The existence and uniqueness of the fully discrete scheme are derived, and the stability and convergence analysis of the fully discrete scheme are proved rigorously

Electronic Journal of Differential Equations, Vol. 2019 (2019), No. 76, pp. 1–17.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

CRANK-NICOLSON LEGENDRE SPECTRAL APPROXIMATION FOR SPACE-FRACTIONAL ALLEN-CAHN EQUATION

WENPING CHEN, SHUJUAN L ¨U, HU CHEN, HAIYU LIU

Abstract. In this article, we consider spectral methods for solving the initial- boundary value problem of the space fractional-order Allen-Cahn equation. A fully discrete scheme based on the modified Crank-Nicolson scheme in time and the Legendre spectral method in space is established. The existence and uniqueness of the fully discrete scheme are derived, and the stability and convergence analysis of the fully discrete scheme are proved rigorously. By constructing a fractional duality argument, the corresponding optimal error estimates inL² andH^α norm are derived, respectively. Also, numerical ex- periments are performed to support the theoretical results.

1. Introduction

Research of fractional differential equations has been a lively topic in mathe- matical theory and real applications in the last few decades. For most fractional differential equations, however, we cannot obtain the exact solutions, it is natural to resort to numerical solutions. Up to now, there are several numerical techniques to solve fractional differential equations, such as finite difference methods [8, 17], finite element methods [7, 15, 21], spectral methods [9, 10, 12, 19].

Allen-Cahn equation was introduced in 1979 [2] to model phase transitions in iron alloys, it has become a basic model equation for the diffuse interface approach developed to study phase transitions and interfacial dynamics in materials science.

There has been an increasing interests in Allen-Cahn equation from local to mem- ory (time fractional) or nonlocal (space fractional) case [6, 20]. The fractional Allen-Cahn equation replaced the standard temporal or/and spatial integer order differential operator by a corresponding fractional order one, such as Riemann- Liouville, Caputo, Riesz, fractional Laplacian operators, etc.

In this article, we study the spectral approximation to the following space- fractional Allen-Cahn equation (SFACE)

ut−²L^(α)u+f(u) = 0, x∈Λ, t∈(0, T], u(x,0) =u0(x), x∈Λ,

u(±1, t) = 0, t∈[0, T],

(1.1)

2010Mathematics Subject Classification. 35R11, 65M06, 65M70, 65M12.

Key words and phrases. Space-fractional Allen-Cahn equation; Legendre spectral method;

modified Crank-Nicolson scheme; stability; convergence.

c

2019 Texas State University.

Submitted August 28, 2018. Published May 31, 2019.

1

where Λ = (−1,1), α ∈ (1/2,1), u= u(x, t) represents the concentration of one of the species of the alloy, the parameter represents the diffuse interface width, f(u) = u³−u, the nonlinear term, is the derivative of a free energy double-well potentialF(u) = ¹₄(u²−1)². OperatorL^(α)in the Riesz case is defined by

L^(α)u= ∂^2αu

∂|x|^2α =− 1

2 cosπα ⁻¹D^2α_x u+_xD^2α₁ u ,

where₋₁D^2α_x ,_xD₁^2αrepresent the left and right Riemann-Liouville (R-L) fractional derivatives operators, respectively. Forn−1< β < n,n∈N, the operators₋₁D^β_x andxD^β₁ are defined as

−1D_x^βu= 1 Γ(n−β)

dⁿ dxⁿ

Z x

−1

(x−s)^n−β−1u(s) ds,

xD₁^βu= (−1)ⁿ Γ(n−β)

dⁿ dxⁿ

Z 1

x

(s−x)^n−β−1u(s) ds,

(1.2)

where Γ(·) is the standard Gamma function.

The fractional Allen-Cahn equation (1.1) can be viewed as anL²-gradient flow of the following fractional version of Ginzburg-Landau-Wilson free energy functional

E(u) = Z

Λ

F(u)−²

2uL^(α)u dx=²

2|u|²_α+ Z

Λ

F(u) dx.

Recently, there have been several studies on the SFACE. Akagi, et al. [1] proved the existence and uniqueness of weak solutions to the related initial-boundary value problem of the SFACE after setting a proper functional framework. Hou et al.

[8] considered Crank-Nicolson scheme in time and second order central difference approach in space for solving the SFACE with small perturbation and strong non- linearity, a nonlinear iteration algorithm is proposed and the unique solvability, en- ergy stability and convergence are proved. Burrage et al. [5] solved the SFACE by implicit finite element method on both structured and unstructured grids. Bueno- Orovio et al. [4] provided a numerical algorithm based on Fourier spectral method in space and backward Euler discretization in time to solve the SFACE. However, there is no theoretical analysis has been provided in [4, 5].

