We obtain the higher order of convergence in both the spatial direction and the temporal direction in L2 normed space for the extrapolated Crank-Nicolson characteristic finite element method

https://doi.org/10.4134/BKMS.b160605 pISSN: 1015-8634 / eISSN: 2234-3016

AN EXTRAPOLATED CRANK-NICOLSON CHARACTERISTIC FINITE ELEMENT METHOD

FOR SOBOLEV EQUATIONS

Mi Ray Ohm and Jun Yong Shin

Abstract. We introduce an extrapolated Crank-Nicolson characteristic finite element method to approximate solutions of a convection dominated Sobolev equation. We obtain the higher order of convergence in both the spatial direction and the temporal direction in L² normed space for the extrapolated Crank-Nicolson characteristic finite element method.

1. Introduction

In this paper, we consider a convection dominated Sobolev equation: Find u(x , t) defined on Ω × [0, T ] such that

(1)

c(x )ut+ d (x ) · ∇u − ∇ · (a(u)∇u) − ∇ · (b(u)∇ut)

= f (x , t, u), in Ω × (0, T ], u(x , t) = 0, on ∂Ω × (0, T ], u(x , 0) = u0(x ), in Ω,

where Ω ⊂ R^m, 1 ≤ m ≤ 3, is a bounded convex domain with its boundary

∂Ω and c, d , a, b and f are known functions. Sobolev equations are used to describe physical phenomena such as thermodynamics [21], the migration of the moisture in soil [18], the flow of fluids through fissured rock [2] and other applications. For the existence, uniqueness, and regularity of Sobolev equa- tions, we refer to [3, 4, 21].

For Sobolev equations with no convection term, many numerical methods, such as classical finite element methods [1, 6, 10, 11, 12], least-squares meth- ods [9, 16, 17, 22, 23], mixed finite element methods [8], or discontinuous finite element methods [13, 14, 19, 20], are used to get numerical results. However, in many situations, the convection term d (x ) · ∇u exists and d (x ) is large in order to describe a convection dominated diffusion. We use a characteristic

Received July 19, 2016; Accepted October 17, 2016.

2010 Mathematics Subject Classification. Primary65M15, 65N30.

Key words and phrases. Sobolev equation, extrapolated Crank-Nicolson characteristic finite element method, higher order of convergence.

This work was supported by a Research Grant of Pukyong National University (2016).

c

2017 Korean Mathematical Society 1409

method to treat effectively both the time derivative term and the convection term. And this method works very well for convection dominated diffusion problems as shown in [5, 7]. In [7], the author construct a characteristic finite element method to obtain the higher order of convergence in the spatial variable and the first order of convergence in the temporal variable. The later makes meaningless the higher order of convergence in the spatial variable. To improve the first order of convergence in the temporal direction, Ohm and Shin in [15]

introduce a Crank-Nicolson characteristic finite element method and obtain the higher order of convergence in both the spatial direction and the temporal direction in L² normed space. We have the difficulty in solving the nonlinear systems when we use the Crank-Nicolson characteristic finite element method in [15] to get the approximate solutions of the Sobolev equation.

In this paper, we introduce an extrapolated Crank-Nicolson characteristic finite element method to avoid the difficulty in solving the nonlinear systems.

We establish the higher order of convergence in both the spatial direction and the temporal direction in L²normed space. Our paper is organized as follows:

In Section 2, we present the smoothness assumptions for u(x , t), the conditions for the given functions, and basic notations. In Section 3, we construct finite element spaces and derive some approximation properties. In Section 4, we construct an extrapolated Crank-Nicolson characteristic finite element approx- imation of u(x , t) and obtain the higher order of convergence in L² and H¹ normed spaces.

2. Assumptions and notations

Throughout this paper, let W^s,p(Ω) be a Sobolev space equipped with its norm k · ks,p for an s ≥ 0 and 1 ≤ p ≤ ∞. For our convenience, we simply denote H^s(Ω) and L²(Ω), instead of W^s,2(Ω) and H⁰(Ω), respectively. And also we write k · k, k · k∞, and k · ks, instead of k · k0,2, k · k0,∞, and k · ks,2, respectively. Let H^s(Ω) = {w = (w1, w2, . . . , wm) | wi ∈ H^s(Ω), 1 ≤ i ≤ m}

be a Sobolev space equipped with its norm kw k²_s=Pm

i=1kwik²_s and H₀¹(Ω) = {w ∈ H¹(Ω) | w(x ) = 0 on ∂Ω}. For a Banach space X and t1, t2∈ [0, T ], we introduce Sobolev spaces with the corresponding norms:

W^s,p(t1, t2; X) =n

w(x , t) | k∂^βw

∂t^β (·, t)kX ∈ L^p(t1, t2), 0 ≤ β ≤ so ,

kwkW^s,p(t₁,t₂;X)=











Ps β=0

Rt₂

t₁ k^∂_∂t^ββ^w(·, t)k^p_Xdt1/p

, 1 ≤ p < ∞,

max0≤β≤sesssup_t∈(t1,t₂)k^∂_∂t^βwβ(·, t)kX, p= ∞.

We simply write L^p(X) and W^s,p(X) instead of W^0,p(0, T ;X) and W^s,p(0, T ;X), respectively.

Assume that u(x , t) and c(x ), d (x ) = (d1(x ), d2(x ), . . . , dm(x ))^T, a(u), b(u) and f (x , t, u) satisfy the following assumptions:

(A1) There exists a positive constant ˜K such that ku(x , t)kL^∞(L^∞)≤ ˜K.

(A2) There exist constants c∗, c^∗, d^∗, a∗, a^∗, b∗, and b^∗ such that 0 < c∗ ≤ c(x ) ≤ c^∗, 0 < |d (x )| ≤ d^∗, 0 < a∗ ≤ a(u) ≤ a^∗, 0 < b∗≤ b(u) ≤ b^∗, for all x ∈ Ω and t ∈ [0, T ], where |d (x )| =Pm

i=1d²_i(x ).

(A3) a(u), au(u), auu(u), b(u), bu(u) and buu(u) are bounded.

(A4) f (x , t, u) is locally Lipschitz continuous in the third variable, that is, if |u(x , t) −u^∗| ≤ ˜K, then

|f (x , t, u(x , t)) − f (x , t, u^∗)| ≤ K(u, ˜K)|u(x , t) − u^∗|.

For each (x , t), let ν = ν(x , t) be the unit vector such that ^∂u_∂ν =_{ψ(x )}^{c(x )∂u}_∂t +

d(x )

ψ(x )· ∇u, where ψ(x ) = [c(x )²+ |d (x )|²]¹². Then the Sobolev equation (1) can be rewritten as follows:

(2)

ψ(x )∂u

∂ν − ∇ · (a(u)∇u) − ∇ · (b(u)∇ut) = f (x , t, u), in Ω × (0, T ], u(x , t) = 0, on ∂Ω × (0, T ], u(x , 0) = u0(x ), in Ω.

Now the variational formulation of the equation (2) is given as follows: Find u(x , t) ∈ H₀¹(Ω) such that

(3)

(ψ(x )∂u

∂ν, τ) + (a(u)∇u, ∇τ ) + (b(u)∇ut,∇τ )

= (f (x, t, u), τ ), ∀τ ∈ H₀¹(Ω), u(x , 0) = u0(x ).

3. Finite element spaces and an elliptic projection

Let {S_h^r} be a family of finite dimensional subspaces of H₀¹(Ω) with approxi- mation and inverse properties: for φ ∈ H₀¹(Ω) ∩ W^s,p(Ω), there exist a positive constant K1, independent of h, φ, and r, and a sequence Phφ∈ S_h^r such that for any 0 ≤ q ≤ s and 1 ≤ p ≤ ∞

kφ − Phφkq,p≤ K1h^µ−qkφks,p,

where µ = min(r + 1, s) and there exist a positive constant K1, independent of hand r, such that

kϕk1≤ K1h⁻¹kϕk and kϕk∞≤ K1h^−m2kϕk, ∀ϕ ∈ S_h^r. Now we introduce bilinear forms A and B defined on H₀¹(Ω) × H₀¹(Ω) by (4) A(u : v, w) = (a(u)∇v, ∇w), B(u : v, w) = (b(u)∇v, ∇w).

It is clear from the assumption (A2) that there exists a unique solution ˜u(t) ∈ S_h^rof the problem

(5) A(u : u − ˜u, χ) + B(u : ut− ˜ut, χ) = 0, ∀χ ∈ Sh^r, (˜u(0), χ) = (u0, χ), ∀χ ∈ Sh^r.

Now, if we let η = u − ˜u, then we have some estimates for η, ηt, ηtt, and ηttt

whose proofs can be found in [15].

