Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 230190, 13 pages
doi:10.1155/2012/230190
Research Article
On the Second Order of Accuracy Stable Implicit
Difference 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, ogercek@fatih.edu.tr
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 under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction 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 solution
of the elliptic-parabolic equation with the nonlocal boundary condition. Well-posedness of this
difference scheme is established. In an application, coercivity estimates in Hölder norms for
approximate solutions of multipoint nonlocal boundary value problems for elliptic-parabolic
differential equations are obtained.
1. Introduction
Methods of solutions of nonlocal boundary value problems for mixed-type differential equations have been studied extensively by various researchers see, e.g., 1–19 and the references
therein.
In 20, we considered the well-posedness of the following multipoint nonlocal boundary value problem:
−
d2 ut
Aut gt,
dt2
dut
− Aut ft,
dt
0 ≤ t ≤ 1,
−1 ≤ t ≤ 0,
J
u1 αi uλi ϕ,
i1
−1 ≤ λ1 < λ2 < · · · < λi < · · · < λJ ≤ 0,
1.1
2
Abstract and Applied Analysis
in a Hilbert space H with the self-adjoint positive definite operator A under assumption
J
|αi | ≤ 1.
1.2
i1
The well-posedness of multipoint nonlocal boundary value problem 1.1 in Hölder
spaces with a weight was established. Moreover, coercivity estimates in Hölder norms for
the solutions of nonlocal boundary value problems for elliptic-parabolic equations were
obtained.
In 21, we studied the well-posedness of the first order of accuracy difference scheme
for the approximate solution of boundary value problem 1.1 under assumption 1.2.
Throughout this work, we consider the following second order of accuracy difference
scheme:
−τ −2 uk1 − 2uk uk−1 Auk gk ,
gk gtk , tk kτ, 1 ≤ k ≤ N − 1, Nτ 1,
τ τ τ −1 uk − uk−1 − I A Auk−1 I A fk ,
fk ftk−1/2 ,
2
2
1
τ, −N − 1 ≤ k ≤ 0,
tk−1/2 k −
2
1.3
u2 − 4u1 3u0 −3u0 4u−1 − u−2 ,
uN
J
λi
τ fλi /τ Auλi /τ
αi uλi /τ λi −
ϕ,
τ
k1
for the approximate solution of boundary value problem 1.1 under assumption 1.2.
The well-posedness of difference scheme 1.3 in Hölder spaces with a weight is
established. As an application, the stability, almost coercivity stability, and coercivity stability
estimates for solutions of second order of accuracy difference scheme for the approximate
solution of the nonlocal boundary elliptic-parabolic problem are obtained.
2. Main Theorems
Throughout the paper, H is a Hilbert space and we denote B 1/2τA A4 τ 2 A,
where A is a self-adjoint positive definite operator. Then, it is clear that B is the self-adjoint
positive definite operator and B ≥ δ1/2 I where δ > δ0 > 0, and R 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 following
operators
τA2
I τA 2
D
Tτ G
,
τ 2A
I−
2
τ P I A ,
2
,
R I τB−1 ,
τ
I B−1 A I τA P −2 K I − R2N−1 GKP −2 R2N−1
2
−1
n
λi
−λi /τ
τ A D
−GKP 2I τBR
αi I λi −
u0
τ
i1
−2
N
2.1
exist and are bounded for a self-adjoint positive operator A. Here,
1
2
B
τA A4 τ A ,
2
K
5
I 2τA τA2
4
−1
.
2.2
Furthermore, positive constants will be indicated by M which can differ in time. On
the other hand Mi α, β, . . . is used to focus on the fact that the constant depends only on
α, 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 essential
below.
Lemma 2.1. For a self-adjoint positive operator A, the following estimates are satisfied:
k
R H →H
k
BR k
D ≤ M1 δ1 δτ−k ,
H →H
−1 P M1 δ
,
kτ
≤
k
D − e−kτA H →H
1/2 k
R − e−kτA H →H
M1 δ
≤
,
k2
H →H
≤
H →H
M1 δ
,
k
≤ M1 δ,
ADk ≤ M1 δ,
−1 I − R2N
H →H
H →H
≤
M1 δ
,
kτ
2.3
≤ M1 δ,
k ≥ 1, δ > 0.
From these estimates, it follows that
τ
I B−1 A I τA P −2 K I − R2N−1 GKP −2 R2N−1 − GKP −2 2I τB
2
−1 n
λi
×RN
αi I λi −
≤ M2 δ.
