MATEMATIQKI VESNIK
originalni nauqni rad
research paper
66, 4 (2014), 418–429
December 2014
ELLIPTIC TRANSMISSION PROBLEM IN DISJOINT DOMAINS
Zorica D. Milovanović
Abstract. In this paper, we investigate an elliptic transmission problem in disjoint domains.
A priori estimate for its weak solution in appropriate Sobolev-like space is proved. A finite
difference scheme approximating this problem is proposed and analyzed. An estimate of the
convergence rate, compatible with the smoothness of the input data (up to a slowly increasing
logarithmic factor of the mesh size), is obtained.
1. Introduction
In applications, especially in engineering, often are encountered composite or
layered structure, where the properties of individual layers can vary considerably
from the properties of the surrounding material. Layers can be structural, thermal,
electromagnetic or optical, etc. Mathematical models of energy and mass transfer in domains with layers lead to so called transmission problems. In this paper
we consider a class of non-standard elliptic transmission problems in disjoint domains [11]. As a model example it is taken an area consisting of two non-adjacent
rectangles. In each subarea was given a boundary problem of elliptic type, where
the interaction between their solutions is described by nonlocal integral conjugation
conditions [9].
2. Formulation of the problem
As a model example,we consider the following boundary-value problem (BVP):
Find functions u1 (x1 , x2 ) and u2 (x1 , x2 ) that satisfy the system of elliptic equations:
Lk uk = f k (x1 , x2 )x = (x1 , x2 ) ∈ Ωk
( k 3−k
r u
, x ∈ Γk1,3−k ,
k k
l u =
0,
x ∈ Γk \ Γk1,3−k ,
(1)
(2)
where k = 1, 2 and Ω1 = (a1 , b1 ) × (c, d), Ω2 = (a2 , b2 ) × (c, d), −∞ < a1 <
S2
b1 < a2 < b2 < +∞ and c < d. We denote Γk = ∂Ωk = i,j=1 Γkij , where
2010 Mathematics Subject Classification: 65N12, 65N15
Keywords and phrases: Elliptic transmission problem; disjoint domains; Sobolev-like space.
418
419
Elliptic transmission problem in disjoint domains
Γ111 = {x = (x1 , x2 ) ∈ Γ1 | x1 = a1 }, Γ112 = {x ∈ Γ1 | x1 = b1 }, Γk21 = {x ∈
Γk | x2 = c}, Γk22 = {x ∈ Γk | x2 = d}, Γ211 = {x = (x1 , x2 ) ∈ Γ2 | x1 = a2 },
Γ212 = {x ∈ Γ2 | x1 = b2 }, ∆k = Γk1,3−k × Γ3−k
1,k (k = 1, 2),
µ
¶
2
k
X
∂
k ∂u
pij
+ q k uk ,
L u := −
∂x
∂x
i
j
i,j=1
k k
lk uk :=
2
X
∂uk
cos (ν, xi ) + αk uk ,
∂xj
(4)
β k (x, x0 ) u3−k (x0 ) dΓ3−k ,
(5)
pkij
i,j=1
¡
k 3−k
r u
¢
Z
(x) :=
Γ3−k
1,k
(3)
and ν is the unit outward normal to Γk (k = 1, 2).
Notice that boundary condition (2) on Γk \ Γk1,3−k reduces to a standard conormal boundary condition while on Γk1,3−k it can be considered as a conjugation
condition of non-local Robin-Dirichlet type. For a special choice of αk and β k such
conjugation conditions describe linearized radiative heat transfer in a system of two
absolutely black bodies [2].
We assume that the standard conditions of regularity and ellipticity are satisfied:
pkij = pkji ∈ L∞ (Ωk ),
ck0
2
P
i=1
ξi2 ≤
2
P
i,j=1
q k ∈ L∞ (Ωk ),
pkij ξi ξj ≤ ck1
2
P
i=1
ξi2 ,
αk ∈ L∞ (Γk ),
0 ≤ q k (x),
β k ∈ L∞ (∆k ), (6)
∀x ∈ Ω̄k ,
∀ξ ∈ R2 .
