PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE
Nouvelle série, tome 95 (109) (2014), 49–62
DOI 10.2298/PIM1409049J
FINITE DIFFERENCE APPROXIMATION
OF A PARABOLIC PROBLEM
WITH VARIABLE COEFFICIENTS
Boško S. Jovanović and Zorica Milovanović
Communicated by Gradimir Milovanović
Abstract. We study the convergence of a finite difference scheme that approximates the third initial-boundary-value problem for a parabolic equation
with variable coefficients on a unit square. We assume that the generalized sos,s/2
lution of the problem belongs to the Sobolev space W2
, s 6 3. An almost
second-order convergence rate estimate (with additional logarithmic factor) in
1,1/2
the discrete W2
norm is obtained. The result is based on some nonstandard
a priori estimates involving fractional order discrete Sobolev norms.
1. Introduction
For a class of finite difference schemes (FDSs) approximating elliptic boundaryvalue problems (BVPs) with generalized solutions, convergence rate estimates compatible with the smoothness of the data
(1.1)
ku − vkWpk (ω) 6 Chs−k kukWps (Ω) ,
s > k,
are of great interest (see [6, 13]). Here u = u(x) denotes the solution of the BVP,
v denotes the solution of the corresponding FDS, h is the discretization parameter,
Wpk (ω) is the Sobolev space of mesh functions, and C is a positive generic constant,
independent of h and u. In the parabolic case, instead of (1.1) it is natural to look
for error bounds of the form
s−k
s > k,
(1.2)
ku − vkW k,k/2 (Q ) 6 C hs−k + τ 2 kukW s,s/2(Q) ,
p
hτ
p
where τ is the temporal mesh-size. A standard technique for establishing estimates
of such types (see [6, 13, 14]) is based on the Bramble–Hilbert lemma [3, 5].
For BVPs with an oblique derivative boundary condition a loss of one half of
an order in the convergence rate (usually O(h3/2 )) is often observed, caused by
2010 Mathematics Subject Classification: 65M15.
Key words and phrases: parabolic initial-boundary-value problem, oblique derivative boundary condition, finite differences, Sobolev spaces, convergence rate estimates.
The research was supported by Ministry of Education and Science of Republic of Serbia
under project 174015.
49
50
JOVANOVIĆ AND MILOVANOVIĆ
the approximation of the boundary condition. Nevertheless, improved results are
obtained in some cases, mainly for elliptic problems (see [4, 7]).
In the present paper, for the FDS approximating initial-boundary value problem (IBVP) for a parabolic equation with variable coefficients and an oblique deriv1,1/2
ative boundary condition a second order error bound in the discrete W2
norm
is obtained under minimal smoothness assumptions on the input data. The result
is based on some nonstandard a priori estimates involving fractional order discrete
Sobolev norms.
2. Formulation of the Problem
As the model problem we consider, in Q = Ω × (0, T ) = (0, 1)2 × (0, T ), the
following initial-boundary value problem for a parabolic equation with variable
coefficients:
∂u
(2.1)
+ Lu = f,
(x, t) = (x1 , x2 , t) ∈ Q,
∂t
(2.2)
lu = 0,
(x, t) ∈ Γ × (0, T ) = ∂Ω × (0, T ),
(2.3)
u(x, 0) = u0 (x),
x ∈ Ω.
where
(2.4)
Lu := −
2
X
∂u ∂ aij
,
∂xi
∂xj
i,j=1
lu :=
2
X
i,j=1
aij
∂u
cos (ν, xi ) + αu
∂xj
and ν is the unit outward normal to Γ. We assume that the conditions of strong
ellipticity are satisfied:
aij = aij (x) = aji ,
(2.5) c0
2
X
i=1
ξi2 6
2
X
i,j=1
aij ξi ξj 6 c1
2
X
ξi2 ,
α = α(x),
∀x ∈ Ω̄, ∀ξ ∈ R2 ,
ci = const. > 0,
i=1
α0 6 α(x) 6 α1 , αi = const. > 0.
S2 S1
Let us denote Γ = i=1 k=0 Γik , where Γ1k = {k} × [0, 1], Γ2k = [0, 1] × {k},
and Σik = Γik × [0, T ].
We also assume that the generalized solution of the problem (2.1)–(2.3) bes,s/2
longs to the Sobolev space W2
(Q), 2 < s 6 3, while the data satisfy the
s−3/2
following smoothness conditions: aij ∈ W2s−1 (Ω), α ∈ W2
(Γik ), α ∈ C(Γ),
s−2,s/2−1
f ∈ W2
(Q) and u0 ∈ W2s−1 (Ω).
3. Finite Difference Approximation
Let n, m ∈ N, n > 2, m > 1, h = 1/n and τ = T /m. We consider the uniform
spatial mesh ω̄ with mesh size h on Ω̄ and the uniform temporal mesh ω̄τ with
mesh size τ on [0, T ]. We also denote ω = ω̄ ∩ Ω, ωτ = ω̄τ ∩ (0, T ), ωτ− = ω̄τ ∩ [0, T ),
ωτ+ = ω̄τ ∩ (0, T ], γ = ω̄ ∩ Γ, γ̄ik = ω̄ ∩ Γik , γik = {x ∈ γ̄ik : 0 < x3−i < 1},
−
+
⋆
γik
= {x ∈ γ̄ik : 0 6 x3−i < 1}, γik
= {x ∈ γ̄ik : 0 < x3−i 6 1}, γik
= γ̄ik r γik ,
FINITE DIFFERENCE APPROXIMATION OF A PARABOLIC PROBLEM
51
S
+
+
γ⋆ = γ r
i,k γik , σik = γik × ωτ , σ̄ik = γ̄ik × ωτ , i = 1, 2, k = 0, 1, and
Q̄hτ = ω̄ × ω̄τ .
The finite difference operators are defined in the usual manner [12]:
vxi = (v +i − v)/h,
vx̄i = (v − v −i )/h,
vt = (v̂ − v)/τ,
vt̄ = (v − v̌)/τ,
where v ±i (x, t) = v(x±hei , t), ei is the unit vector of the axis xi , v̂(x, t) = v(x, t+τ )
and v̌(x, t) = v(x, t − τ ).
We also define the Steklov smoothing operators with the step sizes h and τ
[13]:
Z 1
+
Ti f (x, t) =
f (x + hx′ ei , t) dx′ = Ti− f (x + hei , t) = Ti f (x + h2 ei , t),
0
Ti2± f (x, t) = 2
Tt+ f (x, t) =
Z
Z
0
1
0
1
(1 − x′ )f (x ± hx′ ei ) dx′ ,
i = 1, 2,
f (x, t + τ t′ ) dt′ = Tt− f (x, t + τ ) = Ti f (x, t + τ2 ).
These operators commute and transform derivatives into differences, for example:
2 ∂u
∂u
∂ u
Ti+
= uxi , Ti−
= ux̄i , Ti2
= ux̄i xi , i = 1, 2,
∂xi
∂xi
∂x2i
∂u
∂u
Tt+
= ut , Tt−
= ut̄ .
∂t
∂t
We approximate the IBVP (2.1)–(2.3) with the following implicit FDS:
(3.1)
vt̄ + Lh v = f˜,
x ∈ ω̄,
v(x, 0) = u0 (x),
t ∈ ωτ+ ,
x ∈ ω̄,
where

