On the Second Order of Accuracy Stable Implicit...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2012, Article ID 230190, 13 pagesdoi:10.1155/2012/230190

Research ArticleOn the Second Order of Accuracy Stable ImplicitDifference Scheme for Elliptic-Parabolic Equations

Allaberen Ashyralyev and Okan Gercek

Department of Mathematics, Fatih University, 34500 Buyukcekmece, Istanbul, Turkey

Correspondence should be addressed to Okan Gercek, [email protected]

Received 7 April 2012; Accepted 24 April 2012

Academic Editor: Ravshan Ashurov

Copyright q 2012 A. Ashyralyev and O. Gercek. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We are interested in studying a second order of accuracy implicit difference scheme for the solutionof the elliptic-parabolic equation with the nonlocal boundary condition. Well-posedness of thisdifference scheme is established. In an application, coercivity estimates in Holder norms forapproximate solutions of multipoint nonlocal boundary value problems for elliptic-parabolicdifferential equations are obtained.

1. Introduction

Methods of solutions of nonlocal boundary value problems for mixed-type differential equa-tions have been studied extensively by various researchers (see, e.g., [1–19] and the referencestherein).

In [20], we considered the well-posedness of the followingmultipoint nonlocal bound-ary value problem:

−d2u(t)dt2

+Au(t) = g(t), (0 ≤ t ≤ 1),

du(t)dt

−Au(t) = f(t), (−1 ≤ t ≤ 0),

u(1) =J∑

i=1

αiu(λi) + ϕ,

−1 ≤ λ1 < λ2 < · · · < λi < · · · < λJ ≤ 0,

(1.1)

2 Abstract and Applied Analysis

in a Hilbert spaceH with the self-adjoint positive definite operator A under assumption

J∑

i=1

|αi| ≤ 1. (1.2)

The well-posedness of multipoint nonlocal boundary value problem (1.1) in Holderspaces with a weight was established. Moreover, coercivity estimates in Holder norms forthe solutions of nonlocal boundary value problems for elliptic-parabolic equations wereobtained.

In [21], we studied the well-posedness of the first order of accuracy difference schemefor the approximate solution of boundary value problem (1.1) under assumption (1.2).

Throughout this work, we consider the following second order of accuracy differencescheme:

−τ−2(uk+1 − 2uk + uk−1) +Auk = gk,

gk = g(tk), tk = kτ, 1 ≤ k ≤N − 1, Nτ = 1,

τ−1(uk − uk−1) −(I +

τ

2A)Auk−1 =

(I +

τ

2A)fk, fk = f(tk−1/2),

tk−1/2 =(k − 1

2

)τ, −(N − 1) ≤ k ≤ 0,

u2 − 4u1 + 3u0 = −3u0 + 4u−1 − u−2,

uN =J∑

k=1

αi

(u[λi/τ] +

(λi −

[λiτ

]τ

)(f[λi/τ] +Au[λi/τ]

))+ ϕ,

(1.3)

for the approximate solution of boundary value problem (1.1) under assumption (1.2).The well-posedness of difference scheme (1.3) in Holder spaces with a weight is

established. As an application, the stability, almost coercivity stability, and coercivity stabilityestimates for solutions of second order of accuracy difference scheme for the approximatesolution of the nonlocal boundary elliptic-parabolic problem are obtained.

2. Main Theorems

Throughout the paper, H is a Hilbert space and we denote B = (1/2)(τA +√A(4 + τ2A)),

where A is a self-adjoint positive definite operator. Then, it is clear that B is the self-adjointpositive definite operator and B ≥ δ1/2I where δ > δ0 > 0, andR = (I+τB)−1, which is defined

Abstract and Applied Analysis 3

on the whole space H, is a bounded operator. Here, I is the identity operator. The followingoperators

D =

(I + τA +

(τA)2

2

), G =

(I − τ2A

2

), P =

(I +

τ

2A), R = (I + τB)−1,

Tτ =

(I + B−1A

(I + τA +

τ

2P−2

)K(I − R2N−1

)+GKP−2R2N−1

−GKP−2(2I + τB)RN

[n∑

i=1

αi

(I +

(λi −

[λiτ

]τ

)A

)D −[λi/τ]u0

])−1

(2.1)

exist and are bounded for a self-adjoint positive operator A. Here,

B =12

(τA +

√A(4 + τ2A)

), K =

(I + 2τA +

54(τA)2

)−1. (2.2)

Furthermore, positive constants will be indicated by M which can differ in time. Onthe other hand Mi(α, β, . . .) is used to focus on the fact that the constant depends only onα, β, . . . and the subindex i is used to indicate a different constant.