τ A D−λi /τ
τ
i1
H →H
2.4
4
Abstract and Applied Analysis
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:
−1
uk I − R2N
× Rk − R2N−k u0 RN−k − RNk
⎛
⎞
⎡
⎡
n
0
λ
i
τ A ⎝D −λi /τ u0 − τ
P Ds−λi /τ fs ⎠
× ⎣ αi ⎣ I λi −
τ
i1
sλ /τ1
i
λi
λi −
τ fλi /τ ϕ
τ
N−1
RN−s − RNs gs τ
− RN−k − RNk I τB2I τB−1 B−1
s1
I τB2I τB−1 B−1
N−1
R|k−s| − Rks gs τ,
1 ≤ k ≤ N,
s1
uk D−k u0 − τ
0
−N ≤ k ≤ −1,
P Ds−k fs ,
sk1
u0 1
Tτ KP −2
2
2
×
2I − τ A × 2 τBRN
n
λi
×
αi I λi −
τ A
τ
i1
⎛
0
× ⎝D −λi /τ u0 − τ
⎞
P Ds−λi /τ fs ⎠
sλi /τ1
λi
λi −
τ fλi /τ ϕ
τ
−R
N−1
B
−1
N−1
N−s
R
s1
−R
Ns
N−1
gs τ I − R2N B−1
Rs−1 gs τ
s1
I − R2N I τB τB−1 g1 − 4P B−1 f0 P DB−1 f0 P B−1 f−1 ,
Tτ τ
I B−1 A I τA P −2 K I − R2N−1 GKP −2 R2N−1
2
−1
n
λi
−2
N
−λi /τ
− GKP 2I τBR
αi I λi −
u0
.
τ A D
τ
i1
2.5
Abstract and Applied Analysis
5
Now, we study well-posedness of problem 1.3. Let Fτ H Fa, bτ , H be the
linear space of mesh functions ϕτ {ϕk }N
defined on a, bτ {tk kh, N ≤
N
Nτ
a, Nτ
b} with values in the Hilbert space H. Next, on Fτ H
k ≤ N,
α
α
α −1, 1τ , H,
−1, 1τ , H, C0,1
−1, 0τ , H, C0α 0, 1τ , H, C
we denote Ca, bτ , H, C0,1
0,1
α −1, 0τ , H, 0 < α < 1 Banach spaces with the following norms:
and C
0
τ
ϕ Ca,b
max ϕk H ,
τ ,H
Na ≤k≤Nb
τ
ϕ α
C
ϕτ C−1,1
0,1 −1,1τ ,H
τ ,H
ϕkr − ϕk −kα r −α
E
sup
−N≤k<kr≤0
ϕkr − ϕk k rτα N − kα r −α ,
E
sup
1≤k<kr≤N−1
τ
ϕ α
C −1,0
ϕτ C−1,0
τ
ϕ α
C
ϕτ C0,1
0
τ ,H
τ ,H
0,1 0,1τ ,H
ϕkr − ϕk −kα r −α ,
E
sup
−N≤k<kr≤0
τ ,H
ϕkr − ϕk E k rτα N − kα r −α ,
sup
1≤k<kr≤N−1
ϕτ C−1,1
τ
ϕ α
C
0,1 −1,1τ ,H
τ ,H
ϕk2r − ϕk −kα 2r−α
E
sup
−N≤k<k2r≤0
ϕkr − ϕk k rτα N − kα r −α ,
E
sup
1≤k<kr≤N−1
ϕτ C−1,0
τ
ϕ α
C −1,0
0
2.6
τ ,H
τ ,H
sup
ϕk2r − ϕk −kα 2r−α , respectively.
E
−N≤k<k2r≤0
Theorem 2.3. Nonlocal boundary value problem 1.3 is stable in C−1, 1τ , H space.
Proof. By 22, we have
{uk }N−1
1
C0,1τ ,H
≤ M3 δ g τ C0,1
τ
ψ ,
ξ
H
,H
H
2.7
for the solution of the following boundary value problem:
−τ −2 uk1 − 2uk uk−1 Auk gk ,
gk gtk ,
2.8
tk kτ, 1 ≤ k ≤ N − 1,
u0 ξ,
uN ψ.
By 24, we have
{uk }0−N C−1,0τ ,H
≤ M4 δ f τ C−1,0
τ
ξH
,H
2.9
6
Abstract and Applied Analysis
for the solution of an inverse Cauchy difference problem:
τ τ τ −1 uk − uk−1 − I A Auk−1 I A fk ,
2
2
−N − 1 ≤ k ≤ 0,
2.10
u0 ξ.