Lemma 3.1. Let u0 ∈ H^s(Ω), ut, utt, uttt ∈ H^s(Ω), and ut ∈ L²(H^s(Ω)).

Then there exists a constant K, independent of h, such that (i) kηk + hkηk1≤ Kh^µ(kutkL²(H^s(Ω))+ ku0ks),

(ii) kηtk + hkηtk1≤ Kh^µ(kutkL²(H^s(Ω))+ ku0ks+ kutks), (iii) kηttk1≤ Kh^µ−1(kutkL²(H^s(Ω))+ ku0ks+ kutks+ kuttks),

(iv) kηtttk1≤ Kh^µ−1(kutkL²(H^s(Ω))+ ku0ks+ kutks+ kuttks+ kutttks), where µ= min(r + 1, s) and s ≥ 2.

Lemma 3.2. Let u0 ∈ H^s(Ω), u, ut, utt, uttt∈ L^∞(H^s(Ω)) ∩ L^∞(W^1,∞(Ω)), ut∈ L²(H^s(Ω)) and s ≥ 2. If µ ≥ 1 +^m₂, then the following statements hold:

max{kηk∞, k∇ηk∞, k∇∂tηk∞, k∇ηtk∞, k∇ηttk∞, k∇ηtttk∞} ≤ ˜K.

Throughout this paper, a generic positive constant K depends on the domain Ω, ˜K, and u(x , t) but is independent of the discretization magnitudes of the spatial and the temporal directions. So any K in the different places does not need to be the same.

4. Optimal L^∞(L²) and L^∞(H¹) error estimates

For a given positive integer N , let ∆t = T /N , tⁿ = n∆t and uⁿ = u(x , tⁿ) for 0 ≤ n ≤ N , and tⁿ⁻¹² = ¹₂(tⁿ + tⁿ⁻¹) and uⁿ⁻¹² = ¹₂(uⁿ + uⁿ⁻¹) for 1 ≤ n ≤ N . And let ˇx = x + ¹₂d˜(x )∆t, ˆx = x − ¹₂d˜(x )∆t, ˜d(x ) = ^d_{c(x )}^{(x )}, ˇ

uⁿ= u(ˇx, tⁿ) and ˆuⁿ⁻¹= u(ˆx, tⁿ⁻¹) for 1 ≤ n ≤ N . For 1 ≤ n ≤ N , from (3) and the definitions of bilinear forms A and B, we obtain

(6)

ψ(x )∂u(tⁿ⁻¹²)

∂ν , χ

+ A(u(tⁿ⁻¹²) : u(tⁿ⁻¹²), χ)

+ B(u(tⁿ⁻¹²) : ut(tⁿ⁻¹²), χ) = (f (x , tⁿ⁻¹², u(tⁿ⁻¹²)), χ), ∀χ ∈ S_h^r and so

(7)

c(x )uˇⁿ− ˆuⁿ⁻¹

∆t , χ

+ A(u(tⁿ⁻¹²) : uⁿ⁻¹², χ) + B(u(tⁿ⁻¹²) : uⁿ− uⁿ⁻¹

∆t , χ)

= (f (x , tⁿ⁻¹², u(tⁿ⁻¹²)), χ) + Q1+ Q2+ Q3, ∀χ ∈ Sh^r,

where Q1 = (c(x )^uˇn−_∆t^uˆn−1 − ψ(x )^∂u(t_∂ν^{n− 12)}, χ) , Q2 = A(u(tⁿ⁻¹²) : uⁿ⁻¹² − u(tⁿ⁻¹²), χ), and Q3= B(u(tⁿ⁻¹²) : ^un−u_∆tⁿ⁻¹− ut(tⁿ⁻¹²), χ). Then an extrapo- lated Crank-Nicolson characteristic finite element scheme for the equation (2)

is given as follows: Find {uⁿ_h} ∈ S_h^r such that

(8)

c(x )ˇuⁿ_h− ˆuⁿ⁻¹_h

∆t , χ

+ A(Euⁿ_h: uⁿ⁻_h ¹², χ) + B(Euⁿ_h: uⁿ_h− uⁿ⁻¹_h

∆t , χ)

= (f (x , tⁿ⁻¹², Euⁿ_h), χ), ∀χ ∈ Sh^r, n= 2, . . . , N,

(9)

c(x )uˇ¹_h− ˆu⁰_h

∆t , χ

+ A(u_h¹² : u_h¹², χ) + B(u_h¹² : u¹_h− u⁰_h

∆t , χ)

= (f (x , t¹², u_h¹²), χ), ∀χ ∈ S_h^r,

(10) u⁰_h(x ) = ˜u(x , 0),

where ˇuⁿ_h = uⁿ_h(ˇx), ˆuⁿ⁻¹_h = uⁿ⁻¹_h (ˆx), Euⁿ_h = ³₂uⁿ⁻¹_h − ¹₂uⁿ⁻²_h , and uⁿ⁻_h ¹² =

1

2(uⁿ_h+ uⁿ⁻¹_h ).