First of all, let us start with some auxiliary lemmas from [16, 22–24] that are essentialbelow.

Lemma 2.1. For a self-adjoint positive operator A, the following estimates are satisfied:

∥∥∥Rk∥∥∥H→H

≤M1(δ)(1 + δτ)−k,

∥∥∥Dk∥∥∥H→H

≤M1(δ),

∥∥∥BRk∥∥∥H→H

≤ M1(δ)kτ

,∥∥∥P−1

∥∥∥H→H

≤M1(δ),∥∥∥ADk

∥∥∥H→H

≤ M1(δ)kτ

,

∥∥∥Dk − e−kτA∥∥∥H→H

≤ M1(δ)k2

,

∥∥∥∥(I − R2N

)−1∥∥∥∥H→H

≤M1(δ),

∥∥∥Rk − e−kτA1/2∥∥∥H→H

≤ M1(δ)k

, k ≥ 1, δ > 0.

(2.3)

From these estimates, it follows that

∥∥∥∥∥

(I + B−1A

(I + τA +

τ

2P−2

)K(I − R2N−1

)+GKP−2R2N−1 −GKP−2(2I + τB)

×RN

[n∑

i=1

αi

(I +

(λi −

[λiτ

]τ

)A

)D−[λi/τ]

])−1∥∥∥∥∥∥H→H

≤M2(δ).

(2.4)


Lemma 2.2. For any gk, 1 ≤ k ≤N −1 and fk,−N +1 ≤ k ≤ 0, the solution of problem (1.3) exists,and the following formulas hold:

uk =(I − R2N

)−1

×{[Rk − R2N−k

]u0 +

[RN−k − RN+k

]

×⎡

⎣n∑

i=1

αi

⎡

⎣(I +

(λi −

[λiτ

]τ

)A

)⎛

⎝D −[λi/τ]u0 − τ0∑

s=[λi/τ]+1

PDs−[λi/τ]fs

⎞

⎠

+(λi −

[λiτ

]τ

)f[λi/τ]

]+ ϕ

]

−[RN−k − RN+k

](I + τB)(2I + τB)−1B−1

N−1∑

s=1

[RN−s − RN+s

]gsτ

}

+ (I + τB)(2I + τB)−1B−1N−1∑

s=1

[R|k−s| − Rk+s

]gsτ, 1 ≤ k ≤N,

uk = D−ku0 − τ0∑

s=k+1

PDs−kfs, −N ≤ k ≤ −1,

u0 =12TτKP

−2

×{(

2I − τ2A)

×{(2 + τB)RN

×[

n∑

i=1

αi

[(I +

(λi −

[λiτ

]τ

)A

)

×⎛

⎝D −[λi/τ]u0 − τ0∑

s=[λi/τ]+1

PDs−[λi/τ]fs

⎞

⎠

+(λi −

[λiτ

]τ

)f[λi/τ]

]+ ϕ

]

−RN−1B−1N−1∑

s=1

[RN−s − RN+s

]gsτ +

(I − R2N

)B−1

N−1∑

s=1

Rs−1gsτ

}

+(I − R2N

)(I + τB)

(τB−1g1 − 4PB−1f0 + PDB−1f0 + PB−1f−1

)},

Tτ =

(I + B−1A

(I + τA +

τ

2P−2

)K(I − R2N−1

)+GKP−2R2N−1

− GKP−2(2I + τB)RN

[n∑

i=1

αi

(I +

(λi −

[λiτ

]τ

)A

)D −[λi/τ]u0

])−1.