In this article, we construct a numerical approach by applying the modified Crank-Nicolson scheme in temporal and the Legendre Galerkin spectral method in spatial discretizations to (1.1). The existence and uniqueness of the fully discrete scheme are proved. The stability and convergence are derived strictly by introducing a fractional duality argument. It will be shown that the convergence rate of the numerical scheme is O(τ²+N^−m) inL²-norm.

The organization of this article is as follows. We commence by reviewing some preliminaries of fractional order functional spaces endowed with inner products and norms, and give some useful lemmas in the next section. In section 3, The fully discrete spectral scheme is constructed by applying Crank-Nicolson difference scheme to temporal discretization and Legendre spectral method to the spatial component, the existence and uniqueness of the fully discrete scheme are derived.

In section 4, the stability and convergence analysis of the fully discrete scheme are strictly proved, respectively. We present some numerical experiments in section 5, which support the theoretical results. We conclude by summary and discussion in the last section.

2. Preliminaries

In this section, we introduce some definitions and notation of fractional derivative spaces endowed with inner products and norms, then give some basic properties of fractional derivative and some lemmas, which will be used in the context.

The L²(Λ) inner product is denoted by (·,·) and the L^p(Λ) norm by k · kL^p

with the special case ofL²(Λ) andL^∞(Λ) norms being written ask · kandk · k_∞, respectively. For k ∈ N, we denote the seminorm and norm associated with the Sobolev spaceH^k(Λ) by| · |k andk · kk, respectively. For nonnegative real number r∈R⁺\Z⁺, we useH^r(Λ) to denote the fractional Sobolev spaces, the semi-norm

| · |r and norm k · kr will defined below. Let E be a Sobolev space, we define space-time functional spaceL²(0, T;E) as

L²(0, T;E) :=

u: (0, T)7→E: Z T

0

kuk²_Edt <∞, uis measurableo , and similarly we can define some other spaces for space-time functions. Throughout this article we use C to denote a generic nonnegative constant whose actual value may change from line to line.

Definition 2.1 (see [7, 15]). Let r >0. Define the semi-norm

|u|r= |ω|^ruˆ

_L2(

R)=Z

R

|ω|^2r|ˆu|²dω^1/2 , and the norm

kuk_r= kuk²+|u|²_r^1/2 ,

whereuˆ denote the Fourier transform ofu. Define H₀^r(Λ)as the closure ofC₀^∞(Λ) in H^r(Λ)with respect to norm k · kr, and useH^−r(Λ) to denote the dual space of H₀^r(Λ), with norm denoted by k · k_−r.

Remark 2.2 (see [15]). Let ˜u be the expansion of u by zero outside of Λ, then

|u|_r=|u|˜_Hr(R).

Next, we introduce some useful fractional derivative spaces and related proper- ties, which are used in the formulation of the numerical analysis, one can refer to [7, 15] for more details.

Definition 2.3 (see [7, 15]). Letµ >0. Define the semi-norm

|u|_J^µ

L(Λ)=k₋₁D^µ_xuk, and the norm

kuk_J^µ

L(Λ)=

kuk²+|u|²_Jµ L(Λ)

1/2

.

DenoteJ_L,0^µ (Λ) as the closure ofC₀^∞(Λ) with respect to normk · k_J^µ

L(Λ). Definition 2.4 (see [7, 15]). Letµ >0. Define the semi-norm

|u|_J^µ

R(Λ)=kxD₁^µuk, and the norm

kuk_J^µ

R(Λ)= kuk²+|u|²_J^µ

R(Λ)

1/2

.

DenoteJ_R,0^µ (Λ) as the closure ofC₀^∞(Λ) with respect to normk · k_J^µ

R(Λ).

Definition 2.5 (see [7, 15]). Letµ >0,µ6=n−¹₂, n∈N. Define the semi-norm

|u|_J^µ

S(Λ)=

(₋₁D_x^µu,xD₁^µu)

1/2, and the norm

kuk_J^µ

S(Λ)= kuk²+|u|²_J^µ

S(Λ)

1/2

.

DefineJ_S,0^µ (Λ) as the closure ofC₀^∞(Λ) with respect to normk · k_J^µ

S(Λ).

Remark 2.6. If the domain Λ in definitions 2.3–2.5 replaced by the entire lineR, the corresponding semi-norms should be denoted, respectively, by

|u|_J^µ

L(R)=k−∞D_x^µukL²(R),

|u|_J^µ

R(R)=kxD_∞^µukL²(R),

|u|_J^µ

S(R)= |(_−∞D^µ_xu,xD_∞^µu)|^1/2 .

LetJ_L^µ(R),J_R^µ(R),J_S^µ(R), andH^µ(R) denote the closure ofC₀^∞(R) with respect tokuk_J^µ

L(R),kuk_J^µ

R(R), kuk_J^µ

S(R)andkuk_Hµ(R), respectively.