(7)
By C, ci and cki we denote positive constants, independent of the solution of the
boundary-value problem and the mesh-sizes. In particular, C may take different
values in the different formulas.
3. Existence and uniqueness of weak solutions
We introduce the product space
L = L2 (Ω1 ) × L2 (Ω2 ) = {v = (v 1 , v 2 )|v k ∈ L2 (Ωk )},
endowed with the inner product and associated norm
(u, v)L = (u1 , v 1 )L2 (Ω1 ) + (u2 , v 2 )L2 (Ω2 ) ,
where
1/2
kvkL = (v, v)L ,
ZZ
(uk , v k )L2 (Ωk ) =
uk v k dxdy,
k = 1, 2.
Ωk
We also define the spaces
H s = {v = (v 1 , v 2 )| v k ∈ H s (Ωk )},
s = 1, 2, . . .
420
Z.D. Milovanović
endowed with the inner product and associated norm
1/2
(u, v)H s = (u1 , v 1 )H s (Ω1 ) + (u2 , v 2 )H s (Ω2 ) ,
kvkH s = (v, v)H s ,
where H s (Ωk ) are the standard Sobolev spaces [1]. Finally, with u = (u1 , u2 ) and
v = (v 1 , v 2 ) we define the following bilinear form:
à ZZ
Z
2
2
³X
´
k
k
X
k ∂u ∂v
k k k
pij
A(u, v) =
+ q u v dx1 dx2 +
αk uk v k dΓk
∂xj ∂xi
k
k
Ω
Γ
i,j=1
k=1
!
Z
Z
−
Γk
1,3−i
Γ3−k
1,i
β k u3−k v k dΓ3−k dΓk . (8)
Lemma 1. Under the conditions (6), the bilinear form A, defined by (8), is
bounded on H 1 ×H 1 . If in adition the conditions (7) are fulfilled, this form satisfies
the Gårding’s inequality on H 1 , i.e. there exist positive constants m and κ such that
A(u, u) + κkuk2L ≥ mkuk2H 1 ,
∀ u ∈ H 1.
If β k are sufficiently small and αk > 0 (k = 1, 2), then the bilinear form A is
coercive (i.e. κ = 0). Sufficient conditions are
p
2 α1 (x) α2 (x0 )
1
0
2 0
|β (x, x ) + β (x , x)| ≤
, ∀ x ∈ Γ112 , ∀ x0 ∈ Γ211 .
(9)
d−c
Proof. The proof is analogous to the proof of Lemma 3.8 in [7]. Boundedness
of A follows from (6) and the trace theorem for the Sobolev spaces
kuk kL2 (∂Ωk ) ≤ Ckuk kH 1 (Ωk ) .
Gårding’s inequality follows from (7), (8), multiplicative trace inequality (see, e.g.,
Proposition 1.6.3 in [3])
kuk k2L2 (∂Ωk ) ≤ Ckuk kL2 (Ωk ) kuk kH 1 (Ωk ) ,
Cauchy-Schwartz and ²-inequalities, for sufficiently small ² > 0.
Theorem 1. Let the assumptions (6), (7) and (9) hold. Then the BVP (1)–(5)
has a unique weak solution u ∈ H 1 (Ω), and it depends continuously on f .
Proof. The proof is an easy consequence of Lemma 1 and the Lax-Milgram
Lemma (see Theorems 17.9 and 17.10 in [15]).
4. Finite difference approximation
Let ω hk be a uniform mesh in [ak , bk ], with the step size hk = hk1 = (bk −
ak )/nk , k = 1, 2. We denote ωhk := ω hk ∩(ak , bk ). Analogously we define a uniform
mesh ω h3 in [c, d], with the step size h3 = hk2 = (d − c)/n3 (k = 1, 2) and its
421
Elliptic transmission problem in disjoint domains
submesh ωh3 = ω h3 ∩ (c, d). We assume that h1 ³ h2 ³ h3 ³ h = max{h1 , h2 , h3 }.
k
We also define the following meshes: ω k = ω hk × ω h3 , γ k = ω̄ k ∩ Γk , γ̄ij
= ω̄ k ∩ Γkij ,
k−
k+
k
k
k
=
γ1j = {x ∈ γ̄1j : c < x2 < d}, γ1j = {x ∈ γ̄1j : c ≤ x2 < d}, γ1j
k
k
k
k
k?
k
{x ∈ γ̄1j : c < x2 ≤ d}, γ1j = γ̄1j \ γ1j , γ2j = {x ∈ γ̄2j : ak < x1 < bk },
k−
k+
k
k
k
k
k?