2 h
i
P


aij vxj x̄i + aij vx̄j xi ,
− 21



i,j=1




vx2 +vx̄2
a11 +a+1

2
11

−
v
−
a
+
α̃v
− a12 vx̄2 x1
x
12

1
h
2
2




− a21 vx1 x̄2 − 12 a22 vx2 x̄2 − 12 a22 vx̄2 x2 ,





 h a +a+1
a +a+2
11
2
11
vx1 − a12 vx2 − a21 vx1 − 22 2 22 vx2
Lh v = h −
2
i




+(α̃1 + α̃2 )v ,




h


a11 +a+1
a +a−2

2
11

vx1 − a12 vx̄2 + a21 vx1 + 22 2 22 vx̄2

h −
2

i




+(α̃1 + α̃2 )v − 2 a12 vx̄2 x1 − 2 a21 vx1 x̄2 ,




and analogously at the other boundary nodes,
x∈ω
x ∈ γ10
x = (0, 0)
x = (0, 1)
x ∈ γ r γ̄10
52
JOVANOVIĆ AND MILOVANOVIĆ

2 2 −

T1 T2 Tt f,


2
f˜ = Ti2± T3−i
Tt− f,



T 2± T 2± T − f,
t
1
2
2
α̃ = T3−i
α,
x∈ω
x ∈ γi, 0.5∓0.5
x = (0.5 ∓ 0.5, 0.5 ∓ 0.5) ∈ γ ⋆
x ∈ γi0 ∪ γi1 ,
i = 1, 2
and
α̃i = Ti2± α ,
x ∈ γ⋆,
xi = 0.5 ∓ 0.5 ,
i = 1, 2 .
4. Error Analysis
Let u be the solution of the IBVP (2.1)-(2.3), and let v denote the solution of
the FDS (3.1). The error z = u − v is defined on Q̄hτ and satisfies the following
conditions
zt̄ + Lh z = ψ,
(4.1)
z(x, 0) = 0,
x ∈ ω̄,
t ∈ ωτ+ ,
x ∈ ω̄,
where

P2
ξt̄ + i,j=1 ηij, x̄i ,







 ξ˜ + 2 η11 + 2 η12 + η̃21, x̄ + η̃22, x̄ + 2 ζ,
t̄
2
2
h
h
h
ψ=


2
2
2
2
2
˜
 ξ˜ + η̃ + η̃ + η̃ + η̃ + (ζ + ζ ) ,

11
12
21
22
1
2

t̄

h
h
h
h
h


and analogously at the other boundary nodes,
ξ = u − T12 T22 u,
2±
ξ̃ = u − Ti2 T3−i
u,
ξ̃˜ = u − T12± T22± u,
x∈ω
x ∈ γ10
x = (0, 0)
−
x ∈ γ r γ10
x∈ω
x ∈ γ3−i, 0.5∓0.5
x = (0.5 ∓ 0.5, 0.5 ∓ 0.5) ∈ γ ⋆
∂u 1
+i
2
ηij = Ti+ T3−i
Tt− aij
−
aij uxj + a+i
ij ux̄j ,
∂xj
2
x∈ω
∂u aii + a+i
2± −
−
ii
η̃ii = Ti+ T3−i
Tt aii
−
u xi ,
x ∈ γ3−i,
0.5∓0.5
∂xi
2

