optimal error estimates of linearized crank-nicolson galerkin

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

SIAM J. NUMER. ANAL. c© 2014 Society for Industrial and Applied MathematicsVol. 52, No. 3, pp. 1183–1202

OPTIMAL ERROR ESTIMATES OF LINEARIZEDCRANK-NICOLSON GALERKIN FEMs FOR THE

TIME-DEPENDENT GINZBURG–LANDAU EQUATIONS INSUPERCONDUCTIVITY∗

HUADONG GAO† , BUYANG LI‡ , AND WEIWEI SUN†

Abstract. In this paper, we study linearized Crank–Nicolson Galerkin finite element methodsfor time-dependent Ginzburg–Landau equations under the Lorentz gauge. We present an optimalerror estimate for the linearized schemes (almost) unconditionally (i.e., when the spatial mesh sizeh and the temporal step τ are smaller than a given constant), while previous analyses were givenonly for some schemes with strong restrictions on the time step-size. The key to our analysis isthe boundedness of the numerical solution in some strong norm. We prove the boundedness for thecases τ ≥ h and τ ≤ h, respectively. The former is obtained by a simple inequality, with which theerror functions at a given time level are bounded in terms of their average at two consecutive timelevels, and the latter follows a traditional way with the induction/inverse inequality. Two numericalexamples are investigated to confirm our theoretical analysis and to show clearly that no time stepcondition is needed.

Key words. optimal error estimates, finite element methods, Ginzburg–Landau equations,Crank–Nicolson scheme, superconductivity, unconditional stability

AMS subject classifications. 65N12, 65N30, 35K61

DOI. 10.1137/130918678

1. Introduction. In this paper, we consider the time-dependent Ginzburg–Landau (TDGL) equations under the Lorentz gauge

η∂ψ

∂t− iηκ(divA)ψ +

(i

κ∇+A

)2

ψ + (|ψ|2 − 1)ψ = 0,(1.1)

∂A

∂t−∇divA+ curl curlA+

i

2κ(ψ∗∇ψ − ψ∇ψ∗) + |ψ|2A = curlH(1.2)

for x ∈ Ω and t ∈ (0, T ], where ψ is the order parameter, a complex scalar function,and A is the magnetic potential, a real vector-valued function. Physically, |ψ| denotesthe relative density of the superconducting electron pairs. |ψ| = 1 and |ψ| = 0represent the perfectly superconducting state and the normal state, respectively, while0 < |ψ| < 1 represents a mixed state. The real vector-valued function H is theapplied magnetic field, κ is the Ginzburg–Landau parameter, and η is a dimensionlessconstant. Here we take η = 1 for the sake of simplicity. We assume that Ω is abounded domain in Rd(d = 2, 3). The boundary and initial conditions are given by

∗Received by the editors April 26, 2013; accepted for publication (in revised form) February 25,2014; published electronically May 8, 2014.

http://www.siam.org/journals/sinum/52-3/91867.html†Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong (hdgao2-c@

my.cityu.edu.hk, [email protected]). The work of these authors was supported in part bya grant from the Research Grants Council of the Hong Kong Special Administrative Region, China(Project CityU 102613).

‡Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China([email protected]). The work of this author was supported in part by a grant from NSF of China(Project 11301262).

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http://www.siam.org/journals/sinum/52-3/91867.html

mailto:[email protected]





1184 HUADONG GAO, BUYANG LI, AND WEIWEI SUN

∂ψ

∂n= 0, curlA× n = H× n, A · n = 0, on ∂Ω× (0, T ],(1.3)

ψ(x, 0) = ψ0(x), A(x, 0) = A0(x), in Ω,(1.4)

where n is the unit outer normal vector. The above TDGL equations were first deducedby Gor’kov and Eliashberg in [16] from the microscopic BCS theory. The detaileddescription for the physical background of the TDGL equations and the superconduc-tivity phenomena can be found in the review articles [3, 11] and the monograph [33].

Theoretical analysis for the TDGL equations has been well done [7, 21, 25, 29, 31].The global existence and uniqueness of the strong solution was proved in [7] for theTDGL equations with the Lorentz gauge. Numerical methods for solving the TDGLtype equations have also been investigated extensively; see [4, 5, 6, 8, 9, 14, 17,23, 26, 27, 28, 35, 36]. A review article was presented by Du [10]. However, erroranalysis of numerical methods for the TDGL equations seems incomplete mainly dueto the strong nonlinearity and coupling and complex structure of equations. A semi-implicit Euler scheme with a finite element approximation was first proposed by Chenand Hoffmann [6] for the TDGL equations with the Lorentz gauge. A suboptimalL2 error estimate was obtained for the equations in two-dimensional space. Thesemi-implicit scheme with a mixed finite element method (FEM) was studied in [4]and a similar L2 suboptimal error estimate was derived. Later an adaptive methodand its posteriori error estimates were presented in [5] for the TDGL equations withthe Lorentz gauge. More recently, a decoupled alternating Crank–Nicolson Galerkinmethod was proposed by Mu and Huang [28]. Optimal error estimates were proved

under the time step restrictive conditions τ ≤ O(h1112 ) for the two-dimensional model

and τ ≤ O(h2) for the three-dimensional model, where h and τ are the mesh sizein the spatial direction and the time direction, respectively. All these schemes in[4, 5, 6, 28] are nonlinear. At each time step, one has to solve a system of nonlinearequations. Clearly, more commonly used approximations in the time direction fornonlinear parabolic equations are linearized semi-implicit schemes [20, 30, 32], whichonly require the solution of a linear system. However, the time step-size restriction isalways a key issue for the linearized schemes, and one often suffers from the restrictedtime step-size in practical computations. Among these linearized schemes, the Crank–Nicolson scheme is a popular one since it provides a second-order accuracy in the timedirection. A linearized Crank–Nicolson type scheme was proposed in [27] for slightlydifferent TDGL equations and a systematic numerical simulation was made there.Numerical results show that the linearized Crank–Nicolson is very effective. However,no analysis has been done. The major difficulty in the error analysis of the linearizedCrank–Nicolson type schemes lies in the fact that the error function in an energynorm from this scheme is defined by its average at two consecutive time levels, i.e.,‖∇en+∇en−1‖, which is much weaker than the energy estimate ‖∇en‖ obtained fromthe Euler scheme. Therefore, the classical energy estimate approach cannot providea real estimate in L2 ×H1 norm to bound the nonlinear (or linearized) term. Thispaper focuses on analysis of linearized Crank–Nicolson Galerkin FEMs for the TDGLequations in d-dimensional space, d = 2, 3. We provide an optimal L2 error estimate(almost) unconditionally. The analysis for τ ≥ h is based on a simple estimate, inwhich the error functions at a given time level are bounded, in terms of their averageat two consecutive time levels, and the analysis for τ ≤ h is obtained by using theinduction and inverse inequalities.

The rest of this paper is organized as follows. In section 2, we introduce twolinearized Crank–Nicolson schemes with Galerkin finite element approximations in

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OPTIMAL ERROR ESTIMATES OF C-N SCHEME FOR TDGL 1185

spatial direction for the TDGL equations and we present our main results. In section 3,we define a Ritz projection and analyze the error bounds at the initial step. In section4, we prove that the optimal L2 error estimate holds without any restriction on thetime step. In section 5, we provide some numerical examples to confirm our theoreticalanalysis and show that no time step restriction is needed.

2. Linearized Crank–Nicolson Galerkin FEMs. In this section, we presenttwo linearized Crank–Nicolson Galerkin FEMs for the TDGL equations. For simplic-ity, we introduce some notation below.