,
γ2j
: ak ≤ x1 < bk }, γ2j
: ak < x1 ≤ bk }, γ2j
= γ̄2j
\ γ2j
= {x ∈ γ̄2j
= {x ∈ γ̄2j
©
ª
k
k
k
γ? = γ \ ∪i,j γij , i, j, k = 1, 2.
We shall consider vector-functions of the form v = (v 1 , v 2 ) where v k is a mesh
function defined on ω k , k = 1, 2.
The finite difference operators are defined in the usual manner [14]:
(v k )+i − v k
,
hki
vxki =
v k − (v k )−i
,
hki
vx̄ki =
where (v k )±i (x) = v k (x ± hki ei ), ei is the unit vector of the axis xi , i, k = 1, 2.
We define the following discrete inner products and norms:
[v k , wk ]k = hk h3
X
v k (x)wk (x) +
x∈ω k
[v k , wk )k,i = hk h3
x∈γ \γ?
X
k
x∈ω k ∪γi1
(v k , wk ]k,i = hk h3
hk h3 X k
hk h3 X k
v (x)wk (x) +
v (x)wk (x),
2
4
k
k
k
X
hk h3
v k (x)wk (x) +
2
X
hk h3
2
X
[v k , wk )k = hk h3
v k (x)wk (x),
k−
k−
x∈γ3−i,1
∪γ3−i,2
v k (x)wk (x) +
k
x∈ω k ∪γi2
|[v k ]|2k = [v k , v k ]k ,
x∈γ?
v k (x)wk (x),
k+
k+
x∈γ3−i,1
∪γ3−i,2
|[v k k2k,i = [v k , v k )k,i , kv k ]|2k,i = (v k , v k ]k,i ,
X
v k (x)wk (x), |[v k k2k = [v k , v k )k ,
k−
k−
∪γ21
x∈ω k ∪γ11
X
(v k , wk ]k = hk h3
v k (x)wk (x),
kv k ]|2k = (v k , v k ]k ,
k+
k+
x∈ω k ∪γ12
∪γ22
|[v k ]|2H 1 (ω̄k ) = |[v k ]|2k + |[vxk1 ||2k,1 + |[vxk2 k2k,2 ,
[v k , wk ]γ̄ij
k = hk,3−i
X
v k (x)wk (x) +
k
x∈γij
(v k , wk )γij
k = hk,3−i
X
hk,3−i
2
X
X
ij
ij
k?
x∈γij
v k (x)wk (x),
kv k k2γ k = (v k , v k )γij
k ,
v k (x)wk (x),
|[v k ||2γ k− = [v k , v k )γ k− ,
ij
ij
k−
x∈γij
|v k |2H 1/2 (γ k− ) =
ij
x∈ω̄ k
v k (x)wk (x), |[v k ]|2γ̄ k = [v k , v k ]γ̄ij
k ,
k
x∈γij
[v k , wk )γ k− = hk,3−i
|[v k ]|C(ω̄k ) = max |v k (x)|,
h2k,3−i
X
k−
x, x0 ∈γij
, x0 6=x
·
¸2
v k (x) − v k (x0 )
,
x3−i − x03−i
ij
422
Z.D. Milovanović
|[v k k2H 1/2 (γ k− ) = |v k |2H 1/2 (γ k− ) + |[v k k2γ k− ,
ij
|[v k k2Ḧ 1/2 (γ k− )
ij
hk,3−i
X µ
k−
x∈γij
ij
=
ij
|v k |2H 1/2 (γ k− ) +
ij
¶
1
1
+
|v k (x)|2 ,
x3−i + hk,3−i /2 lki − x3−i − hk,3−i /2
where lk1 = d − c and lk2 = bk − ak .
