∥1 ≤ Chp+1 + C τ 2 .
(3.20)
Proof. From (2.1) and (3.1), we can get
i ∂t θ
n− 12
n− 12 t
, vh + i ρ
1 1 1 1 , vh + i uIt ·, tn− 1 − ∂t uI ·, tn− 1 , vh = An− 2 θ n− 2 , vh + An− 2 ρ n− 2 , vh , 2
2
(3.21) noticing that
n− 12
i ρt
, vh
1 1 = An− 2 ρ n− 2 , vh ,
(3.22)
from (3.21) and (3.22), we have
1
i ∂t θ n− 2 , vh + i uIt ·, tn− 1 2
1 1 1 − ∂t uI ·, tn− 1 , vh = ∇θ n− 2 , ∇vh + V ·, tn− 1 θ n− 2 , vh , 2
2
1
taking vh = ∂t θ n− 2 in (3.23), we get
1 + i uIt ·, tn− 1 − ∂t uI ·, tn− 1 , ∂t θ n− 2 2 2 1 1 1 1 1 = ∇θ n− 2 , ∇∂t θ n− 2 + V ·, tn− 1 θ n− 2 , ∂t θ n− 2 , 1
1
i ∂t θ n− 2 , ∂t θ n− 2 2
2
2
(3.23)
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that is
1
1
i ∂t θ n− 2 , ∂t θ n− 2
i I ut ·, tn− 1 − ∂t uI ·, tn− 1 , θ n − θ n−1
+
τ
2
2
1 1 = V ·, tn− 1 (θ n + θ n−1 ), θ n − θ n−1 . (∇θ n + ∇θ n−1 , ∇θ n − ∇θ n−1 ) + 2 4τ 2τ
(3.24)
Noticing that
∥∇θ n ∥2 − ∥∇θ n−1 ∥2 = Re{(∇θ n + ∇θ n−1 , ∇θ n − ∇θ n−1 )}, comparing the real parts of (3.24), we get
∥∇θ n ∥2 − ∥∇θ n−1 ∥2 = Im 4 ∂t uI ·, tn− 1 − uIt ·, tn− 1 , θ n − θ n−1 2
2
− Re 2 V ·, tn− 1 (θ n + θ n−1 ), θ n − θ n−1 2
,
(3.25)
summing up for n in (3.25), we have
n ∥∇θ ∥ − ∥∇θ ∥ = Im 4 ∂t uI ·, tj− 1 − uIt ·, tj− 1 , θ j − θ j−1 n 2
0 2
2
j =1
−2
2
n V ·, tj− 1 (∥θ j ∥2 − ∥θ j−1 ∥2 ), j=1
(3.26)
2
from (3.17) and (3.26), we can get n 2
j
∥∇θ ∥ ≤ 8 max ∥θ ∥ j ≤n
n j =1
n j ∥η ∥ + 2 ∥η2 ∥ , j =1 j 1
(3.27)
where, n j=1
n I ∂t u ·, tj− 1 − uIt ·, tj− 1 2 2 j =1 tj n 1 tj− 21 2 I 2 I = (t − tj−1 ) uttt (·, t )dt + (t − tj ) uttt (·, t )dt 2τ j=1 tj−1 t 1 j− 2 n t j ∥uIttt (·, t )∥dt ≤ Cτ
∥η1j ∥ =
≤ Cτ
j=1tn
tj−1
∥uIttt (·, t )∥dt ,
(3.28)
0
n n j j 2 j −1 2 ∥η2 ∥ = V ·, tj− 1 (∥θ ∥ − ∥θ ∥ ) 2 j =1 j =1 n −1 n 2 0 2 j 2 = V ·, tn− 1 ∥θ ∥ − V ·, t 1 ∥θ ∥ + ∥θ ∥ V ·, tj+ 1 − V ·, tj− 1 2 2 2 2 j =1 n −1 ≤ V ·, tn− 1 ∥θ n ∥2 + max ∥θ j ∥2 V ·, tj+ 1 − V ·, tj− 1 j ≤n
2
≤ V ·, tn− 1 ∥θ n ∥2 + max ∥θ j ∥2 2
2
j =1
2
n −1
j ≤n
≤ C max ∥θ j ∥2 .
τ |Vt (·, tj )|
j =1
(3.29)
j ≤n
From (3.2) and (3.27)–(3.29), we can get
∥∇θ n ∥2 ≤ C (hp+1 + τ 2 )(hp+1 + τ 2 + τ ),
(3.30)
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Fig. 1. The log of errors at t = 0.01.
Fig. 2. The log of errors at t = 0.1.
that is 3
∥∇θ n ∥ ≤ Chp+1 + C τ 2 , which completes the proof. 4. Numerical examples In this section, we present a numerical example with p = 1 to confirm our theoretical analysis. We consider the following problem of the Schrödinger equation:
1 iut = − 1u + Vu + f ,
∀(x, y, t ) ∈ Ω × [0, T ],
u(x, y, t ) = 0, u(x, y, 0) = u0 (x, y),
∀(x, y, t ) ∈ ∂ Ω × [0, T ], ∀(x, y) ∈ Ω ,
2
where V (x, y, t ) = 1, Ω = [0, 1] × [0, 1], T = 1 and the function f (x, y, t ) is chosen corresponding to the exact solution u = et sin π x sin π y + iet sin π x sin π y. h,1
Ω is uniformly divided into families Th of quadrilaterals with mesh size h, and V0 is the bilinear rectangular finite element space defined on Th . We solved the above problem on uniformly refined rectangular meshes with time steps τ = 10−4 and various space steps h = 1/8, 1/16, 1/32, 1/64, respectively. Then, the convergence and superconvergence results with respect to different time are listed in Tables 1–4. Moreover, in Figs. 1–4, we show the convergence orders by slopes. From our numerical results, we
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Fig. 3. The log of errors at t = 0.5.
Fig. 4. The log of errors at t = 1.0.
Table 1 Numerical results at t = 0.01. Mesh h h h h
= 1/8 = 1/16 = 1/32 = 1/64
∥uI − U n ∥
Ratio
∥uI − U n ∥1
Ratio
1.2796E−04 3.2521E−05 8.3400E−06 2.2747E−06
/ 3.9347 3.8994 3.6664
5.7219E−04 1.4472E−04 3.7068E−05 1.0107E−05
/ 3.9540 3.9042 3.6676
∥uI − U n ∥
Ratio
∥uI − U n ∥1
Ratio
1.3387E−03 3.4023E−04 8.7252E−05 2.3797E−05
/ 3.9347 3.8994 3.6665
5.9863E−03 1.5140E−03 3.8781E−04 1.0574E−04
/ 3.9540 3.9040 3.6676
Table 2 Numerical results at t = 0.1. Mesh h h h h
= 1/8 = 1/16 = 1/32 = 1/64
can see that ∥uI − U n ∥ and ∥uI − U n ∥1 are convergent in terms of h at rate of O(h2 ), which coincide with our theoretical analysis in the paper.