(2.5)


Now, we study well-posedness of problem (1.3). Let Fτ(H) = F([a, b]τ ,H) be the

linear space of mesh functions ϕτ = {ϕk}ÑN

defined on [a, b]τ = {tk = kh, N ≤k ≤ Ñ, Nτ = a, Ñτ = b} with values in the Hilbert space H. Next, on Fτ(H)we denote C([a, b]τ ,H), Cα

0,1([−1, 1]τ ,H), Cα0,1([−1, 0]τ ,H), Cα

0 ([0, 1]τ ,H), Cα0,1([−1, 1]τ ,H),

and Cα0 ([−1, 0]τ ,H), 0 < α < 1 Banach spaces with the following norms:

∥∥ϕτ∥∥C([a,b]τ ,H) = max

Na≤k≤Nb

∥∥ϕk∥∥H,

∥∥ϕτ∥∥Cα

0,1([−1,1]τ ,H) =∥∥ϕτ

∥∥C([−1,1]τ ,H) + sup

−N≤k<k+r≤0

∥∥ϕk+r − ϕk∥∥E(−k)αr−α

+ sup1≤k<k+r≤N−1

∥∥ϕk+r − ϕk∥∥E((k + r)τ)α(N − k)αr−α,

∥∥ϕτ∥∥Cα

0 ([−1,0]τ ,H) =∥∥ϕτ

∥∥C([−1,0]τ ,H) + sup

−N≤k<k+r≤0

∥∥ϕk+r − ϕk∥∥E(−k)αr−α,

∥∥ϕτ∥∥Cα

0,1([0,1]τ ,H) =∥∥ϕτ

∥∥C([0,1]τ ,H)


‖ϕk+r − ϕk‖E((k + r)τ)α(N − k)αr−α,∥∥ϕτ

∥∥Cα

0,1([−1,1]τ ,H) =∥∥ϕτ

∥∥C([−1,1]τ ,H) + sup

−N≤k<k+2r≤0

∥∥ϕk+2r − ϕk∥∥E(−k)α(2r)−α


∥∥ϕk+r − ϕk∥∥E((k + r)τ)α(N − k)αr−α,

∥∥ϕτ∥∥Cα

0 ([−1,0]τ ,H) =∥∥ϕτ

∥∥C([−1,0]τ ,H)

+ sup−N≤k<k+2r≤0

∥∥ϕk+2r − ϕk∥∥E(−k)α(2r)−α, respectively.

(2.6)

Theorem 2.3. Nonlocal boundary value problem (1.3) is stable in C([−1, 1]τ ,H) space.

Proof. By [22], we have

∥∥∥{uk}N−11

∥∥∥C([0,1]τ ,H)

≤M3(δ)[∥∥gτ

∥∥C([0,1]τ ,H) + ‖ξ‖H +

∥∥ψ∥∥H

], (2.7)

for the solution of the following boundary value problem:

−τ−2(uk+1 − 2uk + uk−1) +Auk = gk,

gk = g(tk), tk = kτ, 1 ≤ k ≤N − 1,

u0 = ξ, uN = ψ.

(2.8)

By [24], we have

∥∥∥{uk}0−N∥∥∥C([−1,0]τ ,H)

≤M4(δ)[∥∥fτ

∥∥C([−1,0]τ ,H) + ‖ξ‖H

](2.9)


for the solution of an inverse Cauchy difference problem:

τ−1(uk − uk−1) −(I +

τ

2A)Auk−1 =

(I +

τ

2A)fk,

−(N − 1) ≤ k ≤ 0, u0 = ξ.(2.10)

Then, the proof of Theorem 2.3 is based on stability inequalities (2.7) and (2.9) and on thefollowing estimates:

‖ξ‖H ≤M5(δ)[∥∥fτ

∥∥C([−1,0]τ ,H) +

∥∥gτ∥∥C([0,1]τ ,H) +

∥∥ϕ∥∥H

],

∥∥ψ∥∥H ≤M6(δ)

[∥∥fτ∥∥C([−1,0]τ ,H) +

∥∥gτ∥∥C([0,1]τ ,H) +

∥∥ϕ∥∥H

],

(2.11)

for the solution of boundary value problem (1.3). Estimates (2.11) follow from estimates (2.3)and (2.4) and formula (2.5). This finishes the proof of Theorem 2.3.

Theorem 2.4. Assume that ϕ ∈ D(A) and f0, f−1, g1 ∈ D(I+τB). Then, for the solution of differenceproblem (1.3), the following almost coercivity inequality holds:

∥∥∥∥{τ−2(uk+1 − 2uk + uk−1)

}N−1

1

∥∥∥∥C([0,1]τ ,H)

+∥∥∥∥{τ−1(uk − uk−1)

}0−N+1

∥∥∥∥C([−1,0]τ ,H)

+∥∥∥{Auk}N−1

1

∥∥∥C([0,1]τ ,H)

+∥∥∥∥{(I +

τ

2A)Auk−1

}0−N+1

∥∥∥∥C([−1,0]τ ,H)

≤M7(δ)[min

{ln

1τ, 1 + |ln ‖A‖H→H |

}[∥∥fτ∥∥C([−1,0]τ ,H) +

∥∥gτ∥∥C([0,1]τ ,H)

]

+∥∥Aϕ

∥∥H +

∥∥(I + τB)f0∥∥H +

∥∥(I + τB)g1∥∥H +

∥∥(I + τB)f−1∥∥H

].