For v = (v 1 , v 2 ) and w = (w1 , w2 ) we denote
[v, w] = [v 1 , w1 ]1 + [v 2 , w2 ]2 ,
|[v]|2 = [v, v],
|[v]|2H 1 = |[v 1 ]|2H 1 (ω̄1 ) + |[v 2 ]|2H 1 (ω̄2 ) ,
h
We also define the Steklov smoothing operators (see [4]):
Z 1
+ k
− k
Tki
f (x) =
f k (x + hki x0i ei ) dx0i = Tki
f (x + hki ei ) = Tki f k (x + 0.5hki ei ),
0
Z
2± k
Tki
f (x) = 2
0
1
(1 − x0i )f k (x ± hki x0i ei ) dx0i ,
k, i = 1, 2.
These operators commute and transform derivatives into differences, for example:
µ k¶
µ k¶
µ 2 k¶
∂u
∂u
∂ u
+
−
2
Tki
= ukxi , Tki
= ukx̄i , Tki
= ukx̄i xi .
∂xi
∂xi
∂x2i
We approximate the boundary-value problem (1)–(5) with the following finite
difference scheme:
x ∈ ω̄ k
(10)
Lkh v = f˜k ,
where L1h v is equal to:
−
2
¡
¢ i
1 X h¡ 1 1 ¢
pij vxj x̄ + p1ij vx̄1j x + q̃ 1 v 1 ,
i
i
2 i,j=1
x ∈ ω1
i ¡
¢
v 1 + vx̄12
2 h p111 + (p111 )+1 1
−
vx1 − p112 x2
+ α̃1 v 1 − p112 vx̄12 x1
h1
2
2
¡ 1 1 ¢
1¡ 1 1 ¢
1¡ 1 1 ¢
1
− p21 vx1 x̄2 −
p v
−
p v
+ q̃ 1 v 1 ,
x ∈ γ11
2 22 x2 x̄2 2 22 x̄2 x2
i
2 h p111 + (p111 )+1 1
−
vx1 − p112 vx12 + α̃11 v 1
h1
2
i
2h
p1 + (p122 )+2 1
+
x = (a1 , c)
− p121 vx11 − 22
vx2 + α̃21 v 1 + q̃ 1 v 1 ,
h3
2
i
¡
¢
2 h p111 + (p111 )+1 1
−
vx1 − p112 vx̄12 + α̃11 v 1 − 2 p112 vx̄12 x1
h1
2
i
¡
¢
2h 1
p1 + (p122 )−2 1
+
p21 vx1 + 22
vx̄2 + α̃21 v 1 − 2 p121 vx11 x̄2 + q̃ 1 v 1 , , x = (a1 , d)
h3
2
423
Elliptic transmission problem in disjoint domains
v 1 + vx̄12
2 h p111 + (p111 )−1 1
vx̄1 + p112 x2
+ α̃1 v 1
h1
2
2
i ¡
¢
2
− [β̃ 1 (x, ·), v 2 (·)]γ̄11
− p112 vx12 x̄1
¡
¢
1¡ 1 1 ¢
1¡ 1 1 ¢
− p121 vx̄11 x2 −
p22 vx2 x̄2 −
p v
+ q̃ 1 v 1 ,
2
2 22 x̄2 x2
i
2 h p111 + (p111 )−1 1
2
vx̄1 + p112 vx12 + α̃11 v 1 − [β̃ 1 (x, ·), v 2 (·)]γ̄11
h1
2
i
2h
p1 + (p122 )+2 1
+
vx2 + α̃21 v 1
− p121 vx̄1 − 22
h3
2
¡
¢
¡
¢
− 2 p112 vx12 x̄ − 2 p121 vx̄11 x + q̃ 1 v 1 ,
1
2
i
2 h p111 + (p111 )−1 1
1 1
2
vx̄1 + p12 vx̄2 + α̃11 v 1 − [β̃ 1 (x, ·), v 2 (·)]γ̄11
h1
2
i
p1 + (p122 )−2 1
2h 1 1
+
p21 vx̄1 + 22
vx̄2 + α̃21 v 1 + q̃ 1 v 1 ,
h3
2
and analogously at the other boundary nodes,
L2h v is defined in an analogous manner,
 2 2 k

 Tk1 Tk2 f ,
2± 2
k
Tki
Tk,3−i f k ,
f˜ =

 2± 2± k
T Tk2 f ,
 k1
2
2 k

 Tk1 Tk2 q ,
2± 2
Tki
Tk,3−i q k ,
q̃ k =

 2± 2± k
Tk1 Tk2 q ,
2
αk ,
α̃k = Tk,3−i
α̃ik
=
2±
Tk,3−i
αk ,
and
(
β̃ k =
2 k
Tk2
β ,
x = (b1 , c)
x = (b1 , d)
1
1
x ∈ γ21
∪ γ22
x ∈ ωk
k
k
/ x ∈ γi2
x ∈ γi1
x ∈ γ?k
x ∈ ωk
k
k
/ x ∈ γi2
x ∈ γi1
x ∈ γ?k
k
k
,
∪ γi2
x ∈ γi1
x∈
1
x ∈ γ12
γ?k ,
i = 1, 2,
i = 1, 2
k
x ∈ γ1,3−k
,
2± k
k?