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Table 3 Numerical results at t = 0.5. Mesh h h h h
= 1/8 = 1/16 = 1/32 = 1/64
∥uI − U n ∥
Ratio
∥uI − U n ∥1
Ratio
8.2490E−03 2.0964E−03 5.3762E−04 1.4663E−04
/ 3.9348 3.8994 3.6665
3.6909E−02 9.3294E−03 2.3895E−03 6.5153E−04
/ 3.9562 3.9043 3.6675
∥uI − U n ∥
Ratio
∥uI − U n ∥1
Ratio
2.1829E−02 5.5466E−03 1.4224E−03 3.8795E−04
/ 3.9356 3.8995 3.6666
9.8048E−02 2.4690E−02 6.3223E−03 1.7238E−03
/ 3.9712 3.9052 3.6677
Table 4 Numerical results at t = 1.0. Mesh h h h h
= 1/8 = 1/16 = 1/32 = 1/64
5. Conclusions In this paper, we considered the two-dimensional time-dependent Schrödinger equation with the finite element method. We presented a finite element semi-discrete scheme and a Crank–Nicolson finite element fully discrete scheme in rectangular Lagrange type finite element space of order p. We also provided the error analysis and obtained the superconvergence results in the H 1 norm of the semi-discrete scheme and the fully discrete scheme. A numerical example with the order p = 1 is provided to partly verify our theoretical results. Acknowledgments The authors thank the referees for their valuable suggestions which have helped to improve the presentation of this paper. Wang’s research was supported by Hunan Provincial Innovation Foundation for Postgraduate (Grant No. CX2014B252). Huang’s research was supported by the NSFC Key Project (Grant No. 91430213). Tian’s research was supported by the Project of Scientific Research Fund of Hunan Provincial Education Department (Grant No. 14B044). Zhou’s research was supported by the National Natural Science Foundation of China (Grant No. 11501485). References [1] W. Bao, Y. Cai, Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator, SIAM J. Numer. Anal. 50 (2012) 492–521. [2] M. Delfour, M. Fortin, G. Payr, Finite-difference solutions of a non-linear Schrödinger equation, J. Comput. Phys. 44 (1981) 277–288. [3] G.D. Akrivis, V.A. Dougalis, O.A. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math. 59 (1991) 31–53. [4] J. Jin, X. Wu, Convergence of a finite element scheme for the two-dimensional time-dependent Schrödinger equation in a long strip, J. Comput. Appl. Math. 234 (2010) 777–793. [5] O.A. Karakashian, C. Makridakis, A space–time finite element method for the nonlinear Schrödinger equation: the continuous Galerkin method, SIAM J. Numer. Anal. 36 (1999) 1779–1807. [6] I. Kyza, A posteriori error analysis for the Crank–Nicolson method for linear Schrödinger equations, ESAIM: Math. Model. Numer. Anal. 45 (2011) 761–778. [7] H.Y. Lee, Fully discrete methods for the nonlinear Schrödinger equation, Comput. Math. Appl. 28 (1994) 9–24. [8] Q. Lin, X. Liu, Global superconvergence estimates of finite element method for Schrödinger equation, J. Comput. Math. 6 (1998) 521–526. [9] D. Shi, P. Wang, Y. Zhao, Superconvergence analysis of anisotropic linear triangular finite element for nonlinear Schrödinger equation, Appl. Math. Lett. 38 (2014) 129–134. [10] J. Wang, A new error analysis of Crank–Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation, J. Sci. Comput. 60 (2014) 390–407. [11] M.D. Feit, J.A. Fleck, A. Steiger, Solution of the Schrödinger equation by a spectral method, J. Comput. Phys. 47 (1982) 412–433. [12] D. Pathria, J.L. Morris, Pseudo-spectral solution of nonlinear Schrödinger equations, J. Comput. Phys. 87 (1990) 108–125. [13] W. Lu, Y. Huang, H. Liu, Mass preserving discontinuous Galerkin methods for Schrödinger equations, J. Comput. Phys. 282 (2015) 210–226. [14] Y. Xu, C.W. Shu, Local discontinuous Galerkin methods for nonlinear Schrödinger equations, J. Comput. Phys. 205 (2005) 72–97. [15] M. Dehghan, D. Mirzaei, The meshless local Petrov–Galerkin (MLPG) method for the generalized two-dimensional non-linear Schrödinger equation, Eng. Anal. Bound. Elem. 32 (2008) 747–756. [16] J. Jin, S. Shu, J. Xu, a two-grid discretization method for decoupling systems of partial differential equations, Math. Comp. 75 (2006) 1617–1626. [17] L. Wu, Two-grid mixed finite-element methods for nonlinear Schrödinger equations, Numer. Methods Partial Differential Equations 28 (2012) 63–73. [18] H. Zhang, J. Jin, J. Wang, Two-grid finite-element method for the two-dimensional time-dependent Schrödinger equation, Adv. Appl. Math. Mech. 5 (2013) 180–193. [19] D.N. Arnold, J. Douglas, V. Thomee, Superconvergence of a finite element approximation to the solution of a Sobolev equation in a single space variable, Math. Comp. 36 (1981) 53–63. [20] C. Chen, S. Hu, The highest order superconvergence for bi-k degree rectangular elements at nodes: A proof of 2k-conjecture, Math. Comp. 82 (2013) 1337–1355. [21] C. Chen, Y. Huang, High Accuracy Theory of Finite Element Methods, Hunan Science Press, Changsha, 1995, (in Chinese). [22] J. Douglas, A superconvergence result for the approximate solution of the heat equation by a collocation method, in: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Academic Press, New York, 1972, pp. 475–490. [23] Y. Huang, J. Li, Q. Lin, Superconvergence analysis for time-dependent Maxwell’s equations in metamaterials, Numer. Methods Partial Differential Equations 28 (2012) 1794–1816.