(2.12)

Proof. We have

∥∥∥∥{τ−2(uk+1 − 2uk + uk−1)

}N−1

1

∥∥∥∥C([0,1]τ ,H)

+∥∥∥{Auk}N−1

1

∥∥∥C([0,1]τ ,H)

≤M8(δ)[min

{ln

1τ, 1 + |ln ‖A‖H→H |

}∥∥gτ∥∥C([0,1]τ ,H) + ‖Aξ‖H +

∥∥Aψ∥∥H

],

(2.13)

for the solution of boundary value problem (2.8) (see [22]), and we get

∥∥∥∥{τ−1(uk − uk−1)

}0−N+1

∥∥∥∥C([−1,0]τ ,H)

+∥∥∥∥{(I +

τ

2A)Auk−1

}0−N+1

∥∥∥∥C([−1,0]τ ,H)

≤M9(δ)[min

{ln

1τ, 1 + |ln ‖A‖H→H |

}∥∥fτ∥∥C([−1,0]τ ,H) + ‖Aξ‖H

],

(2.14)


for the solution of inverse Cauchy difference problem (2.10) (see [24]). Then, the proof ofTheorem 2.4 is based on almost coercivity inequalities (2.13) and (2.14) and on the followingestimates:

‖Aξ‖H ≤ M10(δ)[∥∥Aϕ

∥∥H +

∥∥(I + τB)f0∥∥H +min

{ln

1τ, 1 + |ln ‖A‖H→H |

}

×[∥∥fτ

∥∥C([−1,0]τ ,H) +

∥∥gτ∥∥C([0,1]τ ,H)

]],

∥∥Aψ∥∥H ≤ M11(δ)

[∥∥Aϕ∥∥H +

∥∥(I + τB)f0∥∥H +min

{ln

1τ, 1 + |ln ‖A‖H→H |

}

×[∥∥fτ

∥∥C([−1,0]τ ,H) +

∥∥gτ∥∥C([0,1]τ ,H)

]]

(2.15)

for the solution of boundary value problem (1.3). Proofs of these estimates follow thescheme of the papers [23, 24] and rely on both formula (2.5) and estimates (2.3) and (2.4).Theorem 2.4 is proved.

Theorem 2.5. Let assumptions of Theorem 2.5 be satisfied. Then, boundary value problem (1.3) iswell-posed in Holder spaces Cα

0,1([−1, 1]τ ,H), and Cα0,1([−1, 1]τ ,H), and the following coercivity

inequalities hold:

∥∥∥∥{τ−2(uk+1 − 2uk + uk−1)

}N−1

1

∥∥∥∥Cα

0,1([0,1]τ ,H)+∥∥∥∥{τ−1(uk − uk−1)

}0−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,H)

+∥∥∥{Auk}N−1

1

∥∥∥Cα

0,1([0,1]τ ,H)+∥∥∥∥{(I +

τ

2A)Auk−1

}0−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,H)

≤M12(δ)[

1α(1 − α)

[∥∥fτ∥∥Cα

0 ([−1,0]τ ,H) +∥∥gτ

∥∥Cα

0,1([0,1]τ ,H)

]+∥∥Aϕ

∥∥H +

∥∥(I + τB)f0∥∥H

+∥∥(I + τB)g1

∥∥H +

∥∥(I + τB)f−1∥∥H

],

∥∥∥∥{τ−2(uk+1 − 2uk + uk−1)

}N−1

1

∥∥∥∥Cα

0,1([0,1]τ ,H)+∥∥∥∥{τ−1(uk − uk−1)

}0−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,H)

+∥∥∥{Auk}N−1

1

∥∥∥Cα

0,1([0,1]τ ,H)+∥∥∥∥{(I +

τ

2A)Auk−1

}0−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,H)

≤M13(δ)[

1α(1 − α)

[∥∥fτ∥∥Cα

0 ([−1,0]τ ,H) +∥∥gτ

∥∥Cα

0,1([0,1]τ ,H)

]+∥∥Aϕ

∥∥H

+∥∥(I + τB)f0

∥∥H +

∥∥(I + τB)g1∥∥H +

∥∥(I + τB)f−1∥∥H

].