Then, the proof of Theorem 2.3 is based on stability inequalities 2.7 and 2.9 and on the
following estimates:
ξH ≤ M5 δ f τ C−1,0
g τ C0,1
ϕH ,
ψ ≤ M6 δ f τ H
C−1,0
g τ C0,1
,H
ϕ ,
,H
H
τ ,H
τ
τ ,H
τ
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 ∈ DA and f0 , f−1 , g1 ∈ DI τB. Then, for the solution of difference
problem 1.3, the following almost coercivity inequality holds:
N−1 −2
τ uk1 − 2uk uk−1 1
C0,1τ ,H
0
−1
τ uk − uk−1 −N1 C−1,0τ ,H
0
τ I A Auk−1
C0,1τ ,H
2
−N1 C−1,0 ,H
τ
!
1
≤ M7 δ min ln , 1 |ln AH → H | f τ C−1,0 ,H g τ C0,1 ,H
τ
τ
τ
"
AϕH I τBf0 H I τBg1 H I τBf−1 H .
{Auk }N−1
1
2.12
Proof. We have
N−1 −2
τ uk1 − 2uk uk−1 1
C0,1τ ,H
{Auk }N−1
1
C0,1τ ,H
!
1
≤ M8 δ min ln , 1 |ln AH → H | g τ C0,1 ,H AξH Aψ H ,
τ
τ
2.13
for the solution of boundary value problem 2.8 see 22, and we get
0
−1
τ uk − uk−1 0
τ I
A
Au
k−1
2
−N1 C−1,0 ,H
C−1,0τ ,H
τ
!
1
≤ M9 δ min ln , 1 |ln AH → H | f τ C−1,0 ,H AξH ,
τ
τ
−N1 2.14
Abstract and Applied Analysis
7
for the solution of inverse Cauchy difference problem 2.10 see 24. Then, the proof of
Theorem 2.4 is based on almost coercivity inequalities 2.13 and 2.14 and on the following
estimates:
!
1
AξH ≤ M10 δ AϕH I τBf0 H min ln , 1 |ln AH → H |
τ
× f τ C−1,0 ,H g τ C0,1 ,H ,
τ
Aψ τ
H
1
≤ M11 δ AϕH I τBf0 H min ln , 1 |ln AH → H |
τ
× f τ C−1,0 ,H g τ C0,1 ,H
τ
!
2.15
τ
for the solution of boundary value problem 1.3. Proofs of these estimates follow the
scheme 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 is
α
α −1, 1τ , H, and the following coercivity
−1, 1τ , H, and C
well-posed in Hölder spaces C0,1
0,1
inequalities hold:
N−1 −2
τ uk1 − 2uk uk−1 1
α
C0,1
0,1τ ,H
0
−1
τ uk − uk−1 −N1 α
C0 −1,0τ ,H
0
I τ A Auk−1
{Auk }N−1
1
α
α
C0,1 0,1τ ,H
2
−N1 C
0 −1,0τ ,H
τ
1
f α
≤ M12 δ
g τ Cα 0,1 ,H AϕH I τBf0 H
C
−1,0
,H
τ
τ
0
0,1
α1 − α
I τBg1 H I τBf−1 H ,
N−1 −2
τ uk1 − 2uk uk−1 1
α
C0,1
0,1τ ,H
0
−1
τ uk − uk−1 −N1 α
C0 −1,0τ ,H
0
τ N−1 {Auk }1 α
I A Auk−1
α
C0,1 0,1τ ,H
2
−N1 C
0 −1,0τ ,H
1
f τ α
≤ M13 δ
g τ Cα 0,1 ,H AϕH
C
−1,0
,H
τ
τ
0
0,1
α1 − α
I τBf0 H I τBg1 H I τBf−1 H .