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Y. Huang, W. Yang, N. Yi, Superconvergence analysis for the explicit polynomial recovery method, J. Comput. Appl. Math. 265 (2014) 187–198. R. Lin, Z. Zhang, Natural superconvergence points in three-dimensional finite elements, SIAM J. Numer. Anal. 46 (2008) 1281–1297. Q. Lin, J. Zhou, Superconvergence in high-order Galerkin finite element methods, Comput. Methods Appl. Mech. Engrg. 196 (2007) 3779–3784. L.B. Wahlbin, Superconvergence in Galerkin Finite Element Methods, Springer, Berlin, 1995. N. Yan, Superconvergence Analysis and a Posteriori Error Estimation in Finite Element Methods, Science Press, Beijing, 2008.
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Contents lists available at ScienceDirect
Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Superconvergence analysis of finite element method for the time-dependent Schrödinger equation Jianyun Wang ∗ , Yunqing Huang, Zhikun Tian, Jie Zhou School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan, China
article
abstract
info
Article history: Received 2 July 2015 Received in revised form 6 March 2016 Accepted 12 March 2016 Available online xxxx
In this paper, we consider the two-dimensional time-dependent Schrödinger equation. Firstly, we use the rectangular Lagrange type finite element of order p to get a semidiscrete scheme of the equation and discuss the superconvergence error estimate in the H 1 norm. Secondly, we use the Crank–Nicolson method in time to get a fully discrete scheme of the equation, and the superconvergence estimate in the H 1 norm can be obtained in this scheme. Finally, a numerical example with the order p = 1 is provided to verify our theoretical results. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Schrödinger equation Finite element method Superconvergence
1. Introduction In this paper, we consider an initial boundary value problem of the two-dimensional time-dependent Schrödinger equation as the following
1 iut = − 1u + Vu + f ,
∀(x, y, t ) ∈ Ω × [0, T ],
u(x, y, t ) = 0, u(x, y, 0) = u0 (x, y),
∀(x, y, t ) ∈ ∂ Ω × [0, T ], ∀(x, y) ∈ Ω ,
2
(1.1)
where Ω ∈ R2 is a rectangular domain, functions u0 (x, y), f (x, y, t ) and unknown function u(x, y, t ) are complex-valued, the potential (real-valued) function V (x, y, t ) is non-negative, functions V (x, y, t ), Vt (x, y, t ) and Vtt (x, y, t ) are bounded for
all (x, y, t ) ∈ Ω × [0, T ], and 1u = ∂∂ x2u + ∂∂ y2u . The Schrödinger equation is widely used in quantum mechanics, optics, seismology, and plasma physics. Numerical methods and analysis for both the linear and nonlinear Schrödinger equation have been investigated extensively. For instance, the finite difference method (see [1,2]), the finite element method (see [3–10]), the spectral method (see [11,12]), the discontinuous Galerkin method (see [13,14]), the meshless local Petrov–Galerkin method (see [15]) and the two-grid method (see [16–18]). In [2], Delfour, Fortin and Payr described a finite-difference method to approximate a Schrödinger equation with a power non-linearity. In [4], Jin and Wu constructed and analyzed a fully discrete finite element scheme for the two-dimensional time-dependent Schrödinger equation on an infinite strip by using artificial boundary conditions. In [6], Kyza proved a posteriori error estimate of optimal order for linear Schrödinger type equations in the L∞ (L2 ) norm and the L∞ (H 1 ) norm. In [11], Feit, Fleck and Steiger described a spectral method to solve the linear Schrödinger equation. In [14], Xu and Shu developed a local discontinuous Galerkin method to solve the generalized nonlinear 2
∗
2
Corresponding author. E-mail addresses: [email protected] (J. Wang), [email protected] (Y. Huang).
http://dx.doi.org/10.1016/j.camwa.2016.03.015 0898-1221/© 2016 Elsevier Ltd. All rights reserved.
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Schrödinger equation and the coupled nonlinear Schrödinger equation. In [15], Dehghan and Mirzaei presented the meshless local Petrov–Galerkin (MLPG) method for the numerical solution of the two-dimensional non-linear Schrödinger equation. In [16], Jin, Shu and Xu firstly solved the coupled system of partial differential equations by the two-grid method and analyzed the convergence. Later in [18], Zhang, Jin and Wang extended this approach to the time-dependent Schrödinger equation. The semi-discrete schemes are proved to be convergent with an optimal convergence order and the full-discrete schemes are verified by a numerical example. In [17], Wu developed the two-grid mixed finite element schemes for solving both steady state and unsteady state nonlinear Schrödinger equations, where the schemes were based on the mixed finite element method. Superconvergence analysis is a powerful tool to improve the approximation accuracy and the efficiency of the finite element method. There are numerous and extensive studies on superconvergence analysis (see [8,9,19–28]). In [22], Douglas firstly proposed the research on superconvergence of the rectangular finite element in 1972. In [8], Lin and Liu considered a kind of initial boundary value problem of the Schrödinger equation and presented superconvergence estimates in semi-discrete and fully discrete schemes. In [9], Shi, Wang and Zhao applied the simplest anisotropic linear triangular finite element to solve the nonlinear Schrödinger equation, and provided the error estimate and superconvergence analysis. In [21], Chen and Huang detailed and summarized various theories of the finite element high accuracy research and superconvergence analysis. In [23], Huang, Li and Lin considered the time-dependent Maxwell’s equations modeling wave propagation in metamaterials. One-order higher global superclose results in the L2 norm are proved for several semi-discrete and fully discrete schemes. Furthermore, L∞ superconvergence at element centers is proved for the lowest order rectangular edge element. In [28], Yan gave an essentially self-contained presentation of the mathematical theory underlying the global superconvergence analysis and the recovery type posteriori error estimates. In this paper, we extend the idea in [8] to the case of a more general time-dependent linear Schrödinger equation (1.1) with the potential function V (x, y, t ). Furthermore, we generalize the rectangular linear finite element to the rectangular finite element of order p, and a numerical example with the order p = 1 is also provided. We derive L2 - and H 1 -error estimates in the fully discrete scheme compare with [18]. The superconvergence results are obtained both in the semi-discrete scheme and the fully discrete scheme by the theory of interpolation error estimation, where the main difficulty to us is the analysis of the H 1 -error in the fully discrete scheme. The paper is organized as follows. In Section 2, we present a finite element semi-discrete scheme with the rectangular Lagrange type finite element of order p. Then, we obtain the error estimate in the L2 norm and the superconvergence estimate in the H 1 norm with the order O(hp+1 ), respectively. In Section 3, we present a Crank–Nicolson finite element fully discrete scheme. We obtain the error estimate in the L2 norm with the order O(hp+1 + τ 2 ) and the superconvergence estimate in the 3
H 1 norm with the order O(hp+1 + τ 2 ). In Section 4, a numerical example with the order p = 1 is presented to demonstrate our theoretical analysis. Conclusions are given in the last section. Throughout this paper, the symbol C is used for a positive constant which may vary with the context but is independent of the mesh size h. 2. Superconvergence analysis for the semi-discrete finite element scheme Let ΩT = Ω × [0, T ]. For any complex-valued function u(x, y), v(x, y) ∈ L2 (Ω ), let (u, v) denote the inner product
(u, v) =
Ω
u(x, y)¯v (x, y)dx,
and ∥u∥ denotes the corresponding norm
∥ u∥ =
(u, u),
where v¯ denotes the complex conjugate of v . Then, we introduce the complex-valued function spaces H 1 (ΩT ) = {w(x, y, t )|w, wt , wx , wy ∈ L2 (ΩT )}, H01 (ΩT ) = {w(x, y, t )|w ∈ H 1 (ΩT ), w|∂ Ω = 0}, and for 1 ≤ p, q < ∞, integers k, l ≥ 0, W k,p (0, T ; W l,q (Ω )) = {w(x, y, t )| ∥w(x, y, t )∥W k,p (0,T ;W l,q (Ω )) < +∞}, with the norm
∥w(x, y, t )∥W k,p (0,T ;W l,q (Ω )) =
k s=0
0
T
s ∂ w(x, y, t ) p ∂ts
W l,q (Ω )
1p .