∂u + 2+ −
−


− ai,3−i ux3−i , x ∈ γ3−i,0
Ti T3−i Tt ai,3−i ∂x
3−i
η̃i,3−i =
∂u 
2− −
+i
−

Ti+ T3−i
Tt ai,3−i
− a+i
x ∈ γ3−i,1
i,3−i ux̄3−i ,
∂x3−i
FINITE DIFFERENCE APPROXIMATION OF A PARABOLIC PROBLEM
ζ = (Ti2 α)u − Ti2 Tt− (αu),
53
x ∈ γ3−i,0 ∪ γ3−i,1
ζi = (Ti2± α)u − Ti2± Tt− (αu),
x ∈ γ ⋆ , xi = 0.5 ∓ 0.5.
We define the following discrete inner products and norms:
X
h2 X
h2 X
v(x)w(x)+
v(x)w(x),
|[v]|2 = [v, v],
[v, w] = h2
v(x)w(x)+
2
4
x∈γrγ ⋆
x∈γ ⋆
x∈ω
[v, w)i = h2
X
v(x)w(x) +
X
v(x)w(x) +
x∈ω∪γi0
(v, w]i = h2
[v, w) = h2
X
v(x)w(x),
X
v(x)w(x),
|[vk2i = [v, v)i ,
−
−
x∈γ3−i,0
∪γ3−i,1
2
x∈ω∪γi1
X
h2
2
h
2
+
+
x∈γ3−i,0
∪γ3−i,1
|[vk2 = [v, v),
v(x)w(x),
kv]|2i = (v, v]i ,
(v, w] = h2
|[v]|2W 1 (ω̄) = |[v]|2 + |[vx1 ||21 + |[vx2 k22 ,
X
x∈γik
[v, w)γ − = h
ik
|[v]|C(ω̄) = max |v(x)|,
x∈ω̄
2
[v, w]γ̄ik = h
X
v(x)w(x),
+
+
x∈ω∪γ11
∪γ21
−
−
x∈ω∪γ10
∪γ20
kv]|2 = (v, v],
X
h X
v(x)w(x),
v(x)w(x) +
2
⋆
|[v]|2γ̄ik
= [v, v]γ̄ik ,
x∈γik
|[v||2γ − = [v, v)γ − ,
v(x)w(x),
ik
ik
−
x∈γik
kvk2γik = h
X
v 2 (x),
x∈γik
2
X
v(x) − v(x )
|v|2W 1/2 (γ − ) = h2
,
|[vk2W 1/2 (γ − ) = |v|2W 1/2 (γ − )+|[vk2γ − ,
′
ik
x
−
x
2
2
2
ik
ik
ik
3−i
3−i
−
x, x′ ∈γik
, x′ 6=x
X
1
1
|[vk2Ẅ 1/2 (γ − ) = |v|2W 1/2 (γ − ) + h
+
v 2 (x),
x3−i + h/2 1 − x3−i − h/2
2
2
ik
ik
−
x∈γik
X
X
kvk2τ = τ
v 2 (t),
|[vk2i,hτ = τ
|[v(·, t)k2i ,
′
t∈ωτ+
kvk2σik = τ
X
t∈ωτ+
kv(·, t)k2γik ,
|[v]|2σ̄ik = τ
t∈ωτ+
τ ,L2 (ω̄))
=τ
X
t∈ωτ+
|[v(·, t)k2Ẅ 1/2 (γ − ) ,
2
ik
|[v(·, t) − v(·, t′ )]|2
,
τ ,L2 (ω̄))
(t − t′ )2
2
t, t′ ∈ω̄τ , t′ 6=t
X 1
1
= |v|2W 1/2 (ω̄ ,L (ω̄)) + τ
+
|[v(·, t)]|2 ,
τ
2
t
+
τ
/2
T
−
t
+
τ
/2
2
t∈ω̄τ
X
2
|[v]|L2 (ω+ , W 1 (ω̄)) = τ
|[v(·, t)]|2W 1 (ω̄) ,
|v|2W 1/2 (ω̄
2
|[v(·, t)]|2γ̄ik ,
t∈ωτ+
|[vk2L (ω+ , Ẅ 1/2 (γ − ))
2
τ
2
ik
|[v]|2Ẅ 1/2 (ω̄
X
= τ2
τ
|[v]|2W 1, 1/2 (Q
2
hτ )
X
2
2
t∈ωτ+
= |[v]|2L2 (ω+ , W 1 (ω̄)) + |v|2W 1/2 (ω̄
τ
2
2
τ ,L2 (ω̄))
.
54
JOVANOVIĆ AND MILOVANOVIĆ
We shall prove a suitable a priori estimate for the FDS (4.1) which will be used
to estimate its convergence rate.
Lemma 4.1. Let aij and α satisfy the assumptions from Section 2. Then, for
a sufficiently small mesh step h, there exist positive constants C1 and C2 such that
C1 |[v]|2W 1 (ω̄) 6 [Lh v, v] 6 C2 |[v]|2W 1 (ω̄) .
2
2
Proof. The proof immediately follows from
2
[Lh v, v] =
=
1 Xn
[aii vxi , vxi )i + (aii vx̄i , vx̄i ]i + [ai,3−i vx3−i , vxi )
2 i=1
o
X
h X
+ (ai,3−i vx̄3−i , vx̄i ] + h
α̃ v 2 +
(α̃1 + α̃2 ) v 2
2
⋆
⋆
x∈γrγ
x∈γ
2
2 X
1
o hX
1 Xn
2
−