We denote ξ = uh− ˜u, ∂tξⁿ = ^ξn−ξ_∆tⁿ⁻¹ to obtain the error estimates of the extrapolated Crank-Nicolson characteristic finite element scheme for the equation (2).

Theorem 4.1. In addition to the assumptions of Lemma 3.2, if µ≥ 1 + ^m₂, u(t) ∈ H^s(Ω), u ∈ L^∞(H³(Ω))∩W^1,∞(H²(Ω))∩W^2,∞(H¹(Ω))∩W^3,∞(L²(Ω)), and∆t = O(h), then

k∇ξ¹k²+ ∆t(k∂tξ¹k²+ k∇∂tξ¹k²) ≤ K∆t(h^2µ+ (∆t)⁴), where µ= min(r + 1, s).

Proof. The proof of this theorem is given in [15].  Theorem 4.2. Under the same assumptions of Theorem 4.1, we have

0≤n≤Nmax h

kuⁿ− uⁿhk + hk∇(uⁿ− uⁿh)ki

≤ K(h^µ+ (∆t)²), where µ= min(r + 1, s).

Proof. To establish this theorem, we prove the following statement by mathe- matical induction: There exist 0 < ˜h <1 and 0 < ˜∆t < 1 such that

(11) k∇ξⁿk²+ ∆t(k∂tξⁿk²+ k∇∂tξⁿk²) ≤ K(h^2µ+ (∆t)⁴)

for any 0 < h < ˜h, 0 < ∆t < ˜∆t and n = 0, 1, . . . , N . For our convenience, we abuse the notations such as Eu¹_h= 0 and ξ⁻¹= 0. Since ξ⁰= 0, (11) trivially holds for n = 0. And by Theorem 4.1, (11) holds for n = 1. Now we assume that (11) holds with n ≤ l − 1. Notice that kξⁿk∞ ≤ ˜K, 0 ≤ n ≤ l − 1.

Subtracting (7) from (8) with 2 ≤ n ≤ l, we get for any χ ∈ S_h^r

(12)

c(x )ξⁿ− ξⁿ⁻¹

∆t , χ

+ (a(Euⁿh)∇ξⁿ⁻¹²,∇χ) +

b(Euⁿ_h)∇ξⁿ− ∇ξⁿ⁻¹

∆t ,∇χ

=

c(x )ξⁿ− ˇξⁿ

∆t , χ +

c(x )ξˆⁿ⁻¹− ξⁿ⁻¹

∆t , χ +

c(x )ηˇⁿ− ˆηⁿ⁻¹

∆t , χ + (a(Euⁿ_h)∇ηⁿ⁻¹²,∇χ) + ([a(u(tⁿ⁻¹²)) − a(Euⁿ_h)]∇uⁿ⁻¹²,∇χ) +

b(Euⁿh)∇ηⁿ− ∇ηⁿ⁻¹

∆t ,∇χ +

[b(u(tⁿ⁻¹²)) − b(Euⁿ_h)]∇uⁿ− ∇uⁿ⁻¹

∆t ,∇χ

− (f (x , tⁿ⁻¹², u(tⁿ⁻¹²)) − f (x , tⁿ⁻¹², Euⁿ_h), χ) − Q1− Q2− Q3

≡ Σ¹¹_i=1Ri.

Now let three terms of the left-hand side of (12) by L1, L2and L3, respectively and choose χ = ∂tξⁿin (12). First we estimate the lower bounds of L1, L2 and L3as follows:

L1= (c(x )∂tξⁿ, ∂tξⁿ) ≥ c∗k∂tξⁿk², L2= A(Euⁿh: ξⁿ⁻¹², ∂tξⁿ)

≥ 1 2∆t(kq

a(Euⁿ_h)∇ξⁿk²− k q

a(Eu_hⁿ⁻¹)∇ξⁿ⁻¹k²) + 1

2∆t(k q

a(Euⁿ⁻¹_h )∇ξⁿ⁻¹k²− kq

a(Euⁿ_h)∇ξⁿ⁻¹k²), L3= B(Euⁿh: ∂tξⁿ, ∂tξⁿ) ≥ b∗k∇∂tξⁿk².