Tk2
β , x ∈ γ1,3−k
.
The FDS (10) may be compactly presented as operator-difference scheme
Lh v = f˜,
(11)
where v = (v 1 , v 2 ), f˜ = (f˜1 , f˜2 ) and Lh v = (L1h v, L2h v).
In the sequel we shall assume that the generalized solution of the problem (1)–
(5) belongs to the Sobolev space H s , 2 < s ≤ 3, while the data satisfy the following
smoothness conditions:
pkij ∈ H s−1 (Ωk ),
αk ∈ H s−3/2 (Γkij ),
f k ∈ H s−2 (Ωk ),
αk ∈ C(Γk ),
q k ∈ H s−2 (Ωk )
β k ∈ H s−1 (∆k ),
k, i, j = 1, 2.
(12)
424
Z.D. Milovanović
Let us consider the bilinear form Ah (v, w) associated with difference operator Lh v:
Ah (v, w) = [Lh v, w] =
2 ½ X
2 h
X
1
k=1
2
[pkii vxki , wxki )k,i + (pkii vx̄ki , wx̄ki ]k,i
i=1
2
i
X
X
+ [pki,3−i vxk3−i , wxki )k + (pki,3−i vx̄k3−i , wx̄ki ]k + [q̃ k v k , wk ]k +
hk,3−i α̃k v k wk
i,j=1 x∈γ k
ij
1 X
(hk2 α̃1k + hk1 α̃2k ) v k wk − h2k2
+
2
k
x∈γ?
−
h2k2
2
X
X
β k (x, x0 )v 3−k (x0 )wk (x)
k
3−k
x∈γ1,3−k
x0 ∈γ1,k
X
X
¾
β k (x, x0 )v 3−k (x0 )wk (x)
k?
(3−k)?
x∈γ1,3−k
x0 ∈γ1,k
Lemma 2. Let pkij , αk > 0 and β k satisfy the assumptions (12), q k satisfy
assumption (7) and let conditions (9) be fulfilled. Then, for a sufficiently small
mesh step h, there exist positive constants c2 and c3 such that
c2 |[v]|2H 1 ≤ Ah (v, v) = [Lh v, v] ≤ c3 |[v]|2H 1 .
h
h
The proof is analogous to the proof of Lemma 1.
5. Convergence of the finite difference scheme
Let u = (u1 , u2 ) be the solution of the BVP (1)–(5), and let v = (v 1 , v 2 ) denote
the solution of the FDS (10). The error z = (z 1 , z 2 ) = u − v satisfies the following
conditions
x ∈ ω̄ k ,
(13)
Lkh z = ψ k ,
where ψ 1 is equal to:
2
X
1
1
ηij,
x̄i + µ ,
x ∈ ω1 ,
i,j=1
2 1
2 1
2 1
1
1
1
η +
η + η̃21,
ζ + µ̃1 ,
x ∈ γ11
,
x̄2 + η̃22, x̄2 +
h1 11 h1 12
h1
¢
¢
2 ¡ 1
2 ¡ 3
1
1
˜1,
η̃11 + η̃12
+ ζ11 + ζ21 +
η̃21 + η̃22
+ ζ21 + µ̃
x = (a1 , c),
h1
h3
2 1 −1 2 1 −1
2 1 2 1
1
1
1
−
(η ) −
(η ) + η̃21,
ζ +
χ + µ̃1 , x ∈ γ12
,
x̄2 + η̃22, x̄2 +
h1 11
h1 12
h1
h1
¤
¢
2 £
2 ¡ 1
1 −1
1 −1
1
˜1,
−(η̃11
) − (η̃12
) + ζ11 + χ1 +
η̃ + η̃22
+ ζ21 + µ̃
x = (b1 , c),
h1
h3 21
and analogously at the other boundary nodes.