[aij vxj , vxi ) + (aij vx̄j , vx̄i ] +
(−1)k [a+i
ii − aii , vxi )γ3−i,
k
2 i,j=1
4 i=1
k=0
X
+
α̃ v 2 +
x∈γrγ ⋆
h X
(α̃1 + α̃2 ) v 2
2 x∈γ ⋆
and a discrete imbedding theorem [12].
Lemma 4.2. [2, 7] The following inequality holds true:
[v, wx3−i ) − 6 C|[vk 1/2 − |[w]|W 1 (ω̄) .
γ
Ẅ
(γ )
2
ik
2
ik
Lemma 4.3. [7] Let v be a mesh function on ω̄, then
q
|[v]|C(ω̄) 6 C log h1 |[v]|W21 (ω̄) .
Lemma 4.4. [8] The solution of the FDS
(x, t) ∈ ω̄ × ωτ+ ;
vt̄ + Lh v = ϕ,
v(x, 0) = 0,
x ∈ ω̄.
satisfies the a priori estimate
|[v]|W 1, 1/2 (Q
hτ )
2
X
1/2
2
6C τ
|[ϕ(·, t)]|−1
t∈ωτ+
where
|[ϕ(·, t)]|−1 := sup
w
[ϕ(·, t), w]
.
|[w]|W21 (ω̄)
Lemma 4.5. [8] The solution of the FDS
(x, t) ∈ ω̄ × ωτ+ ;
vt̄ + Lh v = φt̄ ,
v(x, 0) = 0,
satisfies the a priori estimate
|[v]|W 1, 1/2 (Q
2
hτ )
6 C|[φ]|Ẅ 1/2 (ω̄τ , L2 (ω̄)) .
2
x ∈ ω̄.
FINITE DIFFERENCE APPROXIMATION OF A PARABOLIC PROBLEM
55
Lemma 4.6. Let w ∈ W2r (Γik ), 0 < r 6 0.5. Then
+
|T3−i
w|W 1/2 (γ − ) 6 C(r)hr−1/2 |w|W2r (Γik ) .
2
ik
Proof. Without loss of generality let us set i = 1 and k = 0. Hence
n−1
X n−1
X T2+ w(0, ih) − T2+ w(0, jh) 2
+ 2
2
|T2 w|W 1/2 (γ − ) = h
(ih − jh)2
2
10
i=1
j=1, j6=i
2
n−1
i−1 +
XX
T2 w(0, ih) − T2+ w(0, jh)
2
= 2h
(ih − jh)2
i=1 j=1
( Z
)2
n−1
i−1
ih+h Z jh+h
XX
1
= 2h2
[w(0, x) − w(0, x′ )] dx′ dx (ih − jh)−2
h2 ih
jh
i=1 j=1
(Z
)
n−1
i−1
2
ih+h Z jh+h
XX
1
[w(0, x) − w(0, x′ )]
2
′
dx dx
= 2h
h4
(x − x′ )1+2r
ih
jh
i=1 j=1
(Z
)
ih+h Z jh+h
(x − x′ )1+2r ′
×
dx dx
(ih − jh)2
ih
jh
(
)
n−1
i−1 Z ih+h Z jh+h
2
XX
[w(0, x) − w(0, x′ )]
′
−2 1+2r 2 2r−1
6 2h 2
h h
dx dx
(x − x′ )1+2r
ih
jh
i=1 j=1
=2
1+2r 2r−1
h
2
Z
1
0
Z
x
0
2
[w(0, x) − w(0, x′ )]
dx′ dx = 21+2r h2r−1 |w|2W r (Γ10 ) .
2
(x − x′ )1+2r
Let us rearrange the summands in truncation error ψ in the following manner:
η̃ = η + η ′ ,
ξ˜ = ξ + ξ ′ ,
ξ̃˜ = ξ + ξ ⋆ ,
ij
ij
ij
where
h + −
∂ ∂u = ± Ti Tt
aii
,
x ∈ γ3−i, 0.5∓0.5 ,
3
∂x3−i
∂xi
h + −
∂ ∂u h + −
∂2u
′
ηi,3−i = ± Ti Tt
ai,3−i
∓ Ti Tt ai,3−i 2
3
∂x
∂x3−i
2
∂x3−i
3−i
h + − ∂ ∂u + Ti Tt
ai,3−i
,
x ∈ γ3−i, 0.5∓0.5 ,
2
∂xi
∂x3−i
h 2 ∂u ξ ′ = ∓ T3−i
,
x ∈ γi, 0.5∓0.5 ,
3
∂xi
∂u h
∂u h
ξ ⋆ = ∓ T22±
∓ T12±
,
x = (0.5 ∓ 0.5, 0.5 ∓ 0.5) ∈ γ ⋆ .
3
∂x1
3
∂x2
Using the boundary condition (2.2) we further obtain
′
ηii
ξt̄′ = λi, x̄3−i + µi + νi ,
x ∈ γi, 0.5∓0.5 ,
56
JOVANOVIĆ AND MILOVANOVIĆ
where
h + − ai,3−i ∂u Tt
,
λi = ± T3−i
3
aii ∂t
h 2
∂ ai,3−i ∂u
µi = ∓ T3−i
Tt−
,
3
∂x3−i
aii
∂t
α ∂u h 2
νi = − T3−i
Tt−
.
3
aii ∂t
Similarly, for x = (0, 0), we obtain
2
2
2
2