To simplify the notation, we denote the Sobolev space W l,2 by H l . Then the weak solution u(x, y, t ) of problem (1.1) is defined as follows: for all t ∈ [0, T ], find u(x, y, t ) ∈ H01 (ΩT ) such that
i(ut , v) = a(u, v) + (f , v), u(x, y, 0) = u0 (x, y),
where a(u, v) =
1 2
∀v ∈ H01 (ΩT ), t > 0, ∀(x, y) ∈ Ω ,
(∇ u, ∇v) + (Vu, v), and ∇ u = ( ∂∂ux , ∂∂ uy ).
(2.1)
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Assume that Th be a quasi-uniform rectangular partition of Ω with mesh size h > 0, each element K be a rectangle and
Ω=
K.
K ∈Th
Let the Lagrange type finite element space of order p V h,p = {v ∈ C (ΩT ) : v|K ∈ Qp , ∀K ∈ Th }, where Qp = span{xi yj , 0 ≤ i, j ≤ p}. Moreover h,p
V0
= V h,p
H01 (ΩT ).
Let uI ∈ V h,p be the standard Lagrange interpolating function of u when p = 1. We define uI ∈ V h,p to be the ‘‘point-edgeelement’’ interpolating function of u such that on the element K ∈ Th when p ≥ 2, uI (zi ) = u(zi ),
i = 1, 2, 3, 4,
(uI − u)v dl = 0,
∀v ∈ Pp−2 (li ), i = 1, 2, 3, 4,
(uI − u)v dxdy = 0,
∀v ∈ Qp−2 (K ),
li
K
where zi and li , i = 1, 2, 3, 4, are nodes and edges of the element K respectively, the symbol Pp−2 is the polynomial of order p − 2, and Qp−2 has been defined on the above. h,p
∈ H01 (ΩT ) be the conforming finite element space of order p on the rectangular mesh. Then we define the h,p semi-discrete finite element solution uh (x, y, t ) of problem (1.1) as follows: for all t ∈ [0, T ], find uh (x, y, t ) ∈ V0 Let V0
such that
i((uh )t , vh ) = a(uh , vh ) + (f , vh ), uh (x, y, 0) = uI0 (x, y), h ,p
where uI0 (x, y) ∈ V0
∀vh ∈ V0h,p , t > 0, ∀(x, y) ∈ Ω ,
(2.2)
is the interpolating function of u0 (x, y). h,p
Lemma 1 ([28]). Let u and uh be the solutions of (2.1) and (2.2), respectively, and let uI ∈ V0
be the interpolating function of u.
h ,p
Let V0 be the conforming finite element space of order p on the rectangular mesh. Assume that u ∈ H p+3 (Ω ) and ut ∈ H p+2 (Ω ) for all t ∈ [0, T ]. Then for all t ∈ [0, T ], there hold
|(∇(u − uI ), ∇v)| ≤ Chp+1 ∥u∥p+3 ∥v∥, |(u − u , v)| ≤ Ch I
p+1
∥u∥p+2 ∥v∥,
|((u − uI )t , v)| ≤ Chp+1 ∥ut ∥p+2 ∥v∥,
∀v ∈ V0h,p ,
(2.3)
∀v ∈
h,p V0
,
(2.4)
∀v ∈
h,p V0
.
(2.5)
Theorem 1. Assume that u and uh be the solutions of (2.1) and (2.2), respectively, and let uI be the corresponding interpolating function of u. If u, ut ∈ H p+3 (Ω ) and utt ∈ H p+2 (Ω ) for all t ∈ [0, T ], then we have the following error estimates
∥uI − uh ∥ ≤ Chp+1 , ∥(u − uh )t ∥ ≤ Ch I
p+1
(2.6)
.
(2.7)
Proof. From (2.1) and (2.2), we can get i((u − uh )t , vh ) = a((u − uh ), vh ),
∀vh ∈ V0h,p .
(2.8)
Let u − uh = ( u − uI ) + θ , with θ = uI − uh , then from (2.8), we have i(θt , vh ) =
1 2
1
(∇θ , ∇vh ) + (V θ , vh ) − i((u − uI )t , vh ) + (∇(u − uI ), ∇vh ) + (V (u − uI ), vh ). 2
(2.9)
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Taking vh = θ in (2.9), we can derive i(θt , θ ) =
1 2
1
(∇θ , ∇θ ) + (V θ , θ ) − i((u − uI )t , θ ) + (∇(u − uI ), ∇θ ) + (V (u − uI ), θ ). 2
(2.10)
Noticing that 1 d 2 dt
∥θ∥2 = Re{(θt , θ )},
comparing the imaginary parts of (2.10), we have
1 ∥θ∥2 = −Re{((u − uI )t , θ )} + Im (∇(u − uI ), ∇θ ) + (V (u − uI ), θ ) 2 dt 2 1 I I ≤ |((u − u )t , θ )| + (∇(u − u ), ∇θ ) + |(V (u − uI ), θ )|, 1 d
2
using (2.3)–(2.5), we can get d dt
∥θ ∥ ≤ Chp+1 (∥ut ∥p+2 + ∥u∥p+3 + ∥u∥p+2 ),
integrating from 0 to t, we obtain
∥θ∥ ≤ ∥θ (·, 0)∥ + Chp+1
t
(∥ut ∥p+2 + ∥u∥p+3 + ∥u∥p+2 )dt .