Elliptic transmission problem in disjoint domains
425
ψ 2 is defined in an analogous manner,
¶
µ
i
k
1 h k
+ 2
k
k ∂u
ηij = Tki Tk,3−i pij
−
pij uxj + (pkij )+i (ukx̄j )+i , x ∈ ω k ,
∂xj
2
¶
µ
k
∂u
pk + (pkii )+i k
+ 2±
k−
k−
k
η̃ii
= Tki
Tk,3−i pkii
− ii
uxi , x ∈ γ3−i,1
/ x ∈ γ3−i,2
,
∂xi
2
¶
µ

k−
+ 2+
∂uk
k
k
k


x ∈ γ3−i,1
,
 Tki Tk,3−i pi,3−i ∂x3−i − pi,3−i ux3−i ,
k
¶
µ
η̃i,3−i =


k−
+ 2−
∂uk
 Tki
− (pki,3−i )+i (ukx̄3−i )+i , x ∈ γ3−i,2
,
Tk,3−i pki,3−i ∂x
3−i
2 k k
2
ζ k = (Tki
α )u − Tki
(αk uk ),
2± k k
2±
ζik = (Tki
α )u − Tki
(αk uk ),
k
k
∪ γ3−i,2
,
x ∈ γ3−i,1
x ∈ γ?k ,
2 2
2 2
µk = (Tki
Tk,3−i q k )uk − Tki
Tk,3−i (q k uk ),
x ∈ ωk ,
2±
2±
2 k k
2
q )u − Tk,3−i
(q k uk ),
µ̃k = (Tk,3−i
Tki
Tki
k−
k−
x ∈ γ3−i,1
/ x ∈ γ3−i,2
,
˜ k = (T 2± T 2± q k )uk − T 2± T 2± (q k uk ), x ∈ γ?k ,
µ̃
k,3−i ki
k,3−i ki
Z
X
2 k
2 k
χk =
Tk2
β (x, x0 )u3−k (x0 ) dΓ3−k
Tk2
β (x, x0 )u3−k (x0 )
1k − h3
Γ3−k
1k
−
h3
2
Z
χk =
Γ3−k
1k
−
h3
2
X
3−k
x0 ∈γ̄1k
2 k
β (x, x0 )u3−k (x0 ),
Tk2
k
,
x ∈ γ1,3−k
3−k
x0 ∈γ̄1k
X
2± k
Tk2
β (x, x0 )u3−k (x0 ) dΓ3−k
1k − h3
X
2± k
Tk2
β (x, x0 )u3−k (x0 )
3−k
x0 ∈γ̄1k
2± k
Tk2
β (x, x0 )u3−k (x0 ),
k?
x ∈ γ1,3−k
.
3−k
x0 ∈γ̄1k
We shall prove a suitable a priori estimate for the FDS (13). For this purpose
we need the following auxiliary results:
Lemma 3. [6] The following inequality holds true:
¯
¯
¯ k k
¯
¯ [v , wx3−i )γ k− ¯ ≤ C|[v k ]|Ḧ 1/2 (γ k− ) |[wk ]|H 1 (ω̄k ) .
ij
ij
Lemma 4. [6] Let v k be a mesh function on ω̄ k , then
r
1
k
|[v ]|C(ω̄k ) ≤ C log |[v k ]|H 1 (ω̄k ) .