(2.16)


Proof. By [22, 24], we have

∥∥∥∥{τ−2(uk+1 − 2uk + uk−1)

}N−1

1

∥∥∥∥Cα

0,1([0,1]τ ,H)+∥∥∥{Auk}N−1

1

∥∥∥Cα

0,1([0,1]τ ,H)

≤M14(δ)[

1α(1 − α)

∥∥gτ∥∥Cα

0,1([0,1]τ ,H) + ‖Aξ‖H +∥∥Aψ

∥∥H

],

(2.17)

for the solution of boundary value problem (2.8), and

∥∥∥∥{τ−1(uk − uk−1)

}0−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,H)+∥∥∥∥{(I +

τ

2A)Auk−1

}0−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,H)

≤M15(δ)[

1α(1 − α)

∥∥fτ∥∥Cα

0 ([−1,0]τ ,H) + ‖Aξ‖H],

(2.18)

∥∥∥∥{τ−1(uk − uk−1)

}0−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,H)+∥∥∥∥{(I +

τ

2A)Auk−1

}0−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,H)

≤M16(δ)[

1α(1 − α)

∥∥fτ∥∥Cα

0 ([−1,0]τ ,H) + ‖Aξ‖H] (2.19)

for the solution of inverse Cauchy difference problem (2.10), respectively. Then, the proof ofTheorem 2.5 is based on coercivity inequalities (2.17)–(2.19) and the following estimates:

‖Aξ‖H ≤ M17(δ)[

1α(1 − α)

[∥∥fτ∥∥Cα

0 ([−1,0]τ ,H) +∥∥gτ

∥∥Cα

0,1([0,1]τ ,H)

]

+∥∥Aϕ

∥∥H +

∥∥(I + τB)f0∥∥H +

∥∥(I + τB)g1∥∥H +

∥∥(I + τB)f−1∥∥H

],

∥∥Aψ∥∥H ≤ M18(δ)

[1

α(1 − α)[‖fτ‖Cα

0 ([−1,0]τ ,H) + ‖gτ‖Cα0,1([0,1]τ ,H)

]

+‖Aϕ‖H + ‖(I + τB)f0‖H + ‖(I + τB)g1‖H + ‖(I + τB)f−1‖H]

(2.20)

for the solution of difference scheme (1.3). Proofs of these estimates follow the scheme of thepapers [22, 24] and rely on both estimates (2.3) and (2.4) and formula (2.5). This concludesthe proof of Theorem 2.5.


3. An Application

In this section, an application of these abstract Theorems 2.3, 2.4, and 2.5 is considered.In [−1, 1] × Ω, let us consider the following boundary value problem for multidimensionalelliptic-parabolic equation:

−utt −n∑

r=1

(ar(x)uxr )xr = g(t, x), 0 < t < 1, x ∈ Ω,

ut +n∑

r=1

(ar(x)uxr )xr = f(t, x), −1 < t < 0, x ∈ Ω,

u(t, x) = 0, x ∈ S, −1 ≤ t ≤ 1,

u(1, x) =J∑

i=1

αiu(λi, x) + ϕ(x),J∑

i=1

|αi| ≤ 1,

−1 ≤ λ1 < λ2 < · · · < λi < · · · < λJ ≤ 0,

u(0+, x) = u(0−, x), ut(0+, x) = ut(0−, x), x ∈ Ω,

(3.1)

where ar(x) (x ∈ Ω), ϕ(x) (ϕ(x) = 0, x ∈ S), g(t, x) (t ∈ (0, 1), x ∈ Ω), and f(t, x) (t ∈(−1, 0), x ∈ Ω) are given smooth functions. Here,Ω is the unit open cube in the n-dimensionalEuclidean space R

n (0 < xk < 1, 1 ≤ k ≤ n)with boundary S, Ω = Ω ∪ S, and ar(x) � a > 0.The discretization of problem (3.1) is carried out in two steps. In the first step, let us

define the following grid sets:

Ωh = {x = xm = (h1m1, . . . , hnmn), m = (m1, . . . , mn),

0 ≤ mr ≤Nr, hrNr = 1, r = 1, . . . , n},

Ωh = Ωh ∩Ω, Sh = Ωh ∩ S.