Lemma 2.7 (see [7, 15]). Letµ >0,µ6=n−1/2,n∈N. Then

(1) J_L,0^µ (Λ), J_R,0^µ (Λ), J_S,0^µ (Λ), and H₀^µ(Λ) are equal, with equivalent semi- norms and norms;

(2) J_L^µ(R), J_R^µ(R), J_S^µ(R), and H^µ(R) are equal, with equivalent semi-norms and norms;

(3) A functionu∈L²(R)belongs toJ_L^µ(R)if and only if|ω|^µuˆ∈L²(R), specif- ically|u|_J^µ

L(R)=k|ω|^µukˆ _L2(R)=|u|_Hµ(R). Similarly,|u|_J^µ

R(R)=|u|_Hµ(R). In what follows, we will use H₀^α(Λ) uniformly by the equivalent property of J_L,0^α (Λ), J_R,0^α (Λ) and H₀^α(Λ), and make no distinction between the three of them unless otherwise stated.

Lemma 2.8 (see [7, 15]). Letµ >0 be given. Then (₋₁D^µ_xu,xD^µ₁u) = (_−∞D^µ_xu,˜ xD^µ_∞˜u)

= cos(πµ)k_−∞D_x^µuk˜ ²_L2(R)

= cos(πµ)kxD^µ_∞uk˜ ²_L2(R). Hence we have the following relations.

Lemma 2.9 (see [7, 15]). Letµ >0,Λ = (−1,1),u∈H₀^µ(Λ). Then (₋₁D_x^µu,_xD₁^µu) = cos(πµ)|u|²_µ.

Proof. We can obtain the result by Remark 2.2 and Lemmas 2.7, 2.8, immediately.

Via integration by parts, one can verify readily the following result.

Lemma 2.10 (see [16]). Let0< s <1,u∈H₀^2s(Λ),v∈H₀^s(Λ). Then we have (₋₁D^2s_xu, v) = (₋₁D_x^su,xD^s₁v), (xD^2s₁ u, v) = (xD^s₁u,₋₁D_x^sv).

Lemma 2.11 (Fractional Poincar´e-Friedrichs inequality [7, 15]). Foru∈H₀^µ(Λ), kuk6C|u|µ,

and for0< s < µ,s6=n−1/2, n∈N,|u|s6C|u|µ.

Lemma 2.12 (Gagliardo-Nirenberg inequality [13]). Let Ω ⊂ Rⁿ be a bounded domain having the cone property and letu∈L^q(Ω) and its derivatives of orderm, D^mu, belong toL^r(Ω),16q, r6∞. For the derivativesD^ju,06j < m, we have

kD^jukL^p 6c kD^mukL^r +kukL^q

s

kuk^1−s_Lq , where

1 p= j

n+s1 r−m

n

+ (1−s)1 q

for allsin the interval _m^j 6s61, (the constantcdepending only onn, m, j, q, r, s), with the following exceptional case:

If 1< r <∞, andm−j−n/r is a nonnegative integer then (2.12) holds only for ssatisfying j/m6s <1.

The following discrete Gronwall’s inequality is also used in the theoretical anal- ysis.

Lemma 2.13 (Discrete Gronwall Lemma [14]). Assume that kn is a non-negative sequence, and that the sequenceφn satisfies

φ₀6g₀, φ_n6g₀+

n−1

X

s=0

p_s+

n−1

X

s=0

k_sφ_s, n>1.

Then ifg0>0 andpn>0forn>0, it follows that φ_n6

g₀+

n−1

X

s=0

p_s

expⁿ⁻¹X

s=0

k_s

, n>1.

The following lemma will be used in the proof of the existence of numerical solutions.

Lemma 2.14([18]). LetX be a finite dimensional Hilbert space with inner product (·,·) and norm k · k, and Let P be a continuous mapping from X into itself such that

(P(ξ), ξ)>0 forkξk=K >0.

Then there existsξ∈X,kξk6K, such thatP(ξ) = 0.

We define a(u, v), 1

2 cosπα (₋₁D^α_xu,_xD^α₁v) + (_xD₁^αu,₋₁D^α_xv)

, ∀u, v∈H₀^α(Λ), (2.1) for convenience. By the linearity of the left and right R-L derivatives, we can verify readily thata(u, v) is a symmetric bilinear form, which has the following property.

Lemma 2.15. The bilinear forma(·,·)is continuous and coercive.

Proof. By H¨older inequality, Lemmas 2.7 and 2.11, yield

|a(u, v)|6 1

2|cosπα|(k−1D^α_xuk kxD^α₁vk+kxD₁^αuk k−1D^α_xv)k) 6C₁

u _α

v

_α, ∀u, v∈H₀^α(Λ), i.e.,a(·,·) is continuous onH₀^α(Λ)×H₀^α(Λ).

On the other hand, by Lemmas 2.9 and 2.11, we have a(u, u) = 1

2 cosπα (−1D_x^αu,xD₁^αu) + (xD^α₁u,−1D_x^αu)

=|u|²_α, ∀u∈H₀^α(Λ), viz.,a(·,·) is coercive onH₀^α(Λ). The proof is complete.