(2.11)
0
From (2.2), we get
θ (·, 0) = 0.
(2.12)
Therefore, (2.6) follows from (2.11) and (2.12). Next we prove (2.7). Taking vh = θt (·, 0) in (2.9) with t = 0, we have i(θt (·, 0), θt (·, 0)) = −i((u(·, 0) − uI (·, 0))t , θt (·, 0)) +
1
(∇(u(·, 0) − uI (·, 0)), ∇θt (·, 0))
2 + (V (·, 0)(u(·, 0) − uI (·, 0)), θt (·, 0)),
then using (2.3)–(2.5), we can get
∥θt (·, 0)∥2 ≤ Chp+1 (∥ut (·, 0)∥p+2 + ∥u(·, 0)∥p+3 + ∥u(·, 0)∥p+2 )∥θt (·, 0)∥, thus
∥θt (·, 0)∥ ≤ Chp+1 .
(2.13)
Differentiating (2.9) with respect to t, we have i(θtt , vh ) =
1 2
(∇θt , ∇vh ) + (Vt θ , vh ) + (V θt , vh ) − i((u − uI )tt , vh ) 1
+ (∇((u − uI )t ), ∇vh ) + (Vt (u − uI ), vh ) + (V (u − uI )t , vh ), 2
(2.14)
taking vh = θt in (2.14), we can derive i(θtt , θt ) =
1 2
(∇θt , ∇θt ) + (Vt θ , θt ) + (V θt , θt ) − i((u − uI )tt , θt ) 1
+ (∇((u − uI )t ), ∇θt ) + (Vt (u − uI ), θt ) + (V (u − uI )t , θt ). 2
Noticing that 1 d 2 dt
∥θt ∥2 = Re{(θtt , θt )},
comparing the imaginary parts of (2.15), we obtain 1 d 2 dt
1
∥θt ∥2 = Im{(Vt θ , θt )} − Re{((u − uI )tt , θt )} + Im{(∇((u − uI )t ), ∇θt )} 2
+ Im{(Vt (u − uI ), θt ) + (V (u − uI )t , θt )},
(2.15)
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using (2.3)–(2.5), we can get d dt
∥θt ∥2 ≤ C (∥θ∥ + hp+1 ∥utt ∥p+2 + hp+1 ∥ut ∥p+3 + hp+1 ∥u∥p+2 + hp+1 ∥ut ∥p+2 )∥θt ∥,
(2.16)
from (2.6) and (2.16), we have d dt
∥θt ∥ ≤ Chp+1 ,
integrating from 0 to t, we have
∥θt ∥ ≤ ∥θt (·, 0)∥ + Chp+1 .
(2.17)
Therefore, (2.7) follows from (2.13) and (2.17). Theorem 2. Assume that u and uh be the solutions of (2.1) and (2.2), respectively, and let uI be the corresponding interpolating function of u. If u ∈ H p+3 (Ω ) and ut ∈ H p+2 (Ω ) for all t ∈ [0, T ], then we have the following superconvergence estimate
∥uI − uh ∥1 ≤ Chp+1 .
(2.18)
Proof. Taking vh = θt in (2.9), we can get i(θt , θt ) =
1 2
1
(∇θ , ∇θt ) + (V θ , θt ) − i((u − uI )t , θt ) + (∇(u − uI ), ∇θt ) + (V (u − uI ), θt ).
(2.19)
2
Noticing that 1 d 2 dt
∥∇θ∥2 = Re{(∇θ , θt )},
comparing the real parts of (2.19), we have 1 d 2 dt
∥∇θ∥2 = −Re{2(V θ , θt ) + (∇(u − uI ), ∇θt ) + 2(V (u − uI ), θt )} − Im{2((u − uI )t , θt )},
using (2.3)–(2.5), we get d dt
∥∇θ∥2 ≤ C (∥θ ∥ + hp+1 ∥u∥p+3 + hp+1 ∥u∥p+2 + hp+1 ∥ut ∥p+2 )∥θt ∥,
(2.20)
from (2.6), (2.7) and (2.20), we get d dt
∥∇θ∥2 ≤ Ch2p+2 ,
integrating from 0 to t, we have
∥∇θ ∥2 ≤ ∥∇θ (·, 0)∥2 + Ch2p+2 ,
(2.21)
from (2.12) and (2.21), we can get
∥∇θ ∥2 ≤ Ch2p+2 , that is
∥∇θ ∥ ≤ Chp+1 , which completes the proof. 3. Superconvergence analysis for the fully discrete scheme In this section, we focus on the fully discrete Crank–Nicolson scheme and discuss the superconvergence error estimate. Let τ = NT be the mesh size of the interval [0, T ], tn = nτ , n = 0, 1, . . . , N be the time nodes, and tn− 1 = 21 (t n + t n−1 ). 2
We introduce some notations. 1
An− 2 (u, v) = An (u, v) =
1 2
1 2
(∇ u, ∇v) + V x, y, tn− 1 u, v , 2
(∇ u, ∇v) + (V (x, y, tn )u, v).
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For function w(x, y, t ) and function series un (x, y), n = 0, 1, . . . , let 1
1
∂t un− 2 (x, y) = 1
un− 2 (x, y) =
1 2
τ
[un (x, y) + un−1 (x, y)], 1
1
∂t w n− 2 (x, y) = 1
wn− 2 (x, y) =
[un (x, y) − un−1 (x, y)],
τ
1 2
[w(x, y, tn ) − w(x, y, tn−1 )],
[w(x, y, tn ) + w(x, y, tn−1 )]. h,p
Then we define the fully discrete finite element solution U n (x, y) ∈ V0 , n = 0, 1, . . . , N of problem (1.1) satisfying the Crank–Nicolson scheme
1 1 1 1 i ∂t U n− 2 , vh = An− 2 U n− 2 , vh + (f n− 2 , vh ),
∀vh ∈ V0h,p , n = 1, 2, . . . , N ,
U 0 (x, y) = uI0 (x, y).
(3.1)
Theorem 3. Let u(x, y, t ) be the solution defined in (2.1), function series U n (x, y) be the fully discrete finite element solution defined in (3.1), and let uI be the corresponding interpolating function of u, then we have the following error estimate
∥uI (·, tn ) − U n ∥ ≤ Chp+1 + C τ 2 .