ξt̄⋆ = λ1 − λ⋆1 + µ1 + ν1 + λ2 − λ⋆2 + µ2 + ν2 ,
h
h
h
h
where λ1 and λ2 are the same as before and
a
h
i,3−i ∂u
,
λ⋆i = ± Tt−
3
aii ∂t
h 2+ −
∂ ai,3−i ∂u
µi = − T3−i Tt
,
3
∂x3−i
aii
∂t
h 2+ − α ∂u Tt
,
νi = − T3−i
3
aii ∂t
with an analogous representation at other nodes from γ ⋆ .
Using Lemmas 4.1–4.5, we obtain the following assertion.
Theorem 4.1. The finite difference scheme (4.1) is stable in the sense of the
a priori estimate
(
2
1 X
2
X
X
|[ηij ki,hτ +
kζkσik
|[z]|W 1, 1/2 (Q ) 6 C |[ξ]|Ẅ 1/2 (ω̄τ , L2 (ω̄)) +
hτ
2
(4.2)
+h
1
X
2
X
2
′
|[ηij
kL2 (ω+ , Ẅ 1/2 (γ −
τ
k=0 i,j=1
+h
1 X
2 X
|[µi ]|σ̄ik +|[νi ]|σ̄ik
k=0 i=1
k=0 i=1
i,j=1
2
3−i,k
))
+h
1 X
2
X
|[λi kL2 (ω+ , Ẅ 1/2 (γ − ))
τ
k=0 i=1
2
ik
)
2 X
q
X
+ h log h1
kζi (x, ·)kτ +kλ⋆i (x, ·)kτ .
i=1 x∈γ ⋆
In accordance with Theorem 4.1, the problem of deriving the convergence rate
estimate for the FDS (3.1) is reduced to estimating the right-hand side terms in
the inequality (4.2).
Let us assume that τ ≍ h2 , i.e., c2 h2 6 τ 6 c3 h2 for some positive constants
c2 and c3 .
The term ηij at the internal mesh nodes can be estimated in the same manner
as in the case of the Dirichlet IBVP (see [6]):
X
X
2
ηij
6 Ch2s−2 kaij k2W s−1 (Ω) kuk2W s,s/2 (Q) , 2 < s 6 3.
(4.3)
τ
h2
t∈ωτ+
x∈ω∪γi0
2
2
In boundary nodes ηii can be decomposed in the following manner
ηii = ηii,1 + ηii,2 + ηii,3 + ηii,4 ,
−
x ∈ γ3−i,
0.5∓0.5 ,
FINITE DIFFERENCE APPROXIMATION OF A PARABOLIC PROBLEM
57
where
∂u ∂u − Ti+ aii Ti+ Tt−
,
ηii,1 = Ti+ Tt− aii
∂xi
∂xi
aii + a+i
∂u ii
ηii,2 = Ti+ aii −
Ti+ Tt−
,
2
∂xi
aii + a+i
∂u + ∂u ii
Ti+ Tt−
− Ti
,
ηii,3 =
2
∂xi
∂xi
∂u ∂u h + −
∂ ∂u + 2± −
+ −
ηii,4 = Ti T3−i Tt aii
− Ti Tt aii
∓ Ti Tt
aii
.
∂xi
∂xi
3
∂x3−i
∂xi
The terms ηii,l for l = 1, 2, 3 satisfy the same conditions as analogous terms in [6]
whereby it follows that
X
X
2
ηii,l
6 Ch2s−2 kaii k2W s−1 (Ω) kuk2W s,s/2 (Q) , 2 < s 6 3.
(4.4) τ
h2
2
−
−
t∈ωτ+ x∈γ3−i,0
∪γ3−i,1
2
s−1,(s−1)/2
∂u
For s > 2.5 the term ηii,4 is a bounded linear functional of w = aii ∂x
∈ W2
i
which vanishes when w = 1, x1 , x2 , t. Using the Bramble–Hilbert lemma [3, 5] and
properties of multipliers in Sobolev spaces [10] we obtain the following result
X
X
∂u 2
2
(4.5) τ
h2
ηii,4
6 Ch2s−2 aii
∂xi W2s−1,(s−1)/2 (Q)
+
−
−
t∈ωτ x∈γ3−i,0 ∪γ3−i,1
6 Ch2s−2 kaii k2W s−1 (Ω) kuk2W s,s/2 (Q) ,
2
2.5 < s 6 3.
2
Similarly, at the boundary nodes ηi,3−i can be decomposed in the following
manner
ηi,3−i = ηi,3−i,1 + ηi,3−i,2 + ηi,3−i,3 + ηi,3−i,4 ,