Let PN(Λ) be the set of all algebraic polynomials defined on domain Λ with the degree less than or equal to N ∈ Z⁺. V_N⁰ = PN(Λ)∩H₀¹(Λ). The following projector Π^1,0_N , which is used below, can be found in [3].

Let Π^1,0_N :H₀¹(Λ)7→V_N⁰ be the orthogonal projection operator such that

∂x(u−Π^1,0_N u), ∂xϕ

= 0, ∀ϕ∈V_N⁰.

Lemma 2.16 (see [3]). Let s be a real number. For any nonnegative real number r,06s616r, there exists a positive constant C depending only on r such that for any functionuin H₀^s(Λ)∩H^r(Λ), the following estimate holds

ku−Π^1,0_N uk_s6CN^s−rkuk_r. We define projection Π^α,0_N :H₀^α(Λ)7→V_N⁰, such that

a(u−Π^α,0_N u, v) = 0, ∀v∈V_N⁰. (2.2) For the operation Π^α,0_N , we have the following result.

Lemma 2.17. Let α∈ (1/2,1), r >1 be a real number. There exists a positive constantCdepending only onr, such that for anyu∈H₀^α(Λ)∩H^r(Λ), the following estimates hold

|u−Π^α,0_N u|α6CN^α−rkukr, ku−Π^α,0_N uk6CN^−rkukr.

Proof. By (2.2) and the continuous and coercivity of the bilinear form a(·,·), we have

|u−Π^α,0_N u|²_α=a(u−Π^α,0_N u, u−Π^α,0_N u)

=a(u−Π^α,0_N u, u−Π^1,0_N u)

6C|u−Π^α,0_N u|_αku−Π^1,0_N uk_α, ∀u∈V_N⁰. Therefore, by Lemma 2.16, we obtain

|u−Π^α,0_N u|α6Cku−Π^1,0_N ukα6CN^α−rkukr, 1

2 < α6r.

Next we estimate the error ku−Π^α,0_N uk by using a duality argument. For any g∈L²(Λ), we consider the auxiliary problem

−L^(α)w=g, in Λ,

w= 0, on∂Λ. (2.3)

By the definition ofL^(α) and Lemma 2.7, we obtain

kwk2α6Ckgk. (2.4)

The weak form of (2.3) is as follows:

a(ϕ, w) = (g, ϕ), ∀ϕ∈H₀^α(Λ).

Takingϕ=u−Π^α,0_N u, we obtain

(g, u−Π^α,0_N u) =a(u−Π^α,0_N u, w)

6C1ku−Π^α,0_N ukαkw−Π^1,0_N wkα

6CN^−rkukrkwk2α.

(2.5)

Using (2.4) and (2.5), we have ku−Π^α,0_N uk= sup

g∈L²(Λ), g6=0

|(g, u−Π^α,0_N u)|

kgk 6CN^−rkuk_r. (2.6)

The proof is complete.

3. Fully discrete scheme

In this section, we study the existence and uniqueness of the fully discrete scheme based on the modified Crank-Nicolson scheme in time and the Legendre spectral method in space .

Firstly, we introduce some notation. Letτ be the step size for timet, tk =kτ, k= 0,1, . . . , nT andT =nTτ,t_k−1

2 = (tk+t_k−1)/2. For convenience, we introduce the following notation for the functionu(x, t),

u^k=u^k(·) =u(·, tk), u^k−12 =u(t_k−1

2), ∂¯_tu^k =u^k−u^k−1

τ , u^k¯=u^k+u^k−1

2 .

The fully discrete spectral method for the problem (1.1) is: find u^k_N ∈ V_N⁰, k= 1, . . . n_T, such that for allϕ∈V_N⁰,

∂¯_tu^k_N, ϕ

+²a(u^¯k_N, ϕ) +1 2

(u^k_N)²+ (u^k−1_N )² u^¯k_N, ϕ

= (u^¯k_N, ϕ), (3.1)

u⁰_N = Π^α,0_N u0. (3.2)

For simplicity in what follows, we denoteg(u) = u³, ˜f(u, v) = ¹₄(u+v)(u²+v²).

A priori estimates are needed in the following analyses.

Lemma 3.1. Suppose that u^k_N (k = 0,1, . . . , n_T) be the solution of (3.1)-(3.2), then we have

²|u^k_N|²_α+1

2ku^k_Nk⁴_L4− ku^k_Nk²6²|u⁰_N|²_α+1

2ku⁰_Nk⁴_L4− ku⁰_Nk². Moreover, if u0∈H^α(Λ), we have

ku^k_Nk∞6C(|u^k_N|α+ku^k_Nk)^2α1 ku^k_Nk^1−2α1 6C|u0|α. Proof. Takingϕ=u^k_N −u^k−1_N in (3.1), we obtain

τk∂¯tu^k_Nk²+²a(u^k_N^¯, u^k_N −u^k−1_N ) +

f˜(u^k_N, u^k−1_N ), u^k_N −u^k−1_N

−1

2(ku^k_Nk²− ku^k−1_N k²) = 0.