(3.2)
Proof. Let u(x, y, tn ) − U n (x, y) = ρ n (x, y) + θ n (x, y) with
ρ n (x, y) = u(x, y, tn ) − uI (x, y, tn ), θ n (x, y) = uI (x, y, tn ) − U n (x, y). From (2.1) and (3.1), we can get
1
1
i ∂t θ n− 2 , vh + i ∂t ρ n− 2 , vh + i ut ·, tn− 1 2
1 1 1 1 − ∂t u ·, tn− 1 , vh = An− 2 θ n− 2 , vh + An− 2 ρ n− 2 , vh , 2
(3.3) taking vh = θ
i ∂t θ
n− 12
n− 12
in (3.3), we get 1
, θ n− 2
1 1 1 = −i ∂t ρ n− 2 , θ n− 2 − i ut ·, tn− 1 − ∂t u ·, tn− 1 , θ n− 2 2 2 1 1 1 1 1 1 + An− 2 θ n− 2 , θ n− 2 + An− 2 ρ n− 2 , θ n− 2 .
(3.4)
Noticing that 1 2τ
(∥θ n ∥2 − ∥θ n−1 ∥2 ) = Re
1 1 ∂t θ n− 2 , θ n− 2 ,
comparing the imaginary parts of (3.4), we get 1 2τ
1 1 1 ∂t ρ n− 2 , θ n− 2 + ut ·, tn− 1 − ∂t u ·, tn− 1 , θ n− 2 2 2 1 1 1 1 n− 2 n− 2 n− 2 n− 12 + Im ∇ρ , ∇θ + V ·, tn− 1 ρ ,θ ,
(∥θ n ∥2 − ∥θ n−1 ∥2 ) = −Re
2
2
thus 1 2τ
1 1 1 1 1 (∥θ n ∥2 − ∥θ n−1 ∥2 ) ≤ ξ1n , θ n− 2 + ∥ξ2n ∥ · θ n− 2 + ∇ξ3n , ∇θ n− 2 + ξ4n , θ n− 2 , 2
where 1
ξ1n = ∂t ρ n− 2 , 1
ξ3n = ρ n− 2 ,
ξ2n = ut ·, tn− 1 − ∂t u ·, tn− 1 , 2 2 n− 21 n ξ4 = V ·, tn− 1 ρ . 2
(3.5)
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Noticing that
|(ξ1n , v)| = ∂t u ·, tn− 1 − ∂t uI ·, tn− 1 , v 2
2
= |(τ −1 (u(·, tn ) − u(·, tn−1 )) − τ −1 (uI (·, tn ) − uI (·, tn−1 )), v)| = τ −1 |(u(·, tn ) − u(·, tn−1 ) − (uI (·, tn ) − uI (·, tn−1 )), v)|, using (2.5), we get
|(ξ1n , v)| ≤ C τ −1 hp+1 ∥u(·, tn ) − u(·, tn−1 )∥p+2 ∥v∥.
(3.6)
Further
∥ξ2n ∥ = ut ·, tn− 1 − ∂t u ·, tn− 1 2 2 tn 1 tn− 12 2 2 ( t − t ) u (·, t ) dt = ( t − t ) u (·, t ) dt + n ttt n−1 ttt 2τ tn−1 t 1 n− 2 tn ≤ Cτ ∥uttt (·, t )∥dt .
(3.7)
tn−1
And using (2.3), we have
|(∇ξ3n , ∇v)| = ∇ u ·, tn− 1 − uI ·, tn− 1 , ∇v 2
1
=
2
2
|(∇(u(·, tn ) + u(·, tn−1 ) − (uI (·, tn ) + uI (·, tn−1 ))), ∇v)|
≤ Chp+1 ∥u(·, tn ) + u(·, tn−1 )∥p+3 ∥v∥,
(3.8)
similarly, using (2.4), we obtain
|(ξ4n , v)| = V ·, tn− 1 u ·, tn− 1 − uI ·, tn− 1 , v 2
2
2
≤ C |(u(·, tn ) + u(·, tn−1 ) − (uI (·, tn ) + uI (·, tn−1 )), v)| ≤ Chp+1 ∥u(·, tn ) + u(·, tn−1 )∥p+2 ∥v∥.
(3.9)
1 n− 21 n n−1 = (θ + θ ) ≤ ∥θ n ∥ + ∥θ n−1 ∥, θ
(3.10)
Noticing that
2
from (3.5)–(3.10), we can get n
∥θ ∥ − ∥θ
n−1
p+1
∥ ≤ Ch
∥u(·, tn ) − u(·, tn−1 )∥p+2 + C τ
2
tn
∥uttt (·, t )∥dt
tn−1
+ C τ hp+1 ∥u(·, tn ) + u(·, tn−1 )∥p+3 + C τ hp+1 ∥u(·, tn ) + u(·, tn−1 )∥p+2 ,
(3.11)
summing up for n in (3.11), we have
∥θ n ∥ − ∥θ 0 ∥ ≤ Chp+1
n
j
T1 + C τ 2
j =1
n j =1
j
T2 + C τ hp+1
n j =1
j
T3 + C τ hp+1
n
j
T4 ,
(3.12)
j=1
where n
j
T1 =
j =1
n j =1 n
≤
∥u(·, tn ) − u(·, tn−1 )∥p+2
j =1 tn
=
0
tj
∥ut (·, t )∥p+2 dt
tj−1
∥ut (·, t )∥p+2 dt ,
(3.13)
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J. Wang et al. / Computers and Mathematics with Applications ( n
n
j
T2 =
j =1
j =1 tn
=
tj
)
–
∥uttt (·, t )∥dt
tj−1
∥uttt (·, t )∥dt ,
(3.14)
0 n
n
j
T3 =
j =1
∥u(·, tn ) + u(·, tn−1 )∥p+3
j =1 n
≤ 2
∥u(·, tj )∥p+3
j=0
≤ ∥u(·, 0)∥p+3 +
tn
(∥ut (·, t )∥p+3 + τ −1 ∥u(·, t )∥p+3 )dt ,
(3.15)
0 n
n
j
T4 =
j =1
∥u(·, tn ) + u(·, tn−1 )∥p+2
j =1 n
≤ 2
∥u(·, tj )∥p+2
j=0
≤ ∥u(·, 0)∥p+2 +
tn
(∥ut (·, t )∥p+2 + τ −1 ∥u(·, t )∥p+2 )dt .
(3.16)
0
From (3.1), we can see that
∥θ 0 ∥ = 0,
(3.17)
from (3.12)–(3.17), we get
∥θ n ∥ ≤ C (hp+1 + τ 2 + τ hp+1 ),
(3.18)
noticing that
τ hp+1 ≤ τ 2 + h2p+2 ,
(3.19)
from (3.18), (3.19) and noticing that 0 ≤ h ≤ 1, we have
∥θ n ∥ ≤ Chp+1 + C τ 2 , which completes the proof. Theorem 4. Let u(x, y, t ) be the solution defined in (2.1), function series U n (x, y) be the fully discrete finite element solution defined in (3.1), and let uI be the corresponding interpolating function of u, then we have the following superconvergence estimate 3
∥uI (·, tn ) − U n ∥1 ≤ Chp+1 + C τ 2 .