−
x ∈ γ3−i,
0.5∓0.5 ,
where
∂u ∂u − Ti+ Tt− ai,3−i
∂x3−i
∂x3−i
∂ ∂u h + −
∓ Ti Tt
ai,3−i
,
3
∂x3−i
∂x3−i
∂u ∂u = Ti+ Tt− ai,3−i
− Tt− ai,3−i
∂x3−i
∂x3−i
∂u h + − ∂ − Ti Tt
ai,3−i
,
2
∂xi
∂x3−i
2 ∂u
∂u
h
∂ u
±
= ai,3−i Tt−
− T3−i
± Ti+ Tt−
,
∂x3−i
∂x3−i
2
∂x23−i
2 h
∂2u
∂ u
= ± Ti+ Tt− ai,3−i 2
− ai,3−i Ti+ Tt−
.
2
∂x3−i
∂x23−i
2± −
Tt ai,3−i
ηi,3−i,1 = Ti+ T3−i
ηi,3−i,2
ηi,3−i,3
ηi,3−i,4
The terms ηi,3−i,1 and ηi,3−i,2 satisfy estimates analogous to (4.5). For s > 2.5 and
s,s/2
ai,3−i ∈ C(Ω̄) the term ηi,3−i,3 is a bounded linear functional of u ∈ W2
which
58
JOVANOVIĆ AND MILOVANOVIĆ
vanishes when w = 1, x1 , x2 , t, x21 , x1 x2 , x22 . Using the Bramble–Hilbert lemma and
the Sobolev imbedding theorem [1] we obtain the following result
(4.6) τ
X
t∈ωτ+
h2
X
2
ηi,3−i,3
6 Ch2s−2 kai,3−i k2C(Ω̄) kuk2W s,s/2 (Q)
2
−
−
x∈γ3−i,0
∪γ3−i,1
6 Ch2s−2 kai,3−i k2W s−1 (Ω) kuk2W s,s/2 (Q) ,
2
2.5 < s 6 3.
2
−
Term ηi,3−i,4 can be estimated directly. Let us set i = 2 and x = (0, x2 ) ∈ γ10
.
Then
h 2
η2,1,4 (0, x2 , t) =
2 hτ
′
xZ2 +h
x2
Zt Zx2
∂a21
x′2 ∂2u
1−
(0, x′′2 ) 2 (0, x′2 , t′ ) dx′′2 dt′ dx′2 ,
h
∂x2
∂x1
t−τ x2
while for other boundary nodes, and also for i = 1, we have analogous integral
representations. Hence
2
2 2
X
X
∂ u
2
2
4 ∂ai,3−i (4.7) τ
h
ηi,3−i,4 6 Ch ∂xi L2 (Γik ) ∂x21 L2 (Σik )
+
−
t∈ωτ
x∈γ3−i,k
6 Ch4 kai,3−i k2W s−1 (Ω) kuk2W s,s/2 (Q) ,
2
s > 2.5 .
2
In such a way, from (4.3)-(4.7) one obtains
(4.8)
|[ηij ki,hτ 6 Chs−1 kaij kW s−1 (Ω) kukW s,s/2 (Q) ,
2
2.5 < s 6 3.
2
The term ξ at the internal mesh nodes is estimated in [6]. At the boundary
nodes ξ admits an analogous integral representation as for x ∈ ω. Hence, one
immediately obtains
q
(4.9) |[ξ]|Ẅ 1/2 (ω̄τ , L2 (ω̄)) 6 Chs−1 log h1 kaij kW s−1 (Ω) kukW s,s/2 (Q) , 2 < s 6 3.
2
2
2
The term ζ on σ3−i,k can be represented in the following manner:
ζ = Ti2 α u − Ti2 Tt− u + Ti2 α Ti2 Tt− u − Ti2 α Tt− u = ζ01 + ζ02 .
For s > 2.5 and α ∈ C(Γ3−i,k ) the term ζ01 is a bounded linear functional of u ∈
s−1,(s−1)/2
W2
(Σ3−i,k ) which vanishes when u = 1 and u = x3−i . Using the Bramble–
Hilbert lemma and the Sobolev imbedding theorem we obtain the following result
(4.10)
kζ01 kσ3−i,k 6 Chs−1 kαkC(Γ3−i,k ) kukW s−1,(s−1)/2 (Σ
3−i,k )
2
6 Ch
s−1
kαkW s−3/2 (Γ
2
3−i,k )
kukW s,s/2 (Q) ,
2.5 < s 6 3.
2
The term ζ02 is a bounded linear functional of (α, Tt− u) ∈ Wqr (Γ3−i,k )×W p2q (Γ3−i,k ),
q−2
q > 2, which vanishes when u = 1 or Tt− u = 1. Using the bilinear version of the