h
426
Z.D. Milovanović
Let us rearrange the summands in truncation error ψ in the following manner:
k
k
k
η̃ij
= ηij
+ η̄ij
,
µ̃k = µk + µ?k ,
˜ k = µk + µ??k
µ̃
where
µ
µ
¶¶
∂uk
hki +
∂
k
k
pkii
, x ∈ γ3−i,1
Ti
/ x ∈ γ3−i,2
3
∂x3−i
∂xi
µ
µ
¶¶
µ
¶
∂
∂uk
hki + k
∂ 2 uk
hki +
Ti
pki,3−i
∓
Ti pi,3−i 2
=±
3
∂x3−i
∂x3−i
2
∂x3−i
µ
µ
¶¶
k
hki + ∂
∂u
k
k
+
Ti
pki,3−i
,
x ∈ γ3−i,1
/ x ∈ γ3−i,2
,
2
∂xi
∂x3−i
hki ³ 2± 2 k ´³ 2 ∂uk ´
k
k
=±
T3−i Ti q
Ti
, x ∈ γ3−i,1
/ x ∈ γ3−i,2
3
∂x3−i
hki ³ 2± 2± k ´³ 2± ∂uk ´ hki ³ 2± 2± k ´³ 2± ∂uk ´
=±
Ti T3−i q
T3−i
±
T3−i Ti q
Ti
, x ∈ γ?k ,
3
∂xi
3
∂x3−i
k
η̄ii
=±
k
η̄i,3−i
µ?k
µ??k
From Lemmas 2–4 one obtains the next assertion.
Theorem 2. The finite difference scheme (13) is stable in the sense of the a
priori estimate
( 2
2
´
X
X³
k
?k
|[z]|Hh1 ≤ C
|[ηij
kk,i + kζ k kγij
kγij
+ |[µk ]|k + |[χk ]|γ̄ k
k + h kµ
k
k=1
1,3−k
i,j=1
r
¶)
2
2
X µX
X
1
k
. (14)
|ζik | + h |µ??k |
+h
|[η̄ij
kḦ 1/2 (γ k− ) + h log
3−i,l
h
k
i=1
i,j,l=1
x∈γ?
Theorem 3. Let the assumptions of Lemma 2 hold. Then the solution of FDS
(10) converges to the solution of BVP (1)–(5) and the convergence rate estimates
r
1³
s−1
|[u − v]|Hh1 ≤ Ch
log
1 + max kpkij kH s−1 (Ωk ) + max kq k kH s−2 (Ωk )
k
i,j,k
h
´
k
k
+ max kα kH s−3/2 (Γkij ) + max kβ kH s−1 (∆k ) kukH s ,
2.5 < s < 3
i,j,k
k
and
µ
|[u − v]|Hh1 ≤ Ch
2
1
log
h
¶3/2 ³
1 + max kpkij kH 2 (Ωk ) + + max kq k kH 1 (Ωk )
i,j,k
k
´
s=3
max kαk kH 3/2 (Γkij ) + max kβ k kH 2 (∆k ) kukH 3 ,
i,j,k
hold.
k
427
Elliptic transmission problem in disjoint domains
k
Proof. The terms ηij
and µk at the internal nodes of the mesh ω̄ k can be
estimated in the same manner as in the case of the Dirichlet BVP [4, 7]:
X ¡ ¢2
k
hk h3
ηij
≤ Ch2s−2 kpkij k2H s−1 (Ωk ) kuk k2H s (Ωk ) , 2 < s ≤ 3.
k
x∈ω k ∪γi1
Analogous result at the boundary nodes is obtained in [5]:
X
¡ k ¢2
hk h3
ηij ≤ Ch2s−2 kpkij k2H s−1 (Ωk ) kuk k2H s (Ωk ) ,
2.5 < s ≤ 3.
k−
k−
x∈γ3−i,1
∪γ3−i,2
From these inequalities, it follows
k
|[ηij
kk,i ≤ Chs−1 kpkij kH s−1 (Ωk ) kuk kH s (Ωk ) ,
2.5 < s ≤ 3.
(15)
Analogously,
|[µk ]|k ≤ Chs−1 kq k kH s−2 (Ωk ) kuk kH s (Ωk ) ,
2.5 < s ≤ 3.