(3.2)


We introduce the Hilbert spaces L2h = L2(Ωh),W12h = W1

2 (Ωh), and W22h = W2

2 (Ωh) ofthe grid functions ϕh(x) = {ϕ(h1m1, . . . , hnmn)} defined on Ωh, equipped with the followingnorms:

∥∥∥ϕh∥∥∥L2h

=

⎛

⎝∑

x∈Ωh

∣∣∣ϕh(x)∣∣∣2h1 · · ·hn

⎞

⎠1/2

,

∥∥∥ϕh∥∥∥W1

2h

=∥∥∥ϕh

∥∥∥L2h

+

⎛

⎝∑

x∈Ωh

n∑

r=1

∣∣∣(ϕh)xr∣∣∣2h1 · · ·hn

⎞

⎠1/2

,

∥∥∥ϕh∥∥∥W2

2h

=∥∥∥ϕh

∥∥∥L2h

+

⎛

⎝∑

x∈Ωh

n∑

r=1

∣∣∣(ϕh)xr∣∣∣2h1 · · ·hn

⎞

⎠1/2

+

⎛

⎝∑

x∈Ωh

n∑

r=1

∣∣∣(ϕh)xrxr ,mr

∣∣∣2h1 · · ·hn

⎞

⎠1/2

.

(3.3)

To the differential operator A generated by problem (3.1), we assign the differenceoperator Ax

h by formula

Axhu

h = −n∑

r=1

(ar(x)uhxr

)

xr ,mr

(3.4)

acting in the space of grid functions uh(x), satisfying the conditions uh(x) = 0 for all x ∈ Sh.With the help of Ax

h, we arrive at the following nonlocal boundary value problem:

−d2uh(t, x)dt2

+Axhu

h(t, x) = gh(t, x), 0 < t < 1, x ∈ Ωh,

duh(t, x)dt

−Axhu

h(t, x) = fh(t, x), −1 < t < 0, x ∈ Ωh,

uh(1, x) =n∑

k=1

αkuh(λk, x) + ϕh(x),

n∑

k=1

|αk| ≤ 1, x ∈ Ωh,

uh(0+, x) = uh(0−, x), duh(0+, x)dt

=duh(0−, x)

dt, x ∈ Ωh,

(3.5)

for an infinite system of ordinary differential equations.


In the second step, we replace problem (3.5) by difference scheme (1.3) accurate to thefollowing second order (see [22, 24]):

−uhk+1(x) − 2uhk(x) + u

hk−1(x)

τ2+Ax

huhk(x) = g

hk (x),

ghk (x) = gh(tk, x), tk = kτ, 1 ≤ k ≤N − 1, Nτ = 1, x ∈ Ωh,

uhk(x) − uhk−1(x)τ

−(Axh +

τ

2(Axh

)2)uhk−1(x) =

(I +

τ

2Axh

)fhk (x),

fhk (x) = fh(tk−1/2, x), tk−1/2 =

(k − 1

2

)τ, −N + 1 ≤ k ≤ 0, x ∈ Ωh,

−uh2(x) + 4uh1(x) − 3uh0(x) = 3uh0(x) − 4uh−1(x) + uh−2(x), x ∈ Ωh,

uhN(x) =J∑

k=1

αi

(uh

[λi/τ](x) +

(λk −

[λiτ

]τ

)(fh

[λi/τ]+Ax

huh[λi/τ]

(x)))

+ ϕh(x), x ∈ Ωh.

(3.6)

Theorem 3.1. Let τ and |h| =√h21 + · · · + h2n be sufficiently small positive numbers. Then, solutions

of difference scheme (3.6) satisfy the following stability and almost coercivity estimates:

∥∥∥∥{uhk

}N−1

−N

∥∥∥∥C([−1,1]τ ,L2h)

≤M19(δ)

[∥∥∥∥{fhk

}−1−N+1

∥∥∥∥C([−1,0]τ ,L2h)

+∥∥∥∥{ghk

}N−1

1

∥∥∥∥C([0,1]τ ,L2h)

+∥∥∥ϕh

∥∥∥L2h

],

∥∥∥∥{τ−2(uhk+1 − 2uhk + u

hk−1)}N−1

1

∥∥∥∥C([0,1]τ ,L2h)