(3.3) For the middle two terms on the left hand side of (3.3), a simple computation yields

a(u^¯k_N, u^k_N−u^k−1_N ) = 1

2a(u^k_N, u^k_N)−1

2a(u^k−1_N , u^k−1_N )

= 1

2(|u^k_N|²_α− |u^k−1_N |²_α),

(3.4)

f˜(u^k_N, u^k−1_N ), u^k_N−u^k−1_N

= 1

4 ku^k_Nk⁴_L4− ku^k−1_N k⁴_L4

. (3.5)

Substituting (3.4) and (3.5) into (3.3), we obtain τk∂¯_tu^k_Nk²+²

2|u^k_N|²_α+1

4ku^k_Nk⁴_L4−1 2ku^k_Nk²

= ²

2|u^k−1_N |²_α+1

4ku^k−1_N k⁴_L4−1

2ku^k−1_N k². Then we can deduce that

²|u^k_N|²_α+1

2ku^k_Nk⁴_L4− ku^k_Nk²6²|u⁰_N|²_α+1

2ku⁰_Nk⁴_L4− ku⁰_Nk². From above inequality, we have

|u^k_N|²_α6|u⁰_N|²_α+ 1

2²(ku⁰_Nk⁴_L4+ 2).

By the definition of Π^α,0_N , we obtain|u⁰_N|α6C|u0|α. Using Lemma 2.12, we have ku⁰_Nk_L4 6C|u₀|_α.

Therefore, we deduce that|u^k_N|²_α6C(|u0|²_α+ 1). By Lemma 2.12, we have ku^k_Nk_∞6C|u0|α,c1.

The proof is complete.

Remark 3.2. Similarly, for the solution u(x, t)of problem (1.1)we havekuk_∞6 C|u0|α.

Theorem 3.3 (Existence). For given{u^j_N}^k−1_j=0, there existsu^k_N satisfying (3.1).

Proof. We defining the mappingP :V_N⁰ →V_N⁰, such that (P(w), ϕ) =(w

τ, ϕ) +²

2a(w+ 2u^k−1_N , ϕ) + ( ˜f(w+u^k−1_N , u^k−1_N ), ϕ)

−1

2(w+ 2u^k−1_N , ϕ), ∀ϕ∈V_N⁰. ObviouslyP is continuous. Lettingϕ=w, we have

(P(w), w) =kwk² τ +²

2a(w+ 2u^k−1_N , w) + ( ˜f(w+u^k−1_N , u^k−1_N ), w)

−1

2(w+ 2u^k−1_N , w).

(3.6)

By a simple calculation, we obtain

(w+ 2u^k−1_N , w) = (w+u^k−1_N ) +u^k−1_N ,(w+u^k−1_N )−u^k−1_N

=kw+u^k−1_N k²− ku^k−1_N k². (3.7) Similarly, we have

a(w+ 2u^k−1_N , w) =|w+u^k−1_N |²_α− |u^k−1_N |²_α, (3.8) ( ˜f(w+u^k−1_N , u^k−1_N ), w) = 1

4 kw+u^k−1_N k⁴_L4− ku^k−1_N k⁴_L4

. (3.9)

Substituting (3.7)-(3.9) into (3.6) yields (P(w), w) =kwk²

τ +1 2

²|w+u^k−1_N |²_α+1

2kw+u^k−1_N k⁴_L4− kw+u^k−1_N k²

−1 2

²|u^k−1_N |²_α+1

2ku^k−1_N k⁴_L4− ku^k−1_N k²

>kwk² τ −1

2

²|u^k−1_N |²_α+1

2ku^k−1_N k⁴_L4− ku^k−1_N k²+ 1 . By Lemma 3.1, we have

(P(w), w)> kwk² τ −1

2

²|u⁰_N|²_α+1

2ku⁰_Nk⁴_L4− ku⁰_Nk²+ 1 .

Thus we have (P(w), w)>0, forkwk=K >[τ /2(²|u⁰_N|²_α+¹₂ku⁰_Nk⁴_L4− ku⁰_Nk²+ 1)]^1/2. By Lemma 2.14, there existsw^k−1∈V_N⁰,kw^k−1k6K, such thatP(w^k−1) = 0. Letu^k_N =u^k−1_N +w^k−1, therefore the existence ofu^k_N is proved.

Theorem 3.4 (Uniqueness). Suppose τ < 1, then the solution of (3.1)-(3.2) is unique.