By applying the lower bounds of L1∼ L3to (12), we get

(13)

c∗k∂tξⁿk²+ b∗k∇∂tξⁿk² + 1

2∆t(kq

a(Euⁿ_h)∇ξⁿk²− k q

a(Euⁿ⁻¹_h )∇ξⁿ⁻¹k²)

≤ 1

2∆t([a(Euⁿ_h) − a(Euⁿ⁻¹_h )]∇ξⁿ⁻¹,∇ξⁿ⁻¹) +

11

X

i=1

Ri. By (11) and ∆t = O(h), we have

kEuⁿh− Euⁿ⁻¹_h k∞

= kE(uⁿ_h− ˜uⁿ) − E(uⁿ⁻¹_h − ˜uⁿ⁻¹) + E ˜uⁿ− E ˜uⁿ⁻¹k∞

≤ ∆t(3

2k∂tξⁿ⁻¹k∞+1

2k∂tξⁿ⁻²k∞) + K∆t (14)

≤ K∆t²¹(h^−m2(h^µ+ (∆t)²)) + K∆t ≤ K∆t.

Hence, by the assumption (A3) and (14), (13) can be estimated as follows:

(15)

c∗k∂tξⁿk²+ b∗k∇∂tξⁿk² + 1

2∆t(k q

a(Euⁿ_h)∇ξⁿk²− k q

a(Euⁿ⁻¹_h )∇ξⁿ⁻¹k²)

≤ Kk∇ξⁿ⁻¹k²+

11

X

i=1

Ri.

By the assumption (A2) and the Taylor expansion, we have the following esti- mates for R1 and R2:

R1= (c(x )ξⁿ− ˇξⁿ

∆t , ∂tξⁿ) ≤ Kk∇ξⁿk²+ ǫk∂tξⁿk², R2=

c(x )ξˆⁿ⁻¹− ξⁿ⁻¹

∆t , ∂tξⁿ

≤ Kk∇ξⁿ⁻¹k²+ ǫk∂tξⁿk². Notice that

ηⁿ− ˇηⁿ

∆t = −1

2∇η(˜x₁, tⁿ) · ˜d(x ), ηⁿ− ηⁿ⁻¹

∆t = ηt(tⁿθ), ηⁿ⁻¹− ˆηⁿ⁻¹

∆t =1

2∇η(˜x₂, tⁿ⁻¹) · ˜d(x )

for some tⁿ_θ ∈ (tⁿ⁻¹, tⁿ), ˜x₁ ∈ (x , ˇx) and ˜x₂ ∈ (ˆx, x). Since we can split R3

into three terms as follows R3=

c(x )ηˇⁿ− ηⁿ

∆t , ∂tξⁿ +

c(x )ηⁿ− ηⁿ⁻¹

∆t , ∂tξⁿ +

c(x )ηⁿ⁻¹− ˆηⁿ⁻¹

∆t , ∂tξⁿ

=

3

X

j=1

Sj, by integration by parts, we have

S1≤ Kkηⁿk²+ ǫk∇∂tξⁿk²+ ǫk∂tξⁿk², S2≤ Kkηtⁿk²+ ǫk∂tξⁿk²,

S3≤ Kkηⁿ⁻¹k²+ ǫk∇∂tξⁿk²+ ǫk∂tξⁿk². Therefore,

R3≤ K(kηⁿk²+ kηⁿ_tk²+ kηⁿ⁻¹k²) + ǫk∇∂tξⁿk²+ ǫk∂tξⁿk². The term R4 can be written as follows:

R4= ([a(Euⁿh) − a(u(tⁿ⁻¹²)]∇ηⁿ⁻¹²,∇∂tξⁿ) + (a(u(tⁿ⁻¹²))[∇ηⁿ⁻¹² − ∇η(tⁿ⁻¹²)], ∇∂tξⁿ)

+ (a(u(tⁿ⁻¹²))∇η(tⁿ⁻¹²), ∇∂tξⁿ)

=

3

X

j=1

Tj. Notice that

(16)