2.16
8
Abstract and Applied Analysis
Proof. By 22, 24, we have
N−1 −2
τ uk1 − 2uk uk−1 1
α
C0,1
0,1τ ,H
{Auk }N−1
1
α
C0,1
0,1τ ,H
2.17
τ
1
g Cα 0,1 ,H AξH Aψ H ,
≤ M14 δ
τ
0,1
α1 − α
for the solution of boundary value problem 2.8, and
0
−1
τ uk − uk−1 −N1 α
0
τ I A Auk−1
α
2
−N1 C
C0 −1,0τ ,H
0 −1,0τ ,H
τ
f α
AξH ,
C −1,0 ,H
1
≤ M15 δ
α1 − α
0
−1
τ uk − uk−1 −N1 α
1
≤ M16 δ
α1 − α
2.18
τ
0
0
τ I A Auk−1
α
2
−N1 C
C0 −1,0τ ,H
0 −1,0τ ,H
τ
f α
AξH
C −1,0 ,H
0
2.19
τ
for the solution of inverse Cauchy difference problem 2.10, respectively. Then, the proof of
Theorem 2.5 is based on coercivity inequalities 2.17–2.19 and the following estimates:
AξH ≤ M17 δ
1
f τ α
g τ Cα 0,1 ,H
C
−1,0
,H
τ
τ
0
0,1
α1 − α
Aϕ H I τBf0 H I τBg1 H I τBf−1 H ,
Aψ ≤ M18 δ
H
1
α
f τ C α −1,0τ ,H g τ C0,1
0,1τ ,H
0
α1 − α
2.20
AϕH I τBf0 H I τBg1 H I τBf−1 H
for the solution of difference scheme 1.3. Proofs of these estimates follow the scheme of the
papers 22, 24 and rely on both estimates 2.3 and 2.4 and formula 2.5. This concludes
the proof of Theorem 2.5.
Abstract and Applied Analysis
9
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 multidimensional
elliptic-parabolic equation:
−utt −
n
ar xuxr xr gt, x,
0 < t < 1, x ∈ Ω,
r1
ut n
ar xuxr xr ft, x,
−1 < t < 0, x ∈ Ω,
r1
ut, x 0,
u1, x J
x ∈ S, −1 ≤ t ≤ 1,
αi uλi , x ϕx,
i1
3.1
J
|αi | ≤ 1,
i1
−1 ≤ λ1 < λ2 < · · · < λi < · · · < λJ ≤ 0,
u0, x u0−, x,
ut 0, x ut 0−, x,
x ∈ Ω,
where ar x x ∈ Ω, ϕx ϕx 0, x ∈ S, gt, x t ∈ 0, 1, x ∈ Ω, and ft, x t ∈
−1, 0, x ∈ Ω are given smooth functions. Here, Ω is the unit open cube in the n-dimensional
Euclidean space Rn 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:
h {x xm h1 m1 , . . . , hn mn , m m1 , . . . , mn ,
Ω
0 ≤ mr ≤ Nr , hr Nr 1, r 1, . . . , n},
h ∩ Ω,
Ωh Ω
h ∩ S.
Sh Ω
3.2
10
Abstract and Applied Analysis
1
2
W21 Ωh , and W2h
W22 Ωh of
We introduce the Hilbert spaces L2h L2 Ωh , W2h
the grid functions ϕh x {ϕh1 m1 , . . . , hn mn } defined on Ωh , equipped with the following
norms:
h
ϕ L2h
h
ϕ 1
W2h
h
ϕ 2
W2h
⎞1/2
#2
##
#
h
⎝
#ϕ x# h1 · · · hn ⎠ ,
⎛
x∈Ωh
ϕh L2h
ϕh L2h
⎞1/2
n #
#2
# h #
⎝
#ϕ xr # h1 · · · hn ⎠ ,
⎛
x∈Ωh r1
⎛
⎝
⎞1/2
3.3
n #
#
# h #2
#ϕ xr # h1 · · · hn ⎠
x∈Ωh r1
⎞1/2
n #
#2
# h
#
⎝
#ϕ xr xr ,mr # h1 · · · hn ⎠ .
⎛
x∈Ωh r1
To the differential operator A generated by problem 3.1, we assign the difference
operator Axh by formula
Axh uh −
n r1
ar xuhxr
3.4
xr ,mr
acting in the space of grid functions uh x, satisfying the conditions uh x 0 for all x ∈ Sh .
With the help of Axh , we arrive at the following nonlocal boundary value problem:
−
d2 uh t, x
Axh uh t, x g h t, x,
dt2
duh t, x
− Axh uh t, x f h t, x,
dt
uh 1, x 0 < t < 1, x ∈ Ωh ,
−1 < t < 0, x ∈ Ωh ,
n
αk uh λk , x ϕh x,
n
k1
k1
uh 0, x uh 0−, x,
3.5
|αk | ≤ 1,
duh 0, x duh 0−, x
,
dt
dt
for an infinite system of ordinary differential equations.