Proof. Let{u^k_N},{v^k_N}be the two solutions of the discrete scheme (3.1)-(3.2) with the same initial condition. Letw_N^k =u^k_N −v^k_N, thus we have

( ¯∂tw^k_N, ϕ) +²a(w^k_N^¯, ϕ) + f˜(u^k_N, u^k−1_N )−f˜(v_N^k, v_N^k−1), ϕ

= (w^k_N^¯, ϕ), ∀ϕ∈V_N⁰. Settingϕ= ¯∂tw^k_N, we obtain

k∂¯_tw^k_Nk²+ ²

2τ |w^k_N|²_α− |w^k−1_N |²_α

+ f˜(u^k_N, u^k−1_N )−f˜(v_N^k, v_N^k−1),∂¯_tw^k_N

(3.10)

= (w_N^¯k,∂¯_tw^k_N). (3.11)

Now, we estimate the last two terms of the above equation. For the last term, by Young inequality, we have

(w^¯k_N,∂¯tw^k_N) =1

2 w_N^k −w_N^k−1,∂¯tw^k_N

+ (w^k−1_N ,∂¯tw^k_N) 6τ

2k∂¯tw^k_Nk²+1

4k∂¯tw^k_Nk²+kw^k−1_N k².

(3.12) For the penultimate term, noting that

f˜(u, v) =1

4(u+v)(u²+v²) = Z 1

0

g(v+s(u−v)) ds, g⁰(s) = 3s²>0;

therefore, utilizing the mean-value theorem of differentials, we obtain f˜(u^k_N, u^k−1_N )−f˜(v^k_N, v_N^k−1),∂¯_tw^k_N

=τZ 1 0

g⁰(ξ)sds,( ¯∂tw^k_N)² +Z 1

0

g⁰(ξ)w^k−1_N ds,∂¯tw_N^k

>Z 1 0

g⁰(ξ)w_N^k−1ds,∂¯tw^k_N ,

(3.13)

whereξlies in the interval with endpointsu^k−1_N +s(u^k_N−u^k−1_N ) andv_N^k−1+s(v_N^k − v_N^k−1). By Lemma 3.1, H¨older inequality and Young inequality yield

Z 1

0

g⁰(ξ) dsw_N^k−1,∂¯tw^k_N 6 1

4k∂¯tw_N^kk²+ 9c⁴₁kw^k−1_N k². (3.14) Substituting (3.12), (3.14) into (3.10), noticing thatτ <1, we deduce that

²

2τ |w^k_N|²_α− |w^k−1_N |²_α

6(9c⁴₁+ 1)kw^k−1_N k².

Summing fork from 1 ton, and by using Lemma 2.11, we deduce that

|wⁿ_N|²_α6|w_N⁰|²_α+C(9c⁴₁+ 1)τ

n−1

X

i=0

|w_Nⁱ |²_α. Thus by Lemma 2.13, we obtain

|w^k_N|²_α6e^{T C(9c41+1)}|w⁰_N|²_α= 0.

Finally, using Lemma 2.11 once again, we have kw^k_Nk = 0, i.e., u^k_N = v^k_N, k = 0,1, . . . , nT. The proof of the uniqueness is complete.

4. Stability and convergence of the fully discrete scheme In this section, we give the stability and convergence analysis for the fully discrete scheme (3.1)–(3.2).

Theorem 4.1 (Stability). Assume τ < 1, u^k_N,v_N^k (k= 1,2, . . . , n_T) be the solu- tions of the fully discrete scheme (3.1) with the initial valueu⁰_N, v⁰_N, respectively.

Then we have

ku^k_N−v_N^kkα6e^{T C(9c41+1)}ku⁰_N −v⁰_Nkα, k= 1,2, . . . , nT. Proof. Letw_N^k =u^k_N −v^k_N in (3.1), then w_N^k satisfies

( ¯∂_tw^k_N, ϕ) +²a(w^k_N^¯, ϕ) + f˜(u^k_N, u^k−1_N )−f˜(v_N^k, v_N^k−1), ϕ

= (w^k_N^¯, ϕ), ∀ϕ∈V_N⁰. With the same line of the proof as for Theorem 3.4, we obtain the desired result.

The proof is complete.

Now, we give the convergence result of the fully discrete scheme (3.1)-(3.2).

Theorem 4.2(Convergence). Letuanduⁿ_N (16n6nT)be the solutions of (1.1) and (3.1)-(3.2), respectively. Assume that u ∈ L^∞(0, T;H^m(Λ)), m > 2α, ut ∈ L⁴(0, T;L⁴(Λ))∩L²(0, T;H^m(Λ)), utt ∈L²(0, T;H^2α(Λ)),uttt∈L²(0, T;L²(Λ)).

Then for τ < 1, there exists a positive constant c independent of τ and N, such that

kuⁿ−uⁿ_Nk6c(τ²+N^−m) and |uⁿ−uⁿ_N|_α6c(τ²+N^α−m).