Bramble–Hilbert lemma, the Sobolev imbedding theorem and the Hölder inequality,
FINITE DIFFERENCE APPROXIMATION OF A PARABOLIC PROBLEM
59
after summation over the mesh γ3−i,k we obtain the following result
kζ02 kγ3−i,k 6 Chr+p kαkWqr (Γ3−i,k ) kTt− ukW p
2q/(q−2)
6 Ch
r+p
kαk
r+ 1 − 1
2
q
W2
(Γ3−i,k )
kTt−uk
p+ 1
q
W2
(Γ3−i,k )
(Γ3−i,k )
,
0 6 r 6 1,
1
q
1−
< p 6 1.
Summing over the mesh ωτ+ and using imbedding and trace theorems we obtain
kζ02 kσ3−i,k 6 Chr+p kαk
6 Ch
r+p
6 Ch
r+p
kαk
kαk
r+ 1 − 1
2
q
(Γ3−i,k )
W2
r+ 1 − 1
2
q
(Γ3−i,k )
W2
r+p− 1
2
W2
(Γ3−i,k )
kuk
kuk
kuk
p+ 1 , 1 (p+ 1 )
q 2
q
W2
(Σ3−i,k )
p+ 1 + 1 , 1 (p+ 1 + 1 )
2
q 2
2
q
W2
(Q)
r+p+1, 1 (r+p+1)
2
(Q)
W2
.
Finally, setting r + p = s − 1, we get
(4.11)
kζ02 kσ3−i,k 6 Chs−1 kαkW s−3/2 (Γ
3−i,k )
2
kukW s,s/2 (Q) ,
2 < s 6 3.
2
From (4.10) and (4.11) it follows that
(4.12)
kζkσ3−i,k 6 Chs−1 kαkW s−3/2 (Γ
3−i,k )
2
kukW s,s/2(Q) ,
2.5 < s 6 3.
2
⋆
For x ∈ γ3−i,k
we set
2±
ζi = Ti α u − Tt− u + Ti2± α Tt− u − Ti2± α Tt− u = ζi1 + ζi2 .
For r > 0.5 and α ∈ C(Γ3−i,k ) the term ζi1 is a bounded linear functional of
u ∈ W2r (0, T ) which vanishes when u = 1. Using the Bramble–Hilbert lemma and
the Sobolev imbedding theorem, after summation over the mesh ωτ+ we obtain
kζi1 kτ 6 Cτ r kαkC(Γ3−i,k ) ku(x, ·)kW2r (0,T )
6 Ch2r kαkW 2r−1/2 (Γ
3−i,k )
2
kukW 2r+1,r+1/2 (Q) ,
0.5 < r 6 1,
2
whereby, setting 2r + 1 = s,
(4.13)
kζi1 kτ 6 Chs−1 kαkW s−3/2 (Γ3−i,k ) kukW s,s/2(Q) ,
2
2 < s 6 3.
2
For i = 1 and x = (0, 0) we have
Z
x′ 2 h
1 − 1 α(x′1 , 0) Tt− u(0, 0, t) − Tt− u(x′1 , 0, t) dx′1
ζ12 (0, 0, t) =
h 0
h
Z Z ′
2 h x1 x′ ∂(Tt− u) ′′
=
1 − 1 α(x′1 , 0)
(x1 , 0, t) dx′′1 dx′1 .
h 0 0
h
∂x1
Analogous representations hold for other nodes x ∈ γ ⋆ and also for i = 2. Hence
∂(T − u) t
|ζi2 | 6 Ch kαkC(Γ3−i,k ) 6 Ch kαkW s−3/2 (Γ
kTt− ukW2s (Ω) , s > 2,
3−i,k )
2
∂xi
C(Ω̄)
whereby, after summation over the mesh ωτ+ , one obtains
(4.14)
kζi2 kτ 6 Ch kαkW s−3/2 (Γ
3−i,k )
2
kukW s,s/2 (Q) ,
s > 2.
2
⋆
In such a way, for x ∈ γ3−i,k
, from (4.13) and (4.14) we get
(4.15)
kζi kτ 6 Ch kαkW s−3/2 (Γ
2
3−i,k )
kukW s,s/2(Q) ,
2
s > 2.
60
JOVANOVIĆ AND MILOVANOVIĆ
′
Let us now estimate |[ηij
kL2 (ω+ , Ẅ 1/2 (γ −
τ
2
3−i,k
))
. Using Lemma 4.6 we immediately
obtain
2
3−i,k
)
−
Tt
∂ ∂u aii
6 Ch
∂x3−i
∂xi W r (Γ3−i,k )
2
∂ ∂u r+1/2 −
6 Ch
Tt ∂x3−i aii ∂xi
r+1/2 ,
W
(Ω)
r+1/2
′
|ηii
|W 1/2 (γ −
0 < r 6 0.5.
2
Using the inequality [11]
kF kL2(0,ε)
 r
ε kF kW2r (0,1) ,
0 < r < 0.5,