For any two complex functions u, v ∈ L2(Ω), we denote the L2(Ω) inner productand norm by

(u, v) =

∫Ω

u(x)(v(x))∗dx, ‖u‖ = (u, u)12 ,

where v∗ denotes the conjugate of the complex variable v. Let W k,p(Ω) be the con-ventional Sobolev space defined on Ω, and Hk(Ω) := W k,2(Ω). We denote Hk(Ω) ={u + iv|u, v ∈ Hk(Ω)} for the complex-valued functions and Hk(Ω) = [Hk(Ω)]d forvector-valued functions with d components (d = 2, 3). We introduce a subspace ofHk(Ω): Hk

n(Ω) = {A|A ∈ Hk(Ω),A · n = 0 on ∂Ω}. The corresponding Sobolevnorm ‖A‖H1 and seminorm |A|H1 in the space H1(Ω) are equivalent to the norm

(2.1)(‖A‖2 + ‖divA‖2 + ‖curlA‖2

) 12

and the seminorm

(2.2)(‖divA‖2 + ‖curlA‖2

) 12

,

respectively. By noting the Dirichlet boundary condition A · n = 0, the norm andsemi-norm mentioned above are equivalent [15] in H1

n(Ω).The weak formulation of the TDGL equations (1.1)–(1.2) is to find (ψ,A) ∈

H1(Ω)×H1n(Ω) such that(∂ψ

∂t, ω

)− iκ((divA)ψ, ω) +

((i

κ∇+A

)ψ,

(i

κ∇+A

)ω

)(2.3)

+ ((|ψ|2 − 1)ψ, ω) = 0 ∀ω ∈ H1(Ω), t ∈ (0, T ],(∂A

∂t,v

)+ (divA, div v) + (curlA, curlv) +

i

2κ((ψ∗∇ψ − ψ∇ψ∗),v)(2.4)

+ (|ψ|2A,v) = (H, curlv) ∀v ∈ H1n(Ω) t ∈ (0, T ],

with ψ(x, 0) = ψ0(x), A(x, 0) = A0(x).For the sake of simplicity, we assume that Ω is a convex polygon (or polyhedron).

Let Th be a regular partition of Ω with Ω = ∪eΩe; we denote the mesh size byh = maxΩe∈Th

{diamΩe}. For a given partition Th, we denote Vrh and Vrh as the rth-

order finite element subspaces of H1(Ω) and H1n(Ω), respectively. We denote Ih as

the commonly used Lagrange interpolation on Vrh and Vrh. For 0 ≤ l ≤ m ≤ r + 1,

1 ≤ p ≤ ∞, the following interpolation error estimates hold [1]:

‖v − Ihv‖W l,p(Ω) ≤ Chm−l‖v‖Wm,p(Ω) ∀v ∈ Wm,p(Ω),(2.5)

where C is a positive constant independent of v.

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Let {tn}Nn=0 be a uniform partition in the time direction with the step-size τ = TN .

For a sequence of functions {Un} defined on Ω, we denote

DτUn =

Un − Un−1

τ, U

n=Un + Un−1

2, Un =

3Un−1 − Un−2

2.

A linearized Crank–Nicolson Galerkin approximation is to find (ψnh , Anh) ∈ Vrh ×

Vrh such that for n = 2, 3, . . . , N

(Dτψnh , ω)− iκ

((div An

h

)ψn

h , ω)+

((i

κ∇+ An

h

)ψn

h,

(i

κ∇+ An

h

)ω

)(2.6)

+((∣∣ψnh ∣∣2 − 1

)ψn

h, ω)= 0 ∀ω ∈ Vrh(Ω),

(DτAnh,v) +

(divA

n

h, div v)+(curlA

n

h, curlv)+(|ψnh |2A

n

h ,v)

(2.7)

= (Hn− 12 , curlv)− i

2κ

((ψnh)∗∇ψnh − ψnh∇

(ψnh)∗,v)

∀v ∈ Vrh(Ω).

At the initial time step, we use the linearized backward Euler scheme

(Dτψ1h, ω)− iκ

((divA0

h

)ψ1h, ω

)+

((i

κ∇+A0

h

)ψ1h,

(i

κ∇+A0

h

)ω

)(2.8)

+((|ψ0h|2 − 1

)ψ1h, ω

)= 0 ∀ω ∈ Vrh(Ω),

(DτA1h,v) + (divA1

h, div v) + (curlA1h, curlv) + (|ψ0

h|2A1h,v)(2.9)

= (H1, curlv)− i

2κ

((ψ0h

)∗∇ψ0h − ψ0

h∇(ψ0h

)∗,v)

∀v ∈ Vrh(Ω),

where ψ0h = Ihψ

0 and A0h = IhA

0. In the above scheme, a standard extrapolationapproximation [2, 12, 18, 22] is used for the nonlinear terms. Numerical simulationsof the above scheme for a slightly different TDGL equation were made in [27]. Notheoretical analysis was presented.

A decoupled alternating Crank–Nicolson scheme was proposed in [28] for theTDGL equations. An inner iteration is needed since the alternating scheme is nonlin-ear. With the standard extrapolation, the corresponding linearized Crank–Nicolsonscheme is defined by

(Dτψnh , ω)− iκ

((divA

n− 12

h

)ψn

h , ω)+

((i

κ∇+A

n− 12

h

)ψn

h,

(i

κ∇+A

n− 12

h

)ω

)(2.10)

+

((∣∣∣ψnh ∣∣∣2 − 1

)ψn

h, ω

)= 0 ∀ω ∈ Vrh(Ω),

(DτA

n− 12

h ,v)+(divA

n− 12

h , div v)+(curlA

n− 12

h , curlv)+(|ψn−1h |2An− 1

2

h ,v)(2.11)

= (Hn−1, curlv) − i

2κ

((ψn−1h

)∗∇ψn−1h − ψn−1

h ∇(ψn−1h

)∗,v)

∀v ∈ Vrh(Ω)

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for n = 2, 3, . . . , N . The initial solution ψ1h and A

12

h can be obtained by (2.8) and alinearized backward Euler scheme similar to (2.9), respectively.

The linearized Crank–Nicolson Galerkin scheme in (2.6)–(2.9) is fully uncoupledsince these two equations can be solved simultaneously. In the alternating Crank–

Nicolson Galerkin scheme (2.10)–(2.11), one should solve (2.11) first for An− 1

2

h andthen (2.10) for ψnh . In this paper, we only focus on the linearized Crank–NicolsonGalerkin scheme (2.6)–(2.7). Our analysis can be easily extended to the second lin-earized scheme (2.10)–(2.11).

Here we assume that the initial-boundary value problem (1.1)–(1.4) has a uniquesolution satisfying the regularity conditions

ψ, ψt ∈ L∞(0, T ;Hr+1), ψtt ∈ L∞(0, T ;H1), ψttt ∈ L2(0, T ;L2)(2.12)

and

A,At ∈ L∞(0, T ;Hr+1), Att ∈ L∞(0, T ;H1), Attt ∈ L2(0, T ;L2).(2.13)

We present our main results in the following theorem. The proof will be givenin sections 3 and 4. The emphasis of this paper is on the unconditionally optimalerror analysis. The above regularity assumptions may possibly be weakened for theanalysis below.

Theorem 2.1. Suppose that the system (1.1)–(1.4) has a unique solution (ψ,A)satisfying (2.12)–(2.13). Then there exist positive constants h0 and τ0 such that whenh < h0 and τ < τ0, the finite element system (2.6)–(2.9) is uniquely solvable and

(2.14) max1≤n≤N

‖ψnh − ψn‖+ max1≤n≤N

‖Anh −An‖ ≤ C1(τ

2 + hr+1),

where C1 is a positive constant, independent of n, h, and τ .For simplicity of notation, we denote by C a generic positive constant, which is

independent of C1, n, h, and τ .

3. Preliminaries. We define a Ritz projection (see [20, 32, 34]) as follows: forgiven t ∈ [0, T ], find (Rhψ, PhA) ∈ Vrh(Ω)×Vr

h(Ω) such that((i

κ∇+A

)(ψ −Rhψ),

(i

κ∇+A

)ω

)+ (M − 1)(ψ −Rhψ, ω)(3.1)

= 0 ∀ω ∈ Vrh(Ω),(div (A− PhA), div v) + (curl (A− PhA), curlv) + (|ψ|2(A− PhA),v)(3.2)

= 0 ∀v ∈ Vrh(Ω),

where M = supt∈[0,T ] ‖A‖2L∞ + 2 is a positive constant.Let

eψ = ψ −Rhψ, eA = A− PhA.

By standard finite element theory, with the regularity in (2.12)–(2.13), we have

‖eψ‖+ h‖eψ‖H1 ≤ Chr+1,(3.3)

‖eA‖+ h‖eA‖H1 ≤ Chr+1,(3.4)

‖(eψ)t‖ ≤ Chr+1,(3.5)

‖(eA)t‖ ≤ Chr+1.(3.6)

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and under the regularity assumptions (2.12)–(2.13),∥∥∥∥ ∂l∂tl (Rhψ)∥∥∥∥H1

+

∥∥∥∥ ∂l∂tlPhA∥∥∥∥H1

≤ C for l = 0, 1, 2.(3.7)

We will use the following Gagliardo–Nirenberg–Sobolev inequality [13] frequentlyin the rest of this paper:

(3.8) ‖u‖Lp ≤ C‖u‖H1 for 1 ≤ p ≤ 6,

where C is independent of u.By the triangular and Cauchy inequalities, we have the following lemma.Lemma 3.1. Let Un(x), n = 0, 1, 2, . . . , N , be a sequence of functions defined on

Ω. Then for any norm ‖ · ‖, we have

τ‖Un‖ ≤ 2τ

n∑m=2

‖Um‖+ τ‖U1‖ ≤ 2√T

(n∑

m=2

τ‖Um‖2 + τ‖U1‖2) 1

2

.(3.9)

Let

θnψ = Rhψn − ψnh , θnA = PhA

n −Anh.

With the above projection error estimates, we only need to estimate the error functionsθnψ and θnA. We present them at the initial time step below.

Lemma 3.2. If we take ψ0h = Ihψ

0 and A0h = IhA

0 and use the linearizedbackward Euler method (2.8)–(2.9) to obtain ψ1

h and A1h, then there exists a constant

C0 > 0, independent of h and τ such that

‖θ1ψ‖2 + ‖θ1A‖2 + τ(‖θ1ψ‖2H1 + ‖θ1A‖2H1

)≤ 1

2C0(τ

2 + hr+1)2.(3.10)

Proof. Writing the exact solution (ψ1,A1) into the form of the scheme (2.8)–(2.9),and with the projection defined in (3.1)–(3.2), we have

(Dτψ1, ω)− iκ

((divA0

)ψ1, ω

)+

((i

κ∇+A0

)Rhψ

1,

(i

κ∇+A0

)ω

)(3.11)

+ (M − 1)(Rhψ1, ω)−M(ψ1, ω) + (|ψ0|2ψ1, ω) = R1

ψ ∀ω ∈ Vrh,

and

(DτA1,v) + (divPhA

1, div v) + (curlPhA1, curlv) + (|ψ0|2PhA1,v)(3.12)

= (H1, curlv)− i

2κ((ψ0)∗∇ψ0 − ψ0∇(ψ0)∗,v) +R1

A ∀v ∈ Vrh,

where

R1ψ =

(Dτψ

1 − ∂ψ1

∂t, ω

)+ iκ((divA1 − divA0)ψ1, ω) + ((|ψ0|2 − |ψ1|2)ψ1, ω)

+i

κ(ω∗∇Rhψ1 −Rhψ

1∇ω∗,A0 −A1) + ((A0 +A1)Rhψ1, (A0 −A1)ω),

R1A =

(DτA

1 − ∂A1

∂t,v

)+ ((|ψ0|2 − |ψ1|2)PhA1,v)

+i

2κ((ψ0)∗∇ψ0 − ψ0∇(ψ0)∗,v) − i

2κ((ψ1)∗∇ψ1 − ψ1∇(ψ1)∗,v)

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stand for the truncation errors. It is easy to see that

|τRe(R1ψ)| ≤ Cτ4 +

1

8‖ω‖2,

|τR1A| ≤ Cτ4 +

1

8‖v‖2,

where we have used integration by parts for the term iκ (Rhψ

1∇ω∗,A0 −A1).Subtracting (2.8) from (3.11) gives(

θ1ψτ, ω

)+

((i

κ∇+A0

)θ1ψ,

(i

κ∇+A0

)ω

)+ (M − 1)(θ1ψ , ω)(3.13)

= R1ψ +

(ψ0 − ψ0

h − e1ψτ

, ω

)+ iκ((divA0)ψ1 − (divA0

h)ψ1h, ω)

+M(ψ1 − ψ1h, ω) + (|ψ0

h|2ψ1h − |ψ0|2ψ1, ω)

+ ((A0h +A0)ψ1

h, (A0h −A0)ω) +

i

κ(ω∗∇ψ1

h − ψ1h∇ω∗,

(A0h −A0

)):= R1

ψ +

6∑i=1

J1i ∀ω ∈ Vrh.

Taking ω = θ1ψ and using the interpolation error estimates (2.5) and the Ritzprojection error estimates (3.3)–(3.7), we have

|τRe(J11

)| ≤ 1

8‖θ1ψ‖2 + Ch2r+2,

|Re(J13

)| ≤ C‖θ1ψ‖2 + Ch2r+2,

|Re(J14

)| ≤ |

(|ψ0h|2(e1ψ + θ1ψ

), θ1ψ)|+ |

((|ψ0h|2 − |ψ0|2

)ψ1, θ1ψ

)|

≤ C‖θ1ψ‖2 + Ch2r+2,

|Re(J15

)| ≤ C|

(ψ1h,(A0h −A0

)θ1ψ)|

≤ C(|(Rhψ

1,(A0h −A0

)θ1ψ)|+ |

(θ1ψ,(A0h −A0

)θ1ψ)|)

≤ C‖θ1ψ‖2 + Ch2r+2.

Moreover, by the interpolation error estimates (2.5) and inverse inequalities and ap-plying integration by parts, we have further

|Re(J12

)| = |iκ

((divA0

)ψ1 −

(divA0

h

)Rhψ

1, θ1ψ)|

= |iκ((div(A0 −A0

h

))Rhψ

1 +(divA0

)e1ψ, θ

1ψ

)|

≤ C(|((div(A0 −A0

h

))ψ1, θ1ψ

)|+ |

((div(A0 −A0

h

))e1ψ, θ

1ψ

)|)

+ C‖e1ψ‖L2‖divA0‖L3‖θ1ψ‖L6

≤ C(|((A0 −A0

h

)· ∇ψ1, θ1ψ

)|+ |

((A0 −A0

h

)ψ1,∇θ1ψ

)|)

+ C‖div(A0 −A0

h

)‖L3‖e1ψ‖L2‖θ1ψ‖L6 + C‖e1ψ‖L2‖θ1ψ‖H1

≤ C‖A0 −A0h‖L2‖∇ψ1‖L3‖θ1ψ‖L6 + C‖A0 −A0

h‖L2‖∇θ1ψ‖L2

+ C‖e1ψ‖L2‖θ1ψ‖H1

≤ Chr+1‖θ1ψ‖H1

≤ 1

8κ2‖θ1ψ‖

2

H1 + Ch2r+2

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and

|Re(J16 )| ≤ C|

((θ1ψ)∗∇θ1ψ − θ1ψ∇

(θ1ψ)∗,A0 −A0

h

)|+ C|

(Rhψ

1∇(θ1ψ)∗,A0 −A0

h

)|

+ C|((θ1ψ)∗∇Rhψ1,A0 −A0

h

)|

≤ C|((θ1ψ)∗∇θ1ψ ,A0 −A0

h

)|+ C|

(Rhψ

1∇(θ1ψ)∗,A0 −A0

h

)|

+ C(|((θ1ψ)∗∇e1ψ,A0 −A0

h

)|+ |

((θ1ψ)∗∇ψ1,A0 −A0

h

)|)

≤ C‖θ1ψ‖L6‖∇θ1ψ‖L2‖A0 −A0h‖L3 + C‖∇θ1ψ‖‖A0 −A0

h‖+ C‖θ1ψ‖L6‖∇e1ψ‖L3‖A0 −A0

h‖L2 + C‖θ1ψ‖L6‖∇ψ1‖L3‖A0 −A0h‖L2

≤ Ch‖θ1ψ‖2H1 + Chr+1‖θ1ψ‖H1 + Chr−d6 hr+1‖θ1ψ‖H1

≤ 1

8κ2‖θ1ψ‖

2

H1 + Ch2r+2,

where we have used the inequality (3.8).If we take ω = θ1ψ in (3.13) and consider the real part of the equation, we arrive

at

‖θ1ψ‖2+ τ‖

(i

κ∇+A0

)θ1ψ‖2 ≤ 1

4‖θ1ψ‖

2+

τ

4κ2‖θ1ψ‖

2

H1 + Cτ‖θ1ψ‖2+ C(τ2 + hr+1)2.

By noting the fact

(3.14)

∥∥∥∥( iκ∇+A0

)θ1ψ

∥∥∥∥2 ≥(1

κ‖∇θ1ψ‖ − ‖A0θ1ψ‖

)2

≥ 1

2κ2‖θ1ψ‖

2

H1 − C‖θ1ψ‖2,

we see that

(3.15) ‖θ1ψ‖2+ τ‖θ1ψ‖

2

H1 ≤ C(τ2 + hr+1)2

when Cτ ≤ 18 .

Now we turn to the analysis of the linearized backward Euler scheme (2.9). Sub-tracting (2.9) from (3.12) results in(

θ1Aτ,v

)+(div θ1A, div v

)+(curl θ1A, curlv

)+(|ψ0|2θ1A,v

)(3.16)

= R1A +

(A0 −A0

h − e1Aτ

,v

)+((|ψ0h|2 − |ψ0|2

)A1h,v)

+i

2κ

(((ψ0h

)∗∇ψ0h − ψ0

h∇(ψ0h

)∗)− ((ψ0)∗∇ψ0 − ψ0∇

(ψ0)∗)

,v)

:= R1A +

9∑i=7

J1i ∀v ∈ Vr

h.

If we take v = θ1A in the above equation, from the interpolation error estimates (2.5)and Ritz projection error estimate (3.3)–(3.7) we obtain the estimates

|τJ17 | ≤

1

8‖θ1A‖2 + Ch2r+2,

|J18 | ≤ |

((|ψ0h|2 − |ψ0|2

)RhA

1, θ1A)|+ |

((|ψ0h|2 − |ψ0|2

)e1A, θ

1A

)|

≤ C‖θ1A‖2 + Ch2r+2,

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and by applying interpolation error estimates (2.5) and integration by parts, we havefurther

|J19 | ≤ C|

((ψ0h

)∗∇ψ0h −

(ψ0)∗∇ψ0, θ1A

)|

≤ C|((ψ0 − ψ0

h

)∗∇ψ0, θ1A)|+ C|

((ψ0h

)∗∇(ψ0 − ψ0h

), θ1A

)|

≤ C|((ψ0 − ψ0

h

)∗∇ψ0, θ1A)|+ C|

((ψ0 − ψ0

h

)∗∇(ψ0 − ψ0h

), θ1A

)|

+ C|((ψ0)∗∇(ψ0 − ψ0

h

), θ1A

)|

≤ C|((ψ0 − ψ0

h

)∗∇ψ0, θ1A)|+ C|

((ψ0 − ψ0

h

)∗∇(ψ0 − ψ0h

), θ1A

)|

+ C(|((ψ0 − ψ0

h

)∇(ψ0)∗, θ1A

)|+ C|

((ψ0)∗(

ψ0 − ψ0h

), div θ1A

)|)

≤ C‖ψ0 − ψ0h‖L2‖∇ψ0‖L3‖θ1A‖L6 + C‖ψ0 − ψ0

h‖L2‖∇(ψ0 − ψ0

h

)‖L3‖θ1A‖L6

+ C‖ψ0 − ψ0h‖L2‖div θ1A‖L2

≤ Chr+1‖θ1A‖H1

≤ 1

2‖θ1A‖2H1 + Ch2r+2.

With the above estimates, (3.16) reduces to

‖θ1A‖2 + τ

2‖θ1A‖2H1 ≤ 1

4‖θ1A‖2 + Cτ‖θ1A‖2 + C(τ2 + hr+1)2.

With the condition Cτ ≤ 14 , we can obtain

(3.17) ‖θ1A‖2 + τ‖θ1A‖2H1 ≤ C(τ2 + hr+1)2.

Combining (3.15) and (3.17) yields (3.10) with C0 ≥ 4C.

4. The proof of Theorem 2.1. Here we prove a slightly stronger estimate

‖θnψ‖2 + ‖θnA‖2 + τ

(n∑

m=2

(∥∥∥θmψ ∥∥∥2H1

+∥∥∥θmA∥∥∥2

H1

)+∥∥θ1ψ∥∥2H1 +

∥∥θ1A∥∥2H1

)(4.1)

≤ 1

2C1(τ

2 + hr+1)2

by mathematical induction. By Lemma 3.2, if we require C1 ≥ C0, (4.1) holds forn = 1. We assume that (4.1) holds for n ≤ k − 1. In this section, we shall find C1,which is independent of n, h, τ , such that (4.1) holds for n ≤ k.

At t = (n− 12 )τ , with the Ritz projection (3.1)–(3.2), the variational form (2.3)–

(2.4) is defined by

(∂ψ

∂t, ω

)− iκ((divA)ψ, ω) +

((i

κ∇+A

)Rhψ,

(i

κ∇+A

)ω

)(4.2)

+ (M − 1)(Rhψ, ω)−M(ψ, ω) + (|ψ|2ψ, ω) = 0 ∀ω ∈ Vrh

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and (∂A

∂t,v

)+ (div PhA, div v) + (curlPhA, curlv) + (|ψ|2PhA,v)(4.3)

+i

2κ(ψ∗∇ψ − ψ∇ψ∗,v) = (H, curlv) ∀v ∈ Vr

h.

Subtracting the FEM equations (2.6)–(2.7) from (4.2)–(4.3), respectively, we ob-tain the error equations

(Dτθ

nψ, ω

)+

((i

κ∇+An− 1

2

)θn

ψ,

(i

κ∇+An− 1

2

)ω

)+ (M − 1)

(θn

ψ, ω)

(4.4)

=

(DτRhψ

n − ∂ψn−12

∂t, ω

)+ iκ

((divAn− 1

2

)ψn−

12 −

(div An

h

)ψn

h, ω)

+

((i

κ∇+An− 1

2

)(Rhψ

n −Rhψn− 1

2

),

(i

κ∇+An− 1

2

)ω

)+((Anh +An− 1

2

)ψn

h,(Anh −An− 1

2

)ω)+ (M − 1)

(Rhψ

n −Rhψn− 1

2 , ω)

+M(ψn−

12 − ψ

n

h, ω)+(|ψnh |2ψ

n

h − |ψn− 12 |2ψn− 1

2 , ω)

− i

κ

((ω∗∇ψnh − ψ

n

h∇ω∗),An− 12 − An

h

):=

8∑i=1

Jni (ω) ∀ω ∈ Vrh

and

(Dτθ

nA,v

)+(div θ

n

A, div v)+(curl θ

n

A, curlv)+(|ψn− 1

2 |2θnA,v)(4.5)

=

(DτPhA

n − ∂An− 12

∂t,v

)+(div (PhA

n − PhAn− 1

2

), div v

)+(curl (PhA

n − PhAn− 1

2

), curlv

)+(|ψn− 1

2 |2(PhA

n − PhAn− 1

2

),v)

+i

2κ

(((ψnh)∗∇ψnh − ψnh∇

(ψnh)∗)− ((ψn− 1

2

)∗∇ψn− 12 − ψn−

12∇(ψn+

12

)∗),v)

+((|ψnh |2 − |ψn− 1

2 |2)An

h ,v)

:=

14∑i=9

Jni (v) ∀v ∈ Vrh.

In the following two subsections we will analyze the above two error equations,respectively, and in the last subsection we will give a proof of (4.1).

4.1. Estimates for (4.4). Taking ω = θn

ψ in (4.4), we now estimate the realpart of the right-hand side of the error equation (4.4) term by term.

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By noting (3.3)–(3.7), we see that

|Re(Jn1(θn

ψ

))| ≤ C‖θnψ‖

2+ C

(τ2 + hr+1

)2,

|Re(Jn3(θn

ψ

))| ≤ 1

16κ2‖θnψ‖

2

H1 + C‖θnψ‖2+ C

(τ2 + hr+1

)2,

|Re(Jn5(θn

ψ

))| ≤ C‖θnψ‖

2+ C

(τ2 + hr+1

)2,

|Re(Jn6(θn

ψ

))| ≤ C‖θnψ‖

2+ C

(τ2 + hr+1

)2.

By inverse inequalities and applying integration by parts, we have

|Re(Jn2(θn

ψ

))| = |Re

(iκ((divAn− 1

2

)ψn−

12 −

(div An

h

)Rhψ

n, θn

ψ

))|

≤ C|((div(An− 1

2 − Anh

))Rhψ

n+(divAn− 1

2

)(ψn−

12 −Rhψ

n), θn

ψ

)|

≤ C(|((An− 1

2 − Anh

)∇Rhψ

n, θn

ψ

)|+ |

((An− 1

2 − Anh

)Rhψ

n,∇θnψ

)|)

C‖divAn− 12 ‖L3‖ψn− 1

2 −Rhψn‖L2‖θnψ‖L6

≤ C(|((An− 1

2 − Anh

)∇enψ, θ

n

ψ

)|+ |

((An− 1

2 − Anh)∇ψ

n, θn

ψ

)|)

+ C(τ2 + hr+1 + ‖θnA‖

)‖θnψ‖H1

≤ C‖An− 12 − An

h‖L2‖∇enψ‖L3‖θnψ‖L6

+ C‖An− 12 − An

h‖L2‖∇ψn‖L3‖θnψ‖L6

+ C(τ2 + hr+1 + ‖θnA‖

)‖θnψ‖H1

≤ Chr−d6

(τ2 + hr+1 + ‖θnA‖

)‖θnψ‖H1

+ C(τ2 + hr+1 + ‖θnA‖

)‖θnψ‖H1

≤ 1

16κ2‖θnψ‖

2

H1 + C‖θnA‖2+ C

(τ2 + hr+1

)2.

Moreover, for Jn4 and Jn7 , by applying Young’s inequality, we have the estimates

Re(Jn4(θn

ψ

))= −‖

(Anh −An− 1

2

)θn

ψ‖2+Re

(((Anh −An− 1

2

)Rhψ

n,(Anh −An− 1

2

)θn

ψ

))+Re

(2(An− 1

2Rhψn,(Anh −An− 1

2

)θn

ψ

))− Re

(2(An− 1

2 θn

ψ,(Anh −An− 1

2

)θn

ψ

))≤ C‖θnψ‖

2+ C‖θnA‖

2+ C

(τ2 + hr+1

)2and

Re(Jn7(θn

ψ

))= Re

(((|ψnh |2 − |ψn|2

)Rhψ

n, θn

ψ

))− Re

(((|ψn− 1

2 |2 − |ψn|2)Rhψ

n, θn

ψ

))− Re

((|ψn− 1

2 |2(ψn−

12 − ψ

n+ enψ

), θn

ψ

))−(|ψnh |2θ

n

ψ, θn

ψ

)≤ |((ψnh(ψnh − ψn

)∗+(ψnh − ψn

)(ψn)∗)

Rhψn, θn

ψ

)| − ‖ψnhθ

n

ψ‖2

+ C‖θnψ‖2+ C

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≤ C‖ψnh − ψn‖‖ψnhθn

ψ‖+ C‖ψnh − ψn‖‖θnψ‖ − ‖ψnhθn

ψ‖2

+ C‖θnψ‖2+ C

(τ2 + hr+1

)2≤ C‖θnψ‖

2+ C‖θnψ‖

2+ C

(τ2 + hr+1

)2.

Finally, we estimate the term Jn8 by

Re(Jn8(θn

ψ

))≤ C|

((θn

ψ

)∗∇θnψ − θn

ψ∇(θn

ψ

)∗,An− 1

2 − Anh

)|

+ C|((θn

ψ)∗∇Rhψ

n,An− 1

2 − Anh

)|+ C|

(Rhψ

n∇(θn

ψ

)∗,An− 1

2 − Anh

)|

≤ C‖θnψ‖L6‖∇θnψ‖L2

(‖An− 1

2 − An‖L3 + ‖enA‖L3 + ‖θnA‖L3

)+ C|

((θn

ψ

)∗∇Rhψn,An− 12 − An

h

)|+ C‖∇

(θn

ψ

)∗‖‖An− 12 − An

h‖

≤ C‖θnA‖L3‖θnψ‖2H1 +1

32κ2‖θnψ‖2H1 + C‖θnψ‖L6‖∇enψ‖L3‖An− 1

2 − Anh‖L2

+ C‖θnψ‖L6‖∇ψn‖L3‖An− 12 − An

h‖L2 + C‖θnψ‖H1‖An− 12 − An

h‖


32κ2‖θnψ‖2H1 + C‖θnψ‖H1‖An− 1

2 − Anh‖


16κ2‖θnψ‖

2

H1 + C‖θn−1A ‖

2+ C(τ2 + hr+1)2,

where we have used C(‖An− 12 − An‖L3 + ‖enA‖L3) ≤ 1

32κ2 .

With the above estimates, taking ω = θn

ψ in (4.4) gives

Dτ‖θnψ‖2+ ‖

(i

κ∇+An− 1

2

)θn

ψ‖2


16κ2‖θnψ‖

2

H1 + C(‖θnψ‖

2+ ‖θnψ‖

2)

+ C‖θnA‖2+ C(τ2 + hr+1)2,

which with a similar argument as for (3.14) implies

Dτ‖θnψ‖2+

5

16κ2‖θnψ‖

2

H1(4.6)

≤ C‖θnA‖L3‖θnψ‖2H1 + C(‖θnψ‖

2+ ‖θnψ‖

2)+ C‖θnA‖

2+ C(τ2 + hr+1)2.

4.2. Estimates for (4.5). We take v = θn

A in the error equation (4.5) andestimate its right-hand side term by term. By (3.3)–(3.7), we see that

|Jn9(θn

A

)| ≤ C‖θnA‖2 + C(τ2 + hr+1)2,

|Jn10(θn

A

)| ≤ 1

4‖div θnA‖2 + C

(τ2 + hr+1

)2,

|Jn11(θn

A

)| ≤ 1

4‖curl θnA‖2 + C

(τ2 + hr+1

)2,

|Jn12(θn

A

)| ≤ C‖θnA‖2 + C

(τ2 + hr+1

)2.

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By applying Young’s inequality, we have

Jn14(θn

A

)≤ |((|ψnh |2 − |ψn− 1

2 |2)PhA

n, θn

A

)|+ |

(|ψn− 1

2 |2θnA, θn

A

)| −(|ψnh |2θ

n

A, θn

A

)≤ |((|ψnh |2 − |ψn|2

)PhA

n, θn

A

)|+ |

((|ψn|2 − |ψn− 1

2 |2)PhA

n, θn

A

)|

− ‖ψnhθn

A‖2 + C‖θnA‖2

≤ C|((ψnh − ψn

)(ψnh)∗

+ ψn(ψnh − ψn

)∗, θn

A

)| − ‖ψnhθ

n

A‖2

+ C‖θnA‖2 + C(τ2 + hr+1

)2≤ C‖ψnh − ψn‖‖ψnhθ

n

A‖+ C‖ψnh − ψn‖‖θnA‖ − ‖ψnhθn

A‖2

+ C‖θnA‖2 + C(τ2 + hr+1

)2≤ C‖θnA‖2 + C‖θnψ‖

2+ C

(τ2 + hr+1

)2.

Finally we estimate Jn13 by

|Jn13(θn

A

)| ≤ C|

((ψnh)∗∇ψnh −

(ψn−

12

)∗∇ψn− 12 , θ

n

A

)|

≤ C|((ψnh)∗∇ψnh − (ψn)∗∇ψn, θnA)|+C|

((ψn)∗∇ψn− (ψn− 1

2

)∗∇ψn− 12 , θ

n

A

)|

≤ C|((ψnh)∗∇(enψ + θnψ

)+(enψ + θnψ

)∗∇ψn, θnA)|+ Cτ2‖θnA‖≤ C|

((ψnh)∗∇(enψ + θnψ

), θn

A

)|+ C‖enψ + θnψ‖L2‖∇ψn‖L3‖θnA‖L6 + Cτ2‖θnA‖


), θn

A

)|+ C‖enψ + θnψ‖L2‖θnA‖H1 + Cτ2‖θnA‖


), θn

A

)|+ 1

32‖θnA‖2H1 + C‖θnA‖2 + C‖θnψ‖

2

+ C(τ2 + hr+1

)2.

To estimate the term C|((ψnh )∗∇(enψ + θnψ), θn

A)|, we rewrite it by

C|((ψnh)∗∇(enψ + θnψ

), θn

A

)|

= C|((ψn)∗∇(enψ + θnψ

), θn

A

)|+ C|

((enψ + θnψ

)∗∇(enψ + θnψ), θn

A

)|.

Using integration by parts, the first term of the right-hand side of the above equationis estimated by

C|((ψn)∗∇(enψ + θnψ

), θn

A

)|

= C|((∇ψn

)∗(enψ + θnψ

), θn

A

)|+ C|

((ψn)∗(

enψ + θnψ), div θ

n

A

)|

≤ 1


2+ C

(τ2 + hr+1

)2and the second term is bounded by

C|((enψ + θnψ

)∗∇(enψ + θnψ), θn

A

)|

≤ C|((enψ + θnψ

)∇enψ, θ

n

A

)|+ C|

(enψ∇θnψ, θ

n

A

)|+ C|

(θnψ∇θnψ, θ

n

A

)|

≤ C(‖enψ‖L2 + ‖θnψ‖L2

)‖∇enψ‖L3‖θnA‖L6 + C‖enψ‖L2‖∇θnψ‖L3‖θnA‖L6

+ C‖θnψ‖L3‖∇θnψ‖L2‖θnA‖L6

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≤ Chr−d6

(‖enψ‖L2 + ‖θnψ‖L2

)‖θnA‖H1 + Chr−

d6 ‖θnψ‖L2‖θnA‖H1

+ C‖θnψ‖L3‖∇θnψ‖L2‖θnA‖L6

≤ C‖θnψ‖L3‖∇θnψ‖L2‖θnA‖H1 +1


2+ Ch2r+2.

It follows that

|Jn13(θn

A

)| ≤ C‖θnψ‖L3‖∇θnψ‖L2‖θnA‖H1 +

1

8‖θnA‖2H1

+ C‖θnA‖2 + C‖θnψ‖2+ C(τ2 + hr+1)2.

With the above estimates, (4.5) reduces to

Dτ‖θnA‖2 + 5

8‖θnA‖2H1(4.7)

≤ C|θnψ‖L3‖∇θnψ‖‖θn

A‖H1 + C(‖θnA‖2 + ‖θnψ‖

2)+ C(τ2 + hr+1)2.

4.3. The proof of the estimate (4.1). Adding the estimates (4.6) and (4.7)together, we have

Dτ

(‖θnψ‖

2+ ‖θnA‖2

)+

5

16κ2‖θnψ‖

2

H1 +5

8‖θnA‖2H1(4.8)

≤ C(‖θnA‖L3‖θnψ‖2H1 + ‖θnψ‖L3‖∇θnψ‖‖θ

n

A‖H1

)+ C

(‖θnψ‖

2+ ‖θnψ‖

2)+ C

(‖θnA‖2 + ‖θnA‖

2)+ C(τ2 + hr+1)2.

To obtain the optimal error estimate (4.1) unconditionally from the above inequality,we consider two different cases.

For τ ≤ h, by the assumption of the induction and inverse inequalities, we have

C‖θnA‖L3 ≤ Ch−d6 ‖θnA‖L2 ≤ Ch−

d6

√2C

121 (h

2 + hr+1) ≤ 2C√2C1 h

1− d6 ,

which leads to

C‖θnA‖L3‖θnψ‖2H1 ≤ 1

16κ2‖θnψ‖2H1(4.9)

when 2C√2C1 h

1−d6 ≤ 1

16κ2 . Similarly, we have

C‖θnψ‖L3‖∇θnψ‖ ≤ Ch−1− d6 ‖θnψ‖2L2 ≤ Ch−1−d

61

2C1(h

2 + hr+1)(τ2 + hr+1),

from which we have further

(4.10) ‖θnψ‖L3‖∇θnψ‖‖θn

A‖H1 ≤ 1

8‖θnA‖2H1 + C(τ2 + hr+1)2,

where we assume that C1h1−d

6 ≤ 1.For h ≤ τ , by the assumption of the induction and Lemma 3.1, we get the estimate

‖θn−1ψ ‖H1 ≤ 2

√Tτ−1

(n−1∑m=2

τ‖θn−1

ψ ‖2H1 + τ‖θ1ψ‖2H1

) 12

≤ 2√2C1T τ.

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Similarly, we can obtain

‖θn−1A ‖H1 ≤ 2

√2C1T τ.

By the above two inequalities, we have

C‖θnA‖L3 ≤ C‖θnA‖12

L2‖θnA‖12

H1 ≤ C√C1 τ

32 ,

C‖θnψ‖L3‖∇θnψ‖ ≤ ‖θnψ‖12

L2‖θnψ‖32

H1 ≤ CC1τ52 .

Therefore, when C√C1 τ

32 ≤ 1

16κ2 and CC1τ12 ≤ 1

C‖θnA‖L3‖θnψ‖2H1 ≤ 1

16κ2‖θnψ‖2H1 ,(4.11)

C‖θnψ‖L3‖∇θnψ‖‖θn

A‖H1 ≤ 1

8‖θnA‖2H1 + Cτ4.(4.12)

From (4.8), for both the cases τ ≤ h and h ≤ τ , we obtain

Dτ

(‖θnψ‖

2+ ‖θnA‖2

)+

1

4κ2‖θnψ‖

2

H1 +1

2‖θnA‖2H1(4.13)

≤ C(‖θnψ‖

2+ ‖θnψ‖

2+ ‖θnA‖2 + ‖θnA‖

2)+ C(τ2 + hr+1)2,

which in turn implies

‖θnψ‖2 + ‖θnA‖2 + τ

(n∑

m=2

(‖θmψ ‖2

H1 + ‖θmA‖2H1

)+ ‖θ1ψ‖

2

H1 + ‖θmA‖2H1

)(4.14)

≤ τC

n∑m=1

(‖θmψ ‖2 + ‖θmA‖2

)+ τC

n∑m=1

(τ2 + hr+1)2 +1

2C0(τ

2 + hr+1)2.

Lemma 4.1 (discrete Gronwall’s inequality [19]). Let τ , B and ak, bk, ck, γk,for integers k ≥ 0, be nonnegative numbers such that

an + τ

n∑k=0

bk ≤ τ

n∑k=0

γkak + τ

n∑k=0

ck +B for n ≥ 0,

suppose that τγk < 1 for all k, and set σk = (1− τγk)−1. Then

an + τ

n∑k=0

bk ≤ exp

(τ

n∑k=0

γkσk

)(τ

n∑k=0

ck +B

)for n ≥ 0.

Here we let

a0 = 0, an = ‖θnψ‖2+ ‖θnA‖2 for n ≥ 1,

b0 = 0, b1 = ‖θ1ψ‖2

H1 + ‖θ1A‖2H1 , bn = ‖θnψ‖2

H1 + ‖θnA‖2H1 for n ≥ 2,

c0 = 0, cn = C(τ2 + hr+1)2 for n ≥ 1,

γk = C, σk =1

1− Cτ, B =

1

2C0(τ

2 + hr+1)2.

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For Cτ ≤ 12 , from the discrete Gronwall’s inequality, we can deduce that

‖θnψ‖2+ ‖θnA‖2 + τ

(n∑

m=2

(‖θmψ ‖2

H1 + ‖θmA‖2H1

)+ ‖θ1ψ‖

2

H1 + ‖θ1A‖2H1

)(4.15)

≤ exp

(TC

1− Cτ

)(TC +

C0

2

)(τ2 + hr+1)2

≤ exp(2TC)

(TC +

C0

2

)(τ2 + hr+1)2.

Thus, (4.1) holds for n = k if we take C1 ≥ exp(2TC)(2TC + C0). We complete theinduction.

Theorem 2.1 is proved by combining (4.1) and the projection error estimates in(3.3)–(3.7).

5. Numerical results. In this section, we only provide numerical experimentsfor the linearized scheme (2.6)–(2.9) to confirm our error analysis and show the effi-ciency of the scheme. We use the free software FEniCS [24] to do all the numericalsimulations.

Example 5.1. First we consider the equations

η∂ψ

∂t− iηκ(divA)ψ +

(i

κ∇+A

)2

ψ + (|ψ|2 − 1)ψ = g,(5.1)

∂A

∂t−∇divA+ curl curlA+

i

2κ(ψ∗∇ψ − ψ∇ψ∗) + |ψ|2A = curlH + f(5.2)

subject to

∂ψ

∂n= 0, A · n = 0, curlA = H,

ψ(x, 0) = ψ0(x), A(x, 0) = A0(x), in Ω,

where Ω = (0, 1) × (0, 1) and η = κ = 1. The functions f , g, ψ0, and A0 are chosencorrespondingly to the exact solution

ψ = exp (−t)(cos(πx) + i cos(πy)),

A = (exp(y − t) sin(πx), exp(x− t) sin(πy))

with

H = exp(x− t) sin(πy)− exp(y − t) sin(πx).

We use a uniform triangular partition with M + 1 nodes in each direction (seeFigure 1 forM = 20). We solve the system by the proposed linearized Crank–NicolsonGalerkin finite element methods (2.6)–(2.9) with linear elements and quadratic ele-ments, respectively. To confirm our error estimates on the L2 norm, we choose τ = hfor the linear element method and τ = h

32 for the quadratic element method. We

present in Table 1 the errors of the linear Galerkin method and in Table 2 the errorsof the quadratic Galerkin method, respectively, at time T = 0.5, 1.0, 1.5, 2.0. We cansee clearly that the L2 norm errors of both ψ and A are proportional to hr+1, r = 1,2, which confirm our theoretical analysis.

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0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Fig. 1. A uniform mesh with M = 20.

Table 1

L2 norm errors of ψ and A with linear FEM and h = τ = 1M

.

‖ψnh − ψ(·, tn)‖

T M = 20 M = 40 M = 80 Order0.5 7.7016e-03 1.9242e-03 4.7748e-04 1.981.0 1.5487e-02 3.8566e-03 9.5916e-04 2.001.5 2.5079e-02 6.2363e-03 1.5512e-03 2.012.0 4.0260e-02 1.0006e-02 2.4886e-03 2.00

‖Anh −A(·, tn)‖


Table 2

L2 norm errors of ψ and A with quadratic FEM, h = 1M

and τ = h32 .

‖ψnh − ψ(·, tn)‖




To show the unconditional stability of the proposed scheme, we test the quadraticGalerkin method on a fine mesh with h = 1

160 and different time steps τ = 164 ,

132 ,

116 .

The results are presented in Table 3. From Table 3, it is clear that the L2 errorsdecrease in order of O(τ2). Numerical results indicate that the scheme is stablefor the large time steps, although the numerical results with τ = 1

16 seem not veryaccurate.

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Table 3

L2 norm errors of ψ and A with quadratic FEM on a fine mesh with h = 1160

.

‖ψnh − ψ(·, tn)‖

T τ = 164

τ = 132

τ = 116

0.5 2.7352e-04 1.0529e-03 4.0079e-031.0 1.8971e-04 7.0910e-04 2.5771e-031.5 2.1947e-04 8.0547e-04 2.8362e-032.0 3.2194e-04 1.1744e-03 4.0898e-03


T τ = 164

τ = 132

τ = 116

0.5 4.0542e-05 1.5331e-04 5.9704e-041.0 1.2238e-05 4.6947e-05 1.7878e-041.5 7.6931e-06 2.9270e-05 1.1089e-042.0 6.0806e-06 2.1538e-05 7.1775e-05

0

0.5

1

0

0.5

10

0.2

0.4

0.6

0.8

1

XY

|ψfine |2

0 0.2 0.4 0.6 0.8 1

0

0.5

13.1

3.2

3.3

3.4

3.5

3.6

XY

curlAfine

Fig. 2. Surface plots of |ψfine|2 and curlAfine with linear FEM on a fine mesh with M = 256and a small time step τ = 0.002.

Example 5.2. In the second example, we simulate the vortex of TDGL equationswith domain Ω = (0, 1) × (0, 1). Numerical simulations were done with differentmethods [5, 27, 28]. Here, we set the Ginzburg–Landau parameter κ = 10, η = 1. Wetake ψ0 = 0.8+ i0.6 and A0 = (0, 0)T as the initial conditions, and the initial state ispurely in the superconducting state. The applied magnetic field H = 3.5.

In our computations, we find that the full vortex state is stable at T = 20.Therefore, we set the final time T = 20 in this example. We first do simulationwith a linear Galerkin method on a uniform mesh with M = 256 and a small timestep τ = 0.002. We denote this numerical solution by ψfine and Afine. The surfaceplots of the density function |ψfine|2 and the magnetic field curlAfine are shown inFigure 2. Four vortices of |ψfine|2 and the corresponding magnetic field curlAfine

can be observed clearly. Due to the boundary condition (1.3), there are also cornersingularities for the magnetic field curlAfine [27]. To show the unconditional stabilityof the scheme, we use a linear FEM to solve the problem on a fixed uniform mesh ofM = 100 with different time steps τ = 0.2 and 0.01. The computed ψ for τ = 0.2and τ = 0.01 are denoted by ψτ1 and ψτ2 , respectively. It should be noted that theplots of |ψτ1 |2 and |ψτ2 |2 are quite similar to the plots of |ψfine|2 and thus are omittedhere. Moreover, since ψfine is computed with relatively small τ and h, we can take

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00.2

0.40.6

0.81

0

0.5

10

0.005

0.01

0.015

0.02

0.025

XY 0

0.20.4

0.60.8

1

0

0.5

10

0.005

0.01

0.015

0.02

0.025

XY

Fig. 3. Surface plots of the error |ψfine − ψτ1 | (τ = 0.2 left) and |ψfine − ψτ2 | (τ = 0.01 right).

ψfine to be the exact solution. We provide error plots for |ψτ1 −ψfine| and |ψτ2 −ψfine|in Figure 3. Numerical results show that the scheme is unconditionally stable and notime step restriction is needed.

6. Conclusions. We have presented the optimal L2 error estimate of an uncou-pled and linearized Crank–Nicolson Galerkin FEM for the TDGL equations. To thebest of our knowledge, no analysis has been given for the popular linearized Crank–Nicolson Galerkin methods, although numerical simulations for the linearized schemewere made in [27] and the efficiency was observed clearly. More important is thatthe Crank–Nicolson Galerkin method provides the optimal accuracy unconditionally.Numerical experiments presented in this paper show the efficiency of the method andconfirm our theoretical analysis. Also, our analysis can be easily extended to thelinearized scheme (2.10)–(2.11) and the nonlinear scheme proposed in [28] to obtainoptimal L2 error estimate without any restrictions on the time step.

Acknowledgment. The authors would like to thank the referee for several help-ful suggestions.

REFERENCES

[1] S. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods, Springer,New York, 2002.

[2] J. R. Cannon and Y. Lin, A priori L2 error estimates for finite-element methods for nonlineardiffusion equations with memory, SIAM J. Numer. Anal., 27 (1990), pp. 595–607.

[3] S. J. Chapman, S. D. Howison, and J. R. Ockendon, Macroscopic models for superconduc-tivity, SIAM Rev., 34 (1992), pp. 529–560.

[4] Z. Chen, Mixed finite element methods for a dynamical Ginzburg-Landau model in supercon-ductivity, Numer. Math., 76 (1997), pp. 323–353.

[5] Z. Chen and S. Dai, Adaptive Galerkin methods with error control for a dynamical Ginzburg-Landau model in superconductivity, SIAM J. Numer. Anal., 38 (2001), pp. 1961–1985.

[6] Z. Chen and K.-H. Hoffmann, Numerical studies of a non-stationary Ginzburg-Landau modelfor superconductivity, Adv. Math. Sci. Appl., 5 (1995), pp. 363–389.

[7] Z. Chen, K.-H. Hoffmann, and J. Liang, On a non-stationary Ginzburg-Landau supercon-ductivity model, Math. Methods Appl. Sci., 16 (1993), pp. 855–875.

[8] E. Coskun, Computational simulation of flux trapping and vortex pinning in Type-II super-conductors, Appl. Math. Comput., 106 (1999), pp. 31–49.

[9] E. Coskun and M. K. Kwong, Simulating vortex motion in superconducting films with thetime-dependent Ginzburg-Landau equations, Nonlinearity, 10 (1997), pp. 579–593.

[10] Q. Du, Numerical approximations of the Ginzburg-Landau models for superconductivity,J. Math. Phys., 46 (2005), pp. 095–109.

Dow

nloa

ded

02/1

5/15

to 1

44.2

14.7

4.34

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php



[11] Q. Du, M. D. Gunzburger, and J. S. Peterson, Analysis and approximation of the Ginzburg-Landau model of superconductivity, SIAM Rev., 34 (1992), pp. 54–81.

[12] T. Dupont, G. Fairweather, and J. P. Johnson, Three-level Galerkin methods for parabolicequations, SIAM J. Numer. Anal., 11 (1974), pp. 392–410.

[13] L. C. Evans, Partial Differential Equations, Grad. Stud. Math. 19, AMS, Providence, RI, 1998.[14] H. Frahm, S. Ullah, and A. Dorsey, Flux dynamics and the growth of the superconducting

phase, Phys. Rev. Lett., 66 (1991), pp. 3067–3070.[15] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations,

Springer-Verlag, Berlin, 1986.[16] L. P. Gor’kov and G. M. Eliashberg, Generalization of the Ginzburg-Landau equations for

non-stationary problems in the case of alloys with paramagnetic impurities, J. Exp. Theor.Phys., 27 (1968), pp. 328–334.

[17] W. D. Gropp et al., Numerical simulation of vortex dynamics in type-II superconductors,J. Comput. Phys., 123 (1996), pp. 254–266.

[18] Y. He and W. Sun, Stabilized finite element method based on the Crank-Nicolson extrapo-lation scheme for the time-dependent Navier-Stokes equations, Math. Comp., 76 (2007),pp. 115–136.

[19] J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationaryNavier-Stokes problem IV: Error analysis for second-order time discretization, SIAM J.Numer. Anal., 27 (1990), pp. 353–384.

[20] Y. Hou, B. Li, and W. Sun, Error analysis of splitting Galerkin methods for heat and sweattransport in textile materials, SIAM J. Numer. Anal., 51 (2013), pp. 88–111.

[21] C. D. Levermore and M. Oliver, The complex Ginzburg-Landau equation as a model problem,in Dynamical System and Probabilistic Methods for Nonlinear Waves, Lectures Appl. Math.31, AMS, Providence, RI, 1996, pp. 141–190.

[22] B. Li, H. Gao, and W. Sun, Unconditionally optimal error estimates of a Crank–NicolsonGalerkin method for the nonlinear thermistor equations, SIAM J. Numer. Anal., 52 (2014),pp. 933–954.

[23] F. Liu, M. Mondello, and N. Goldenfeld, Kinetics of the superconducting transition, Phys.Rev. Lett., 66 (1991), pp. 3071–3074.

[24] A. Logg, K.-A. Mardal, G. N. Wells, et al., Automated Solution of DifferentialEquations by the Finite Element Method, Springer, Berlin, 2012.

[25] S. Lu and Q. Lu, Exponential attractor for the 3D Ginzburg-Landau type equation, NonlinearAnal., 67 (2007), pp. 3116–3135.

[26] T. Matsuo and D. Furihata, Dissipative or conservative finite-difference schemes forcomplex-valued nonlinear partial differential equations, J. Comput. Phys., 171 (2001),pp. 425–447.

[27] M. Mu, A linearized Crank-Nicolson-Galerkin method for the Ginzburg-Landau model, SIAMJ. Sci. Comput., 18 (1997), pp. 1028–1039.

[28] M. Mu and Y. Huang, An alternating Crank-Nicolson method for decoupling the Ginzburg-Landau equations, SIAM J. Numer. Anal., 35 (1998), pp. 1740–1761.

[29] A. Rodriguez-Bernal, B. Wang, and R. Willie, Asymptotic behavior of the time-dependentGinzburg-Landau equations of superconductivity, Math. Methods Appl. Sci., 22 (1999),pp. 1647–1669.

[30] W. Sun and Z. Z. Sun, Finite difference methods for a nonlinear and strongly coupled heatand moisture transport system in textile materials, Numer. Math., 120 (2012), 153–187.

[31] Q. Tang and S. Wang, Time dependent Ginzburg-Landau equations of superconductivity,Phys. D, 88 (1995), pp. 139–166.

[32] V. Thomee, Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, Berlin,2006.

[33] M. Tinkham, Introduction to Superconductivity, McGraw-Hill, New York, 1975.[34] M. F. Wheeler, A priori L2 error estimates for Galerkin approximations to parabolic partial

differential equations, SIAM J. Numer. Anal., 10 (1973), pp. 723–759.[35] T. Winiecki and C. S. Adams, A fast semi-implicit finite difference method for the TDGL

equation, J. Comput. Phys., 179 (2002), pp. 127–139.[36] Q. Xu and Q. S. Chang, Difference methods for computing the Ginzburg-Landau equation in

two dimensions, Numer. Methods Partial Differential Equations, 27 (2011), pp. 507–528.

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optimal error estimates of linearized crank-nicolson galerkin

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