Proof. Setting u^k −u^k_N = (u^k −Π^α,0_N u^k) + (Π^α,0_N u^k −u^k_N) = θ^k+η^k. By (1.1), (3.1)-(3.2) and the definition of Π^α,0_N , we have the error equation

( ¯∂tη^k, v) +²a(η^k¯, v) = ( ¯∂tu^k−u^k−

1 2

t , v)−( ¯∂tθ^k, v) + (u^k−12 −u^k_N^¯, v) +²a(u^¯k−u^k−12, v) + f˜(u^k_N, u^k−1_N )−g(u^k−12), v

, η⁰= 0.

(4.1)

Takingv= ¯∂tη^k in (4.1), we have k∂¯_tη^kk²+ ²

2τ(|η^k|²_α− |η^k−1|²_α)

= ( ¯∂_tu^k−u^k−_t ¹²,∂¯_tη^k)−( ¯∂_tθ^k,∂¯_tη^k) + (u^k−12 −u^¯k_N,∂¯_tη^k) +²a(u^k¯−u^k−12,∂¯_tη^k) + f˜(u^k_N, u^k−1_N )−g(u^k−12),∂¯_tη^k ,

5

X

i=1

G_i.

(4.2)

Now we estimate terms on the right-hand side of (4.2). Via Taylor’s theorem with integral remainder, H¨older inequality and Young inequality, we deduce that

|G1|

=|( ¯∂tu^k−u^k−

1 2

t ,∂¯tη^k)|

6 1

2τk∂¯tη^kk k

Z t_k−1

2

tk−1

(t_k−1−t)²utttdtk+k Z t_k

t_k−1 2

(tk−t)²utttdtk

61

16k∂¯tη^kk²+ 2 τ²

k Z t_k−1

2

t_k−1

(t_k−1−t)²utttdtk²+k Z t_k

t_k−1

2

(tk−t)²utttdtk² .

(4.3)

By H¨older’s inequality, we obtain k

Z t_k−1

2

tk−1

(t_k−1−t)²utttdtk²6 Z 1

−1

Z t_k−1

2

tk−1

(t_k−1−t)⁴dtZ t_k−1

2

tk−1

|uttt|²dt dx

= τ⁵ 5·2⁵

Z t_k−1 2

t_k−1

kutttk²dt, and

k Z tk

t_k−1 2

(tk−t)²utttdtk²6 τ⁵ 5·2⁵

Z tk

t_k−1 2

kutttk²dt.

Substituting two inequalities above into (4.3) yields

|G1|6 1

16k∂¯tη^kk²+τ³ 80

Z t_k

t_k−1

kutttk²dt. (4.4) By H¨older inequality and Young inequality, we have

|G2|=|( ¯∂tθ^k,∂¯tη^k)|6 1

16k∂¯tη^kk²+ 4k∂¯tθ^kk². From H¨older inequality and Lemma 2.17, we obtain

k∂¯_tθ^kk²= 1 τ²

Z t_k

tk−1

θ_tdt

26 1 τ

Z t_k

tk−1

kθtk²dt6C τN^−2m

Z t_k

tk−1

kutk²_mdt.

Thus we obtain

|G₂|6 1

16k∂¯_tη^kk²+C τN^−2m

Z tk

tk−1

ku_tk²_mdt. (4.5) Next, via a simple derivation, we have

|G3|=|(u^k−12 −u^¯k_N,∂¯tη^k)|6|(u^k−12 −u^¯k,∂¯tη^k)|+|(u^¯k−u^k_N^¯,∂¯tη^k)|. (4.6) Similar to (4.3), by Taylor’s theorem with integral remainder, H¨older and Young inequalities, we find that

|(u^k−12 −u^¯k,∂¯tη^k)|

6 1

2k∂¯_tη^kk k

Z t_k−1 2

tk−1

(t−t_k−1)u_ttdtk+k Z tk

t_k−1 2

(t_k−t)u_ttdtk

6 1

32k∂¯tη^kk²+τ³ 6

Z t_k

tk−1

kuttk²dt

and

|(u^¯k−u^¯k_N,∂¯tη^k)|6|(θ^¯k+η^k−1,∂¯tη^k)|+

τ 2

∂¯tη^k,∂¯tη^k

6 τ 2 + 1

32

k∂¯tη^kk²+ 16(kθ^¯kk²+kη^k−1k²).

Thus, substituting above two inequalities into (4.6), we obtain

|G₃|6 τ 2 + 1

16

k∂¯_tη^kk²+ 16(kθ^¯kk²+kη^k−1k²) +τ³ 6

Z tk

tk−1

ku_ttk²dt. (4.7) Next, by using Lemma2.7 and analogous to the estimation ofG3, we deduce that

|G₄|=²|a(u^k−12−u^¯k,∂¯_tη^k)|

=²

L^(α)(u^k−12 −u^¯k),∂¯_tη^k

6 ²

|cosπα|k∂¯tη^kk|u^k−12 −u^k¯|2α

61

16k∂¯tη^kk²+Cτ³ Z t_k

t_k−1

|utt|²_2αdt.