(16)
s > 2.5,
(17)
Terms µ?k and µ??k can be estimated directly:
k
k
kµ?k kγij
k ≤ Ch kq kH s−2 (Ωk ) ku kH s (Ωk ) ,
|µ??k (x)| ≤ C kq k kH s−2 (Ωk ) kukH s (Ωk ) ,
x ∈ γ?k ,
s > 2.
(18)
k
Terms analogous to ζ k , ζik , η̄ij
, are estimated in [5] whereby it follows that:
s−1
kζ k kγij
kαk kH s−3/2 (Γkij ) kuk kH s (Ωk ) ,
k ≤ Ch
2.5 < s ≤ 3,
(19)
kζik kγ?k ≤ Ch kαk kH s−3/2 (Γk3−i,j ) kukH s (Ωk ) ,
s > 2,
(20)
r
1
k
kη̄ij
kḦ 1/2 (γ k− ) ≤ Chs−2 log kpkij kH s−1 (Ωk ) kuk kH s (Ωk ) , 2.5 < s < 3, (21)
3−i,l
h
The term χk can be estimated in the following manner: let us denote
Z h
h
I1 = I1 (g) =
g(x) dx − [g(0) + g(h)].
2
0
For r > 0.5 I1 (g) is a bounded linear functional of g ∈ H r (0, h) which vanishes
when g(x) = 1 and g(x) = x. Using the Bramble-Hilbert lemma [4] we obtain
|I1 | ≤ Chr+1/2 |g|H r (0,h) ,
0.5 < r ≤ 2,
whereby it follows that
¯Z
·
¸¯
n−1
¯ 1
g(0) X
g(1) ¯¯
¯
g(x) dx − h
+
g(ih) +
¯
¯ ≤ Chr |g|H r (0,1) ,
¯ 0
2
2 ¯
i=1
0.5 < r ≤ 2.
428
Z.D. Milovanović
From this inequality, using properties of multipliers in Sobolev spaces [13], we
immediately obtain
° 2 k
°
|χk (x)| ≤ Chr °Tk2
β (x, ·) u3−k (·)°H r (Γ3−k )
1k
°
°
°
°
r ° 2 k
°
1<r≤2
≤ Ch Tk2 β (x, ·) H r (Γ3−k ) °u3−k (·)°H r (Γ3−k ) ,
1k
1k
k
k?
when x ∈ γ1,3−k
, while for x ∈ γ1,3−k
the analogous inequalities hold. After
k
summation over the mesh γ̄1,3−k we obtain
° °
°
°
|[χk ]|γ̄ k
≤ Chr °β k °H r (∆k ) °u3−k °H r (Γ3−k ) ,
1 < r ≤ 2.
1,3−k
1k
Finally, using the trace theorem for anisotropic Sobolev spaces [12] and denoting
r = s − 1, we get
° °
°
°
|[χk ]|γ̄ k
≤ Chs−1 °β k °H s−1 (∆k ) °u3−k °H s (Ω3−k ) , 2 < s ≤ 3.
(22)
1,3−k
The assertion follows from (13)–(22).
Remark 1. Convergence rate estimates of the form
ku − vkH m (Ωh ) ≤ Chs−m kukH s (Ω) ,
m<s≤m+r
are often called “compatible with the smoothness of the solution” (se e.g. [4]).
Here u is the solution of boundary value problem (defined in the domain Ω), v
is the solution of the corresponding finite difference scheme (defined on the mesh
Ωh ⊂ Ω), h is discretization parameter (mesh size), r is a given constant (the
highest possible order of convergence), H s (Ω) is Sobolev space and H m (Ωh ) is
Sobolev space of mesh-functions. In such a manner, error bounds obtained in
Theorem 3 are compatible with the smoothness of the solution, up to a slowly
increasing logarithmic factor of the mesh size.
Acknowledgement. The research was supported by Ministry of Education
and Science of Republic of Serbia under project 174015.
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(received 11.07.2013; in revised form 26.09.2013; available online 15.11.2013)
University of Belgrade, Faculty of Mathematics, Studentski trg 16, 11000 Beograd, Serbia
E-mail: zorica.milovanovic@gmail.com