x ∈ Ωh ,
x ∈ Ωh ,
Abstract and Applied Analysis
11
In the second step, we replace problem 3.5 by difference scheme 1.3 accurate to the
following second order see 22, 24:
uhk1 x − 2uhk x uhk−1 x
−
τ2
tk kτ,
gkh x g h tk , x,
Axh uhk x gkh x,
1 ≤ k ≤ N − 1,
Nτ 1,
x ∈ Ωh ,
τ x 2 h
τ − Axh uk−1 x I Axh fkh x,
Ah
τ
2
2
1
τ,
−N 1 ≤ k ≤ 0, x ∈ Ωh ,
tk−1/2 k −
fkh x f h tk−1/2 , x,
2
uhk x − uhk−1 x
−uh2 x 4uh1 x − 3uh0 x 3uh0 x − 4uh−1 x uh−2 x,
uhN x 3.6
h,
x∈Ω
J
λi
ϕh x,
τ f hλ /τ Axh uhλ /τ x
αi uhλ /τ x λk −
i
i
i
τ
k1
h.
x∈Ω
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:
h N−1 u
k
−N
C−1,1τ ,L2h h −1 ≤ M19 δ fk
−N1 C−1,0τ ,L2h N−1 −2 h
h
h
τ
u
−
2u
u
k1
k
k−1
1
C0,1τ ,L2h 0
−1 h
h
τ
u
−
u
k
k−1
−N1 ⎡
≤M20 δ⎣f0h L2h
h
f−1
L2h
1
h N−1 u
k
1
C−1,0τ ,L2h g1h h N−1 gk
L2h
C0,1τ ,L2h ϕh L2h
,
2
C0,1τ ,W2h
h 0
uk−1
−N1 ϕh 2
W2h
2
C−1,0τ ,W2h
τ f0h 1
W2h
h
τ f−1
1
W2h
h −1 h N−1 1
f
.
ln
g
τ |h| k −N1 C−1,0τ ,L2h k 1 C0,1τ ,L2h τ g1h 1
W2h
3.7
The proof of Theorem 3.1 is based on Theorem 2.3, Theorem 2.4, the symmetry
property of the difference operator Axh defined by formula 3.4, the estimate
# #!
1
1
# x
#
,
min ln , 1 #ln Ah L2h → L2h # ≤ M21 δ ln
τ
τ |h|
3.8
and the following theorem on the coercivity inequality for the solution of elliptic difference
equation in L2h .
12
Abstract and Applied Analysis
Theorem 3.2. For the solution of the following elliptic difference problem:
Axh uh x ωh x,
x ∈ Ωh ,
uh x 0,
x ∈ Sh ,
3.9
the following coercivity inequality holds [25]:
n h u
xr xr ,mr ≤ M22 δωh L2h
L2h
r1
.
3.10
Theorem 3.3. Let τ and |h| be sufficiently small positive numbers. Then, solutions of difference
scheme 3.6 satisfy the following coercivity stability estimates:
N−1 −2 h
h
h
τ
uk1 − 2uk uk−1
1
α
C0,1
0,1τ ,L2h 0
−1 h
h
uk − uk−1
τ
−N1 α
h 0
uk−1
C0 −1,0τ ,L2h −N1 α
h N−1 uk
1
α
2
C0,1
0,1τ ,W2h
2
C0 −1,0τ ,W2h
≤ M23 δ ϕh 2
W2h
τ f0h 1
W2h
h
τ f−1
1
W2h
τ g1h 1
W2h
h −1 h N−1 1
f
,
g
α
α1 − α k −N1 C0α −1,0τ ,L2h k 1 C0,1
0,1τ ,L2h N−1 −2 h
h 0
h
h
τ
uk1 − 2uk uk−1
uk−1
1
α
C0,1
0,1τ ,L2h 0
−1 h
h
uk − uk−1
τ
−N1 α
C0 −1,0τ ,L2h ≤ M24 δ ϕh 2
W2h
τ f0h 1
W2h
3.11
−N1 C
α −1,0τ ,W 2 0
2h
h N−1 uk
1
h
τ f−1
1
W2h
α
2
C0,1
0,1τ ,W2h
τ g1h 1
W2h
h −1 h N−1 1
f
.
g
α
α1 − α k −N1 C0α −1,0τ ,L2h k 1 C0,1
0,1τ ,L2h The proof of Theorem 3.3 is based on the abstract Theorem 2.5, Theorem 3.2, and the
symmetry property of the difference operator Axh defined by formula 3.4.
References
1 D. G. Gordeziani, On Methods of Resolution of a Class of Nonlocal Boundary Value Problems, Tbilisi
University Press, Tbilisi, Georgia, 1981.
Abstract and Applied Analysis
13
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