Euⁿ_h− u(tⁿ⁻¹²) = Eξⁿ+ E ˜uⁿ− ˜u(tⁿ⁻¹²) − η(tⁿ⁻¹²), Eu˜ⁿ− ˜u(tⁿ⁻¹²) = 3

2

h˜u(tⁿ⁻¹²) −∆t

2 u˜t(tⁿ⁻¹²) +(∆t)²

8 u˜tt(˜t1,θ)i

−1 2

hu(t˜ ⁿ⁻¹²) −3∆t

2 u˜t(tⁿ⁻¹²) +9(∆t)²

8 u˜tt(˜t2,θ)i

− ˜u(tⁿ⁻¹²)

= O((∆t)²)(˜utt(˜t1,θ) + ˜utt(˜t2,θ)),

∇ηⁿ⁻¹² − ∇η(tⁿ⁻¹²) = 1

2(∇ηⁿ+ ∇ηⁿ⁻¹) − ∇η(tⁿ⁻¹²)

= O((∆t)²)(∇ηtt(t1,θ) + ∇ηtt(t2,θ)),

for some t1,θ ∈ (tⁿ⁻¹², tⁿ), ˜t1,θ ∈ (tⁿ⁻¹, tⁿ⁻¹²), t2,θ ∈ (tⁿ⁻¹, tⁿ⁻¹²), and ˜t2,θ ∈ (tⁿ⁻², tⁿ⁻¹²). Hence, by using (A2) and Lemma 3.2, the estimates for T1 and T2 are obtained as follows:

T1≤ K[(∆t)⁴+ kEξⁿk²+ kη(tⁿ⁻¹²)k²] + ǫk∇∂tξⁿk², T2≤ K(∆t)⁴+ ǫk∇∂tξⁿk²,

which implies that

R4≤ K[(∆t)⁴+ kξⁿ⁻¹k²+ kξⁿ⁻²k²+ kη(tⁿ⁻¹²)k²] + ǫk∇∂tξⁿk²+ T3. And, by using (A3) and (16), the estimate for R5 is given as follows:

R5= ([a(u(tⁿ⁻¹²)) − a(Euⁿh)]∇uⁿ⁻¹²,∇∂tξⁿ)

≤ K[(∆t)⁴+ kξⁿ⁻¹k²+ kξⁿ⁻²k²+ kη(tⁿ⁻¹²)k²] + ǫk∇∂tξⁿk². The term R6 can be written as follows:

R6=

b(Euⁿh)h∇ηⁿ− ∇ηⁿ⁻¹

∆t − ∇ηt(tⁿ⁻¹²)i

,∇∂tξⁿ + ([b(Euⁿ_h) − b(u(tⁿ⁻¹²))]∇ηt(tⁿ⁻¹²), ∇∂tξⁿ) + (b(u(tⁿ⁻¹²))∇ηt(tⁿ⁻¹²), ∇∂tξⁿ)

=

3

X

j=1

Vj. Notice that

∇ηⁿ− ∇ηⁿ⁻¹

∆t − ∇ηt(tⁿ⁻¹²) = O((∆t)²)(∇ηttt(t1,θ) + ∇ηttt(t2,θ))

for some t1,θ ∈ (tⁿ⁻¹², tⁿ) and t2,θ ∈ (tⁿ⁻¹, tⁿ⁻¹²). Hence, by using (A3) and (16), the estimates for V1 and V2are obtained as follows:

V1≤ K(∆t)⁴+ ǫk∇∂tξⁿk²,

V2≤ K[(∆t)⁴+ kξⁿ⁻¹k²+ kξⁿ⁻²k²+ kη(tⁿ⁻¹²)k²] + ǫk∇∂tξⁿk², which implies that

R6≤ K[(∆t)⁴+ kξⁿ⁻¹k²+ kξⁿ⁻²k²+ kη(tⁿ⁻¹²)k²] + ǫk∇∂tξⁿk²+ V3. By (5), T3+ V3= 0 and so we have an estimate for the sum of R4 and R6 as follows

R4+ R6≤ K[(∆t)⁴+ kξⁿ⁻¹k²+ kξⁿ⁻²k²+ kη(tⁿ⁻¹²)k²] + ǫk∇∂tξⁿk². By using (A3), (A4), and (16), the estimates for R7 and R8 can be obtained as follows:

R7=

(b(u(tⁿ⁻¹²)) − b(Euⁿh))∇uⁿ− ∇uⁿ⁻¹

∆t ,∇∂tξⁿ

≤ K[(∆t)⁴+ kξⁿ⁻¹k²+ kξⁿ⁻²k²+ kη(tⁿ⁻¹²)k²] + ǫk∇∂tξⁿk² and

R8= −(f (x , tⁿ⁻¹², u(tⁿ⁻¹²)) − f (x , tⁿ⁻¹², Euⁿh), ∂tξⁿ)

≤ K[(∆t)⁴+ kξⁿ⁻¹k²+ kξⁿ⁻²k²+ kη(tⁿ⁻¹²)k²] + ǫk∇∂tξⁿk². By Taylor expansion, the estimates for R9∼ R11 are obtained as follows:

R9=

ψ(x )∂u(tⁿ⁻¹²)

∂ν − c(x )uˇⁿ− ˆuⁿ⁻¹

∆t , ∂tξⁿ

≤ K(∆t)⁴+ ǫk∂tξⁿk², R10= A(u(tⁿ⁻¹²) : u(tⁿ⁻¹²) − uⁿ⁻¹², ∂tξⁿ) ≤ K(∆t)⁴+ ǫk∇∂tξⁿk², R11= B(u(tⁿ⁻¹²) : ut(tⁿ⁻¹²) −uⁿ− uⁿ⁻¹

∆t , ∂tξⁿ) ≤ K(∆t)⁴+ ǫk∇∂tξⁿk². Using the estimates for R1∼ R11 in (15), we get

c∗k∂tξⁿk²+ b∗k∇∂tξⁿk² + 1

2∆t(kq

a(Euⁿ_h)∇ξⁿk²− k q

a(Euⁿ⁻¹_h )∇ξⁿ⁻¹k²)

≤ Kh

k∇ξⁿk²+ k∇ξⁿ⁻¹k²+ kηⁿk²+ kηⁿ⁻¹k²

+ kη(tⁿ⁻¹²)k²+ kη_tⁿk²+ kξⁿ⁻¹k²+ kξⁿ⁻²k²+ (∆t)⁴i + ǫk∂tξⁿk²+ ǫk∇∂tξⁿk².

Since ǫ is sufficiently small, we obtain

(17)

∆tk∂tξⁿk²+ ∆tk∇∂tξⁿk² + (kq

a(Euⁿ_h)∇ξⁿk²− k q

a(Euⁿ⁻¹_h )∇ξⁿ⁻¹k²)

≤ K∆th

k∇ξⁿk²+ k∇ξⁿ⁻¹k²+ kηⁿk²+ kηⁿ⁻¹k²

+ kη(tⁿ⁻¹²)k²+ kηⁿtk²+ kξⁿ⁻¹k²+ kξⁿ⁻²k²+ (∆t)⁴i . Now we add both sides of (17) from n = 2 to l to get

∆t

l

X

n=2

[k∂tξⁿk²+ k∇∂tξⁿk²] + kq

a(Eu^l_h)∇ξ^lk²

≤ K∆t

l

X

n=0

nkξⁿk²+ k∇ξⁿk²o

+ K∆t

l

X

n=1

nkηⁿk²+ kηⁿ_tk²+ kη(tⁿ⁻¹²)k²+ (∆t)⁴o

+ Kk∇ξ¹k². So, by Lemma 3.1 and Theorem 4.1, we have

k∇ξ^lk²+ ∆t{k∂tξ^lk²+ k∇∂tξ^lk²}

≤ Kh

∆t

l−1

X

n=0

{k∇ξⁿk²} + K∆t

l

X

n=1

{h^2µ+ (∆t)⁴}i , for sufficiently small ∆t. Therefore, by Gronwall’s inequality, we have

k∇ξ^lk²+ ∆t{k∂tξ^lk²+ k∇∂tξ^lk²} ≤ K[h^2µ+ (∆t)⁴],

which completes the proof of the statement (11). By the triangle inequality and the Poincare’s inequality, we finally have ku^l− u^l_hk ≤ K(h^µ+ (∆t)²) and k∇(u^l− u^l_h)k ≤ Kh⁻¹(h^µ+ (∆t)²). Thus the result of this theorem hold. 

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Mi Ray Ohm

Division of Mechatronics Engineering Dongseo University

Busan 47011, Korea

E-mail address: mrohm@dongseo.ac.kr Jun Yong Shin

Department of Applied Mathematics Pukyong National University Busan 48513, Korea

E-mail address: jyshin@pknu.ac.kr