+∥∥∥∥{uhk

}N−1

1

∥∥∥∥C([0,1]τ ,W

22h)

+∥∥∥∥{τ−1(uhk − uhk−1

)}0−N+1

∥∥∥∥C([−1,0]τ ,L2h)

+∥∥∥∥{uhk−1

}0−N+1

∥∥∥∥C([−1,0]τ ,W2

2h)

≤M20(δ)

⎡

⎣∥∥∥fh0

∥∥∥L2h

+∥∥∥fh−1

∥∥∥L2h

+∥∥∥gh1

∥∥∥L2h

+∥∥∥ϕh

∥∥∥W2

2h

+ τ∥∥∥fh0

∥∥∥W1

2h

+ τ∥∥∥fh−1

∥∥∥W1

2h

+τ∥∥∥gh1

∥∥∥W1

2h

+ ln1

τ + |h|

[∥∥∥∥{fhk

}−1−N+1

∥∥∥∥C([−1,0]τ ,L2h)

+∥∥∥∥{ghk

}N−1

1

∥∥∥∥C([0,1]τ ,L2h)

]].

(3.7)

The proof of Theorem 3.1 is based on Theorem 2.3, Theorem 2.4, the symmetryproperty of the difference operator Ax

hdefined by formula (3.4), the estimate

min{ln

1τ, 1 +

∣∣∣ln∥∥Ax

h

∥∥L2h →L2h

∣∣∣}

≤M21(δ) ln1

τ + |h| , (3.8)

and the following theorem on the coercivity inequality for the solution of elliptic differenceequation in L2h.


Theorem 3.2. For the solution of the following elliptic difference problem:

Axhu

h(x) = ωh(x), x ∈ Ωh, uh(x) = 0, x ∈ Sh, (3.9)

the following coercivity inequality holds [25]:

n∑

r=1

∥∥∥∥(uh)

xrxr ,mr

∥∥∥∥L2h

≤M22(δ)∥∥∥ωh

∥∥∥L2h. (3.10)

Theorem 3.3. Let τ and |h| be sufficiently small positive numbers. Then, solutions of differencescheme (3.6) satisfy the following coercivity stability estimates:

∥∥∥∥{τ−2(uhk+1 − 2uhk + u

hk−1)}N−1

1

∥∥∥∥Cα

0,1([0,1]τ ,L2h)

+∥∥∥∥{τ−1(uhk − uhk−1

)}0−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,L2h)+∥∥∥∥{uhk

}N−1

1

∥∥∥∥Cα

0,1([0,1]τ ,W22h)

+∥∥∥∥{uhk−1

}0−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,W22h)

≤M23(δ)

[∥∥∥ϕh∥∥∥W2

2h

+ τ∥∥∥fh0

∥∥∥W1

2h

+ τ∥∥∥fh−1

∥∥∥W1

2h

+ τ∥∥∥gh1

∥∥∥W1

2h

+1

α(1 − α)

[∥∥∥∥{fhk

}−1−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,L2h)+∥∥∥∥{ghk

}N−1

1

∥∥∥∥Cα

0,1([0,1]τ ,L2h)

]],

∥∥∥∥{τ−2(uhk+1 − 2uhk + u

hk−1)}N−1

1

∥∥∥∥Cα

0,1([0,1]τ ,L2h)+∥∥∥∥{uhk−1

}0−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,W22h)

+∥∥∥∥{τ−1(uhk − uhk−1

)}0−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,L2h)+∥∥∥∥{uhk

}N−1

1

∥∥∥∥Cα

0,1([0,1]τ ,W22h)

≤M24(δ)

[∥∥∥ϕh∥∥∥W2

2h

+ τ∥∥∥fh0

∥∥∥W1

2h

+ τ∥∥∥fh−1

∥∥∥W1

2h

+ τ∥∥∥gh1

∥∥∥W1

2h

+1

α(1 − α)

[∥∥∥∥{fhk

}−1−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,L2h)+∥∥∥∥{ghk

}N−1

1

∥∥∥∥Cα

0,1([0,1]τ ,L2h)

]].

(3.11)

The proof of Theorem 3.3 is based on the abstract Theorem 2.5, Theorem 3.2, and thesymmetry property of the difference operator Ax

h defined by formula (3.4).

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