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

66, 4 (2014), 418–429 December 2014

research paper

ELLIPTIC TRANSMISSION PROBLEM IN DISJOINT DOMAINS Zorica D. Milovanovi´c

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 trans- fer 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 do- mains [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 functionsu¹(x1, x2) andu²(x1, x2) that satisfy the system of elliptic equations:

L^ku^k =f^k(x1, x2)x= (x1, x2)∈Ω^k (1) l^ku^k =

(r^ku^3−k, x∈Γ^k_1,3−k,

0, x∈Γ^k\Γ^k_1,3−k, (2)

where k = 1,2 and Ω¹ = (a1, b1)×(c, d), Ω² = (a2, b2)×(c, d), −∞ < a1 <

b1 < a2 < b2 < +∞ and c < d. We denote Γ^k = ∂Ω^k = S₂

i,j=1Γ^k_ij, where

2010 Mathematics Subject Classification: 65N12, 65N15

Keywords and phrases: Elliptic transmission problem; disjoint domains; Sobolev-like space.

418

Γ¹₁₁ = {x = (x1, x2) ∈ Γ¹|x1 = a1}, Γ¹₁₂ = {x ∈ Γ¹|x1 = b1}, Γ^k₂₁ = {x ∈ Γ^k|x2 = c}, Γ^k₂₂ = {x ∈ Γ^k|x2 = d}, Γ²₁₁ = {x = (x1, x2) ∈ Γ²|x1 = a2}, Γ²₁₂={x∈Γ²|x₁=b₂}, ∆^k= Γ^k_1,3−k×Γ^3−k_1,k (k= 1,2),

L^ku^k :=− X2

i,j=1

∂

∂xi

µ p^k_ij ∂u^k

∂xj

¶

+q^ku^k, (3)

l^ku^k :=

X2

i,j=1

p^k_ij ∂u^k

∂xj cos (ν, xi) +α^ku^k, (4)

¡r^ku^3−k¢ (x) :=

Z

Γ^3−k_1,k

β^k(x, x⁰)u^3−k(x⁰) dΓ^3−k, (5) andν is the unit outward normal to Γ^k (k= 1,2).

Notice that boundary condition (2) on Γ^k\Γ^k_1,3−k reduces to a standard co- normal boundary condition while on Γ^k_1,3−k it can be considered as a conjugation condition of non-local Robin-Dirichlet type. For a special choice ofα^k andβ^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 satis- fied:

p^k_ij =p^k_ji∈L^∞(Ω^k), q^k ∈L^∞(Ω^k), α^k∈L^∞(Γ^k), β^k ∈L^∞(∆^k), (6) c^k₀ P²

i=1

ξ_i²≤ P²

i,j=1

p^k_ijξiξj ≤c^k₁ P²

i=1

ξ_i², 0≤q^k(x), ∀x∈Ω¯^k, ∀ξ∈R². (7) By C, ci and c^k_i 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= (v¹, v²)|v^k ∈L2(Ω^k)}, endowed with the inner product and associated norm

(u, v)L = (u¹, v¹)_L2(Ω¹₎+ (u², v²)_L2(Ω²₎, kvkL= (v, v)^1/2_L , where

(u^k, v^k)_L2(Ω^k₎= Z Z

Ω^k

u^kv^kdxdy, k= 1,2.

We also define the spaces

H^s={v= (v¹, v²)|v^k ∈H^s(Ω^k)}, s= 1,2, . . .

endowed with the inner product and associated norm

(u, v)_H^s = (u¹, v¹)_Hs(Ω¹)+ (u², v²)_Hs(Ω²), kvk_H^s = (v, v)^1/2_Hs,

whereH^s(Ω^k) are the standard Sobolev spaces [1]. Finally, with u= (u¹, u²) and v= (v¹, v²) we define the following bilinear form:

A(u, v) = X2

k=1

à ZZ

Ω^k

³ X²

i,j=1

p^k_ij∂u^k

∂xj

∂v^k

∂xi +q^ku^kv^k

´

dx1dx2+ Z

Γ^k

α^ku^kv^kdΓ^k

− Z

Γ^k_1,3−i

Z

Γ^3−k_1,i

β^ku^3−kv^kdΓ^3−kdΓ^k

! . (8) Lemma 1. Under the conditions(6), the bilinear form A, defined by (8), is bounded onH¹×H¹. If in adition the conditions(7)are fulfilled, this form satisfies the G˚arding’s inequality onH¹, i.e. there exist positive constantsmandκsuch that

A(u, u) +κkuk²_L≥mkuk²_H1, ∀u∈H¹.

If β^k are sufficiently small and α^k > 0 (k = 1,2), then the bilinear form A is coercive (i.e.κ= 0). Sufficient conditions are

|β¹(x, x⁰) +β²(x⁰, x)| ≤ 2p

α¹(x)α²(x⁰)

d−c , ∀x∈Γ¹₁₂, ∀x⁰∈Γ²₁₁. (9) Proof. The proof is analogous to the proof of Lemma 3.8 in [7]. Boundedness ofAfollows from (6) and the trace theorem for the Sobolev spaces

ku^kk_L2(∂Ω^k₎≤Cku^kk_H¹_(Ω^k₎.