(3.20)
Proof. From (2.1) and (3.1), we can get
i ∂t θ
n− 12
n− 12 t
, vh + i ρ
1 1 1 1 , vh + i uIt ·, tn− 1 − ∂t uI ·, tn− 1 , vh = An− 2 θ n− 2 , vh + An− 2 ρ n− 2 , vh , 2
2
(3.21) noticing that
n− 12
i ρt
, vh
1 1 = An− 2 ρ n− 2 , vh ,
(3.22)
from (3.21) and (3.22), we have
1
i ∂t θ n− 2 , vh + i uIt ·, tn− 1 2
1 1 1 − ∂t uI ·, tn− 1 , vh = ∇θ n− 2 , ∇vh + V ·, tn− 1 θ n− 2 , vh , 2
2
1
taking vh = ∂t θ n− 2 in (3.23), we get
1 + i uIt ·, tn− 1 − ∂t uI ·, tn− 1 , ∂t θ n− 2 2 2 1 1 1 1 1 = ∇θ n− 2 , ∇∂t θ n− 2 + V ·, tn− 1 θ n− 2 , ∂t θ n− 2 , 1
1
i ∂t θ n− 2 , ∂t θ n− 2 2
2
2
(3.23)
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that is
1
1
i ∂t θ n− 2 , ∂t θ n− 2
i I ut ·, tn− 1 − ∂t uI ·, tn− 1 , θ n − θ n−1
+
τ
2
2
1 1 = V ·, tn− 1 (θ n + θ n−1 ), θ n − θ n−1 . (∇θ n + ∇θ n−1 , ∇θ n − ∇θ n−1 ) + 2 4τ 2τ
(3.24)
Noticing that
∥∇θ n ∥2 − ∥∇θ n−1 ∥2 = Re{(∇θ n + ∇θ n−1 , ∇θ n − ∇θ n−1 )}, comparing the real parts of (3.24), we get
∥∇θ n ∥2 − ∥∇θ n−1 ∥2 = Im 4 ∂t uI ·, tn− 1 − uIt ·, tn− 1 , θ n − θ n−1 2
2
− Re 2 V ·, tn− 1 (θ n + θ n−1 ), θ n − θ n−1 2
,
(3.25)
summing up for n in (3.25), we have
n ∥∇θ ∥ − ∥∇θ ∥ = Im 4 ∂t uI ·, tj− 1 − uIt ·, tj− 1 , θ j − θ j−1 n 2
0 2
2
j =1
−2
2
n V ·, tj− 1 (∥θ j ∥2 − ∥θ j−1 ∥2 ), j=1
(3.26)
2
from (3.17) and (3.26), we can get n 2
j
∥∇θ ∥ ≤ 8 max ∥θ ∥ j ≤n
n j =1
n j ∥η ∥ + 2 ∥η2 ∥ , j =1 j 1
(3.27)
where, n j=1
n I ∂t u ·, tj− 1 − uIt ·, tj− 1 2 2 j =1 tj n 1 tj− 21 2 I 2 I = (t − tj−1 ) uttt (·, t )dt + (t − tj ) uttt (·, t )dt 2τ j=1 tj−1 t 1 j− 2 n t j ∥uIttt (·, t )∥dt ≤ Cτ
∥η1j ∥ =
≤ Cτ
j=1tn
tj−1
∥uIttt (·, t )∥dt ,
(3.28)
0
n n j j 2 j −1 2 ∥η2 ∥ = V ·, tj− 1 (∥θ ∥ − ∥θ ∥ ) 2 j =1 j =1 n −1 n 2 0 2 j 2 = V ·, tn− 1 ∥θ ∥ − V ·, t 1 ∥θ ∥ + ∥θ ∥ V ·, tj+ 1 − V ·, tj− 1 2 2 2 2 j =1 n −1 ≤ V ·, tn− 1 ∥θ n ∥2 + max ∥θ j ∥2 V ·, tj+ 1 − V ·, tj− 1 j ≤n
2
≤ V ·, tn− 1 ∥θ n ∥2 + max ∥θ j ∥2 2
2
j =1
2
n −1
j ≤n
≤ C max ∥θ j ∥2 .
τ |Vt (·, tj )|
j =1
(3.29)
j ≤n
From (3.2) and (3.27)–(3.29), we can get
∥∇θ n ∥2 ≤ C (hp+1 + τ 2 )(hp+1 + τ 2 + τ ),
(3.30)
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Fig. 1. The log of errors at t = 0.01.
Fig. 2. The log of errors at t = 0.1.
that is 3
∥∇θ n ∥ ≤ Chp+1 + C τ 2 , which completes the proof. 4. Numerical examples In this section, we present a numerical example with p = 1 to confirm our theoretical analysis. We consider the following problem of the Schrödinger equation:
1 iut = − 1u + Vu + f ,
∀(x, y, t ) ∈ Ω × [0, T ],
u(x, y, t ) = 0, u(x, y, 0) = u0 (x, y),
∀(x, y, t ) ∈ ∂ Ω × [0, T ], ∀(x, y) ∈ Ω ,
2
where V (x, y, t ) = 1, Ω = [0, 1] × [0, 1], T = 1 and the function f (x, y, t ) is chosen corresponding to the exact solution u = et sin π x sin π y + iet sin π x sin π y. h,1
Ω is uniformly divided into families Th of quadrilaterals with mesh size h, and V0 is the bilinear rectangular finite element space defined on Th . We solved the above problem on uniformly refined rectangular meshes with time steps τ = 10−4 and various space steps h = 1/8, 1/16, 1/32, 1/64, respectively. Then, the convergence and superconvergence results with respect to different time are listed in Tables 1–4. Moreover, in Figs. 1–4, we show the convergence orders by slopes. From our numerical results, we
J. Wang et al. / Computers and Mathematics with Applications (
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Fig. 3. The log of errors at t = 0.5.
Fig. 4. The log of errors at t = 1.0.