ε1/2 log 1ε kF kW 1/2 (0,1) , r = 0.5,
6C
2



 ε1/2 kF k 1/2
,
r > 0.5,
W
(0,1)
2
where 0 < ε < 1, we obtain
X 1
1
′ 2
h
+
(ηii
)
x
+
h/2
1
−
x
−
h/2
i
i
−
x∈γ3−i,k
2
∂u ∂xi W r (Γ− )
2
3−i,k
2
∂u ,
∂xi W r+1/2 (Ω)
∂ 1
−
T
aii
6 Ch
log t
h
∂x3−i
1 −
∂ 6 Ch2r+1 log T
aii
h t ∂x3−i
2r+1
and
h
X
−
x∈γ3−i,k
1
1
+
xi + h/2 1 − xi − h/2
0 < r < 0.5
2
2
′
(ηii
)
2
1
∂ ∂u −
6 Ch log
T
aii
1 .
h t ∂x3−i
∂xi
W (Ω)
3
2
2
From the obtained inequalities, summing over the mesh ωτ+ , denoting r + 2.5 = s
and using properties of multipliers in Sobolev spaces, we immediately obtain
q
′
(4.16)
kηii
kL2 (ω+ ,Ẅ 1/2 (γ − )) 6 Chs−2 log h1 kaii kW s−1 (Ω) kukW s,s/2 (Q) ,
τ
2
2
3−i,k
2
for 2.5 < s < 3 and
(4.17)
′
kηii
kL2 (ω+ ,Ẅ 1/2 (γ −
τ
2
))
3−i,k
6 Ch log h1
3/2
kaii kW22 (Ω) kukW 3,3/2 (Q) .
2
′
Analogous estimates hold for ηi,3−i
substituting aii with ai,3−i .
′
The term λi has a similar structure as ηij
. Handling it in the same manner we
obtain for 2.5 < s < 3
q
s−2
1 ai,3−i kλi kL2 (ω+ ,Ẅ 1/2 (γ − )) 6 Ch
log h kukW s,s/2 (Q) ,
τ
2
2
ik
aii M(W s−2 (Ω))
2
FINITE DIFFERENCE APPROXIMATION OF A PARABOLIC PROBLEM
61
where k · kM(W s−2 (Ω)) is the norm in the space of multipliers in W2s−2 (Ω) [10].
2
Further, for 2.5 < s < 3 and q > 2, using the properties of multipliers in Sobolev
spaces and the assumptions (2.4), we obtain
ai,3−i ai,3−i 1 (Ω) + kaii kW 1 (Ω)
6
C
6
C
ka
k
i,3−i
W
aii 1
aii q
q
M(W2s−2 (Ω))
Wq (Ω)
6 C kai,3−i kW s−1 (Ω) + kaii kW s−1 (Ω) .
2
2
In such a way, we finally obtain
q
(4.18) kλi kL2 (ω+ ,Ẅ 1/2 (γ − )) 6 Chs−2 log h1
τ
ik
2
× kaii kW s−1 (Ω) + kai,3−i kW s−1 (Ω) kukW s,s/2 (Q) ,
2
2
2.5 < s < 3
2
and analogously
3/2
(4.19) kλi kL2 (ω+ ,Ẅ 1/2 (γ − )) 6 Ch log h1
τ
2
ik
× kaii kW22 (Ω) + kai,3−i kW22 (Ω) kukW 3,3/2 (Q) ,
s = 3.
2
For r > 0.5 and aii , ai,3−i ∈ C(Ω̄) the term λ⋆i is a bounded linear functional
of u ∈ W2r (0, T ) which vanishes when u = 1. Using the Bramble–Hilbert lemma
and handling it in the same manner as in the case of ζi1 we obtain
kλ⋆i kτ 6 Chs−2 kai,3−i kW s−1 (Ω) kukW s,s/2(Q) ,
(4.20)
2
2 < s 6 3.
2
The terms νi and µi can be estimated directly:
α ∂u α
|[νi ]|σ̄ik 6 Ch 6 Ch kukW s,s/2(Q)
aii 2
a
∂t
ii
(4.21)
L2 (Σik )
C(Γik )
6 Ch kukW s,s/2 (Q) ,
s > 2.5
2
and
(4.22) |[µi ]|σ̄ik 6 Ch kaii kW s−1 (Ω) + kai,3−i kW s−1 (Ω) kukW s,s/2 (Q) ,
2
2
s > 2.5
2
From (4.2), (4.8), (4.9), (4.12) and (4.15)–(4.22) we obtain the following result.
Theorem 4.2. Let the assumptions from Section 2 hold and let τ ≍ h2 . Then
1,1/2
the FDS (3.1) converges in the norm W2
(Qhτ ) and the convergence rate estimates
q
|[z]|W 1,1/2 (Q ) 6 Chs−1 log h1
hτ
2
× 1 + max kaij kW s−1 (Ω) + max kαkW s−3/2 (Γ ) kukW s,s/2 (Q) ,
2.5 < s < 3
i,j
2
i,k
2
ik
2
and
|[z]|W 1,1/2 (Q
2
hold.
hτ )
6 Ch2 log h1
32 1+max kaij kW22 (Ω)+max kαkW 3/2 (Γ
i,j
i,k
2
ik )
kukW 3,3/2 (Q)
2
62
JOVANOVIĆ AND MILOVANOVIĆ
Remark 4.1. Note that the obtained error bounds are ‘almost’ compatible
with the smoothness of the data (up to additional logarithmic factors).
Remark 4.2. The obtained error bounds have been derived under the assumption τ ≍ h2 , which links the temporal mesh-size τ to the spatial mesh-size h,
despite the fact that the implicit FDS is unconditionally stable, and therefore from
the point of view of stability there should be no limitation on the choice of meshsizes. This limitation can be avoided in certain cases by careful study of truncation
error functionals (see e.g. [9]).
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University of Belgrade
Faculty of Mathematics
11000 Belgrade, Serbia
bosko@matf.bg.ac.rs
zorica.milovanovic@gmail.com
(Received 23 03 2013)
(Revised 01 04 2013)