G˚arding’s inequality follows from (7), (8), multiplicative trace inequality (see, e.g., Proposition 1.6.3 in [3])

ku^kk²_L2(∂Ωk)≤Cku^kk_L2(Ω^k₎ku^kk_H¹_(Ω^k₎, Cauchy-Schwartz and²-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 solutionu∈H¹(Ω), and it depends continuously onf.

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 ωhk be a uniform mesh in [ak, bk], with the step size hk =hk1 = (bk − ak)/nk,k= 1,2. We denoteωhk:=ωhk∩(ak, bk). Analogously we define a uniform mesh ωh3 in [c, d], with the step size h3 = hk2 = (d−c)/n3 (k = 1,2) and its

submeshωh3=ωh3∩(c, d). We assume thath1³h2³h3³h= max{h1, h2, h3}.

We also define the following meshes: ω^k =ωhk×ωh3,γ^k = ¯ω^k∩Γ^k, ¯γ_ij^k = ¯ω^k∩Γ^k_ij, γ_1j^k = {x ∈ γ¯_1j^k : c < x2 < d}, γ_1j^k− = {x ∈ γ¯_1j^k : c ≤ x2 < d}, γ_1j^k+ = {x ∈ ¯γ^k_1j : c < x2 ≤ d}, γ_1j^k? = ¯γ_1j^k \γ_1j^k, γ_2j^k = {x ∈ ¯γ^k_2j : ak < x1 < bk}, γ_2j^k−={x∈γ¯_2j^k : ak ≤x1< bk},γ_2j^k+={x∈γ¯_2j^k : ak< x1≤bk},γ_2j^k?= ¯γ_2j^k \γ_2j^k, γ_?^k=γ^k\©

∪i,jγ_ij^kª

, i, j, k= 1,2.

We shall consider vector-functions of the formv= (v¹, v²) wherev^k is a mesh function defined onω^k,k= 1,2.

The finite difference operators are defined in the usual manner [14]:

v^k_xi = (v^k)⁺ⁱ−v^k hki

, v^k_¯xi =v^k−(v^k)⁻ⁱ hki

,

where (v^k)^±i(x) =v^k(x±hkiei), ei is the unit vector of the axisxi, i, k= 1,2.

We define the following discrete inner products and norms:

[v^k, w^k]k =hkh3

X

x∈ω^k

v^k(x)w^k(x) +hkh3

2 X

x∈γ^k\γ^k_?

v^k(x)w^k(x) +hkh3

4 X

x∈γ^k_?

v^k(x)w^k(x), [v^k, w^k)_k,i=h_kh₃ X

x∈ω^k∪γ_i1^k

v^k(x)w^k(x) +hkh3

2

X

x∈γ^k−_3−i,1∪γ_3−i,2^k−

v^k(x)w^k(x),

(v^k, w^k]k,i=hkh3

X

x∈ω^k∪γ_i2^k

v^k(x)w^k(x) +hkh3

2

X

x∈γ^k+_3−i,1∪γ_3−i,2^k+

v^k(x)w^k(x),

|[v^k]|²_k= [v^k, v^k]k, |[v^kk²_k,i = [v^k, v^k)k,i, kv^k]|²_k,i= (v^k, v^k]k,i, [v^k, w^k)k=hkh3

X

x∈ω^k∪γ₁₁^k−∪γ₂₁^k−

v^k(x)w^k(x), |[v^kk²_k = [v^k, v^k)k, (v^k, w^k]_k =h_kh₃ X

x∈ω^k∪γ^k+₁₂∪γ₂₂^k+

v^k(x)w^k(x), kv^k]|²_k= (v^k, v^k]_k,

|[v^k]|²_H1(¯ω^k)=|[v^k]|²_k+|[v_x^k₁||²_k,1+|[v_x^k₂k²_k,2, |[v^k]|_C(¯ω^k₎= max

x∈¯ω^k|v^k(x)|, [v^k, w^k]_γ¯k

ij=hk,3−i

X

x∈γ_ij^k

v^k(x)w^k(x) +hk,3−i

2 X

x∈γ_ij^k?

v^k(x)w^k(x), |[v^k]|²_¯γk

ij= [v^k, v^k]_¯γk ij, (v^k, w^k)_γ^k

ij=hk,3−i

X

x∈γ_ij^k

v^k(x)w^k(x), kv^kk²_γk

ij= (v^k, v^k)_γ^k

ij, [v^k, w^k)_γ^k−

ij =hk,3−i

X

x∈γ_ij^k−

v^k(x)w^k(x), |[v^k||²_γk−

ij = [v^k, v^k)_γ^k−

ij ,

|v^k|²_H1/2(γk−

ij )=h²_k,3−i X

x, x⁰∈γ_ij^k−, x⁰6=x

·v^k(x)−v^k(x⁰) x3−i−x⁰_3−i

¸₂ ,

|[v^kk²_H1/2(γ_ij^k−)=|v^k|²_H1/2(γ_ij^k−)+|[v^kk²_γk−

ij ,

|[v^kk²_H¨1/2(γ^k−_ij )=|v^k|²_H1/2(γ^k−_ij )+ hk,3−i

X

x∈γ^k−_ij

µ 1

x3−i+hk,3−i/2 + 1

lki−x3−i−hk,3−i/2

¶

|v^k(x)|²,

wherelk1=d−candlk2=bk−ak.

Forv= (v¹, v²) andw= (w¹, w²) we denote [v, w] = [v¹, w¹]1+ [v², w²]2, |[v]|²= [v, v], |[v]|²_H1

h=|[v¹]|²_H1(¯ω¹)+|[v²]|²_H1(¯ω²), We also define the Steklov smoothing operators (see [4]):

T_ki⁺f^k(x) = Z ₁

0

f^k(x+hkix⁰_iei) dx⁰_i =T_ki⁻f^k(x+hkiei) =Tkif^k(x+ 0.5hkiei), T_ki^2±f^k(x) = 2

Z ₁

0

(1−x⁰_i)f^k(x±hkix⁰_iei) dx⁰_i, k, i= 1,2.

These operators commute and transform derivatives into differences, for example:

T_ki⁺ µ∂u^k

∂xi

¶

=u^k_xi, T_ki⁻ µ∂u^k

∂xi

¶

=u^k_¯xi, T_ki² µ∂²u^k

∂x²_i

¶

=u^k_x¯ixi.

We approximate the boundary-value problem (1)–(5) with the following finite difference scheme:

L^k_hv= ˜f^k, x∈ω¯^k (10) whereL¹_hv is equal to:

−1 2

X2

i,j=1

h¡p¹_ijv_x¹_j¢

¯ xi+¡

p¹_ijv¹_¯xj¢

xi

i

+ ˜q¹v¹, x∈ω¹

2 h1

h

−p¹₁₁+ (p¹₁₁)⁺¹

2 v_x¹₁−p¹₁₂v_x¹₂+v¹_x¯2

2 + ˜α¹v¹i

−¡

p¹₁₂v_x¹_¯2¢

x1

−¡ p¹₂₁v_x¹₁¢

¯ x2−1

2

¡p¹₂₂v¹_x2¢

¯ x2−1

2

¡p¹₂₂v¹_¯x2¢

x2+ ˜q¹v¹, x∈γ¹₁₁ 2

h1

h

−p¹₁₁+ (p¹₁₁)⁺¹

2 v_x¹₁−p¹₁₂v_x¹₂+ ˜α¹₁v¹ i

+ 2 h3

h

−p¹₂₁v¹_x1−p¹₂₂+ (p¹₂₂)⁺²

2 v¹_x2+ ˜α¹₂v¹ i

+ ˜q¹v¹, x= (a1, c) 2

h1

h

−p¹₁₁+ (p¹₁₁)⁺¹

2 v_x¹₁−p¹₁₂v_x¹_¯2+ ˜α¹₁v¹i

−2¡ p¹₁₂v¹_¯x2¢

x1

+ 2 h3

h

p¹₂₁vx1+p¹₂₂+ (p¹₂₂)⁻²

2 v_x¹_¯2+ ˜α₂¹v¹ i

−2¡ p¹₂₁v¹_x1¢

¯

x2+ ˜q¹v¹, , x= (a1, d)

2 h1

hp¹₁₁+ (p¹₁₁)⁻¹

2 v¹_¯x1+p¹₁₂v¹_x2+v¹_¯x2

2 + ˜α¹v¹

−[ ˜β¹(x,·), v²(·)]_¯γ²

11

i

−¡ p¹₁₂v¹_x2¢

¯ x1

−¡ p¹₂₁v_x¹_¯1¢

x2−1 2

¡p¹₂₂v¹_x2¢

¯ x2−1

2

¡p¹₂₂v¹_¯x2¢

x2+ ˜q¹v¹, x∈γ¹₁₂ 2

h1

hp¹₁₁+ (p¹₁₁)⁻¹

2 v¹_¯x1+p¹₁₂v¹_x2+ ˜α¹₁v¹−[ ˜β¹(x,·), v²(·)]_γ¯2 11

i

+ 2 h₃

h

−p¹₂₁vx¯1−p¹₂₂+ (p¹₂₂)⁺²

2 v_x¹₂+ ˜α¹₂v¹i

−2¡ p¹₁₂v_x¹₂¢

¯ x1−2¡

p¹₂₁v_x¹_¯1¢

x2+ ˜q¹v¹, x= (b1, c)

2 h1

hp¹₁₁+ (p¹₁₁)⁻¹

2 v¹_¯x1+p¹₁₂v¹_¯x2+ ˜α¹₁v¹−[ ˜β¹(x,·), v²(·)]_γ¯²

11

i

+ 2 h3

h

p¹₂₁v_x¹_¯1+p¹₂₂+ (