Table 1 Numerical results at t = 0.01. Mesh h h h h
= 1/8 = 1/16 = 1/32 = 1/64
∥uI − U n ∥
Ratio
∥uI − U n ∥1
Ratio
1.2796E−04 3.2521E−05 8.3400E−06 2.2747E−06
/ 3.9347 3.8994 3.6664
5.7219E−04 1.4472E−04 3.7068E−05 1.0107E−05
/ 3.9540 3.9042 3.6676
∥uI − U n ∥
Ratio
∥uI − U n ∥1
Ratio
1.3387E−03 3.4023E−04 8.7252E−05 2.3797E−05
/ 3.9347 3.8994 3.6665
5.9863E−03 1.5140E−03 3.8781E−04 1.0574E−04
/ 3.9540 3.9040 3.6676
Table 2 Numerical results at t = 0.1. Mesh h h h h
= 1/8 = 1/16 = 1/32 = 1/64
can see that ∥uI − U n ∥ and ∥uI − U n ∥1 are convergent in terms of h at rate of O(h2 ), which coincide with our theoretical analysis in the paper.
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Table 3 Numerical results at t = 0.5. Mesh h h h h
= 1/8 = 1/16 = 1/32 = 1/64
∥uI − U n ∥
Ratio
∥uI − U n ∥1
Ratio
8.2490E−03 2.0964E−03 5.3762E−04 1.4663E−04
/ 3.9348 3.8994 3.6665
3.6909E−02 9.3294E−03 2.3895E−03 6.5153E−04
/ 3.9562 3.9043 3.6675
∥uI − U n ∥
Ratio
∥uI − U n ∥1
Ratio
2.1829E−02 5.5466E−03 1.4224E−03 3.8795E−04
/ 3.9356 3.8995 3.6666
9.8048E−02 2.4690E−02 6.3223E−03 1.7238E−03
/ 3.9712 3.9052 3.6677
Table 4 Numerical results at t = 1.0. Mesh h h h h
= 1/8 = 1/16 = 1/32 = 1/64
5. Conclusions In this paper, we considered the two-dimensional time-dependent Schrödinger equation with the finite element method. We presented a finite element semi-discrete scheme and a Crank–Nicolson finite element fully discrete scheme in rectangular Lagrange type finite element space of order p. We also provided the error analysis and obtained the superconvergence results in the H 1 norm of the semi-discrete scheme and the fully discrete scheme. A numerical example with the order p = 1 is provided to partly verify our theoretical results. Acknowledgments The authors thank the referees for their valuable suggestions which have helped to improve the presentation of this paper. Wang’s research was supported by Hunan Provincial Innovation Foundation for Postgraduate (Grant No. CX2014B252). Huang’s research was supported by the NSFC Key Project (Grant No. 91430213). Tian’s research was supported by the Project of Scientific Research Fund of Hunan Provincial Education Department (Grant No. 14B044). Zhou’s research was supported by the National Natural Science Foundation of China (Grant No. 11501485). References [1] W. Bao, Y. Cai, Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator, SIAM J. Numer. Anal. 50 (2012) 492–521. [2] M. Delfour, M. Fortin, G. Payr, Finite-difference solutions of a non-linear Schrödinger equation, J. Comput. Phys. 44 (1981) 277–288. [3] G.D. Akrivis, V.A. Dougalis, O.A. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math. 59 (1991) 31–53. [4] J. Jin, X. Wu, Convergence of a finite element scheme for the two-dimensional time-dependent Schrödinger equation in a long strip, J. Comput. Appl. Math. 234 (2010) 777–793. [5] O.A. Karakashian, C. Makridakis, A space–time finite element method for the nonlinear Schrödinger equation: the continuous Galerkin method, SIAM J. Numer. Anal. 36 (1999) 1779–1807. [6] I. Kyza, A posteriori error analysis for the Crank–Nicolson method for linear Schrödinger equations, ESAIM: Math. Model. Numer. Anal. 45 (2011) 761–778. [7] H.Y. Lee, Fully discrete methods for the nonlinear Schrödinger equation, Comput. Math. Appl. 28 (1994) 9–24. [8] Q. Lin, X. Liu, Global superconvergence estimates of finite element method for Schrödinger equation, J. Comput. Math. 6 (1998) 521–526. [9] D. Shi, P. Wang, Y. Zhao, Superconvergence analysis of anisotropic linear triangular finite element for nonlinear Schrödinger equation, Appl. Math. Lett. 38 (2014) 129–134. [10] J. Wang, A new error analysis of Crank–Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation, J. Sci. Comput. 60 (2014) 390–407. [11] M.D. Feit, J.A. Fleck, A. Steiger, Solution of the Schrödinger equation by a spectral method, J. Comput. Phys. 47 (1982) 412–433. [12] D. Pathria, J.L. Morris, Pseudo-spectral solution of nonlinear Schrödinger equations, J. Comput. Phys. 87 (1990) 108–125. [13] W. Lu, Y. Huang, H. Liu, Mass preserving discontinuous Galerkin methods for Schrödinger equations, J. Comput. Phys. 282 (2015) 210–226. [14] Y. Xu, C.W. Shu, Local discontinuous Galerkin methods for nonlinear Schrödinger equations, J. Comput. Phys. 205 (2005) 72–97. [15] M. Dehghan, D. Mirzaei, The meshless local Petrov–Galerkin (MLPG) method for the generalized two-dimensional non-linear Schrödinger equation, Eng. Anal. Bound. Elem. 32 (2008) 747–756. [16] J. Jin, S. Shu, J. Xu, a two-grid discretization method for decoupling systems of partial differential equations, Math. Comp. 75 (2006) 1617–1626. [17] L. Wu, Two-grid mixed finite-element methods for nonlinear Schrödinger equations, Numer. Methods Partial Differential Equations 28 (2012) 63–73. [18] H. Zhang, J. Jin, J. Wang, Two-grid finite-element method for the two-dimensional time-dependent Schrödinger equation, Adv. Appl. Math. Mech. 5 (2013) 180–193. [19] D.N. Arnold, J. Douglas, V. Thomee, Superconvergence of a finite element approximation to the solution of a Sobolev equation in a single space variable, Math. Comp. 36 (1981) 53–63. [20] C. Chen, S. Hu, The highest order superconvergence for bi-k degree rectangular elements at nodes: A proof of 2k-conjecture, Math. Comp. 82 (2013) 1337–1355. [21] C. Chen, Y. Huang, High Accuracy Theory of Finite Element Methods, Hunan Science Press, Changsha, 1995, (in Chinese). [22] J. Douglas, A superconvergence result for the approximate solution of the heat equation by a collocation method, in: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Academic Press, New York, 1972, pp. 475–490. [23] Y. Huang, J. Li, Q. Lin, Superconvergence analysis for time-dependent Maxwell’s equations in metamaterials, Numer. Methods Partial Differential Equations 28 (2012) 1794–1816.
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