(4.8)

Now we estimate the last term on the right-hand side of (4.2). It is easy to obtain

G₅= ˜f(u^k_N, u^k−1_N )−g(u^k−12),∂¯_tη^k

= ˜f(u^k_N, u^k−1_N )−f˜(u^k, u^k−1),∂¯tη^k

+ f˜(u^k, u^k−1)−g(u^k−12),∂¯tη^k .

(4.9) For the first term of the right hand side of (4.9), analogous to (3.13) and (3.14), we find that

f˜(u^k_N, u^k−1_N )−f˜(u^k, u^k−1),∂¯_tη^k

=−Z 1 0

3ξ₁²dsθ^k−1,∂¯tη^k

−Z 1 0

3ξ₁²sds(θ^k−θ^k−1),∂¯tη^k

−Z 1 0

3ξ²₁dsη^k−1,∂¯tη^k

−Z 1 0

3ξ₁²sds(η^k−η^k−1),∂¯tη^k 6 3

16k∂¯_tη^kk²+ 36c⁴₁(kθ^k−1k²+kθ^kk²+kη^k−1k²), wherec1depends onu.

For the last term of the right-hand side of (4.9), applying Taylor’s formula for multivariate functions, we find that

f(u˜ ^k, u^k−1)−g(u^k−12),∂¯tη^k

= 3 (u^k−12)²(u^¯k−u^k−12),∂¯tη^k +1

4 (3ξ2+ξ3)(u^k−u^k−12)²,∂¯tη^k +1

4 (ξ2+ 3ξ3)(u^k−1−u^k−12)²,∂¯tη^k +1

2 (ξ2+ξ3)(u^k−1−u^k−12)(u^k−u^k−12),∂¯tη^k 6 1

16k∂¯tη^kk²+Cτ³ Z t_k

tk−1

c²₁kutk⁴_L4+c⁴₁kuttk² dt.

Substituting the two inequalities above into (4.9), we have

|G5|61

4k∂¯tη^kk²+ 36c⁴₁(kθ^k−1k²+kθ^kk²+kη^k−1k²) (4.10) +Cτ³

Z tk

tk−1

c²₁ku_tk⁴_L4+c⁴₁ku_ttk²

dt. (4.11)

Substituting (4.4), (4.5), (4.7), (4.8) and (4.10) into (4.2), in view of τ < 1, we obtain

²

2τ(|η^k|²_α− |η^k−1|²_α)

6C(1 +c⁴₁)kη^k−1k²+CN^−2m

(1 +c⁴₁)kuk²_L∞(0,T;H^m(Λ))+1 τ

Z tk

tk−1

ku_tk²_mdt +Cτ³

Z t_k

t_k−1

c²₁kutk⁴_L4+c⁴₁kuttk²+kuttk²_2α+kutttk² dt.

Summing fork from 1 ton(n6n_T) we have

|ηⁿ|²_α6C(1 +c⁴₁)τ

n−1

X

j=0

kη^jk²+c2(τ⁴+N^−2m), wherec2= max(c3, c4), and

c3=C c²₁kutk⁴_L4(0,T;L⁴(Λ))+c⁴₁kuttk²_L2(0,T;L²(Λ))+kuttk²_L2(0,T;H^2α(Λ))

+ku_tttk²_L2(0,T;L²(Λ))

,

c4=C (1 +c⁴₁)kuk²_L∞(0,T;H^m(Λ))+kutk²_L2(0,T;H^m(Λ))

. Using Lemma 2.13, we have

|ηⁿ|²_α6c2e^CT(1+c41)(τ⁴+N^−2m).

Utilizing Lemma 2.11 and the triangle inequality, we have

kuⁿ−uⁿ_Nk6c(τ²+N^−m), |uⁿ−uⁿ_N|α6c(τ²+N^α−m),

wherec=C(1 +c2e^CT(1+c41)). The proof is complete.

5. Numerical experiments

Example 5.1. To guarantee the exact solution have enough regularity, we add a forcing term on the equation.

u_t−²L^(α)u+u³−u=h(x, t), x∈(−1,1), t∈(0, T], u(x,0) = (1−x²)², x∈(−1,1),

u(±1, t) = 0, t∈[0, T].

(5.1)

where

h(x, t) = 4²e^t

Γ(5−2α) cosπα(1 +x)^2−2α

3(1 +x)²−3(4−2α)(1 +x) + (4−2α)(3−2α)

+ 4²e^t

Γ(5−2α) cosπα(1−x)^2−2α

3(1−x)²

−3(4−2α)(1−x) + (4−2α)(3−2α)] +e^3t(1−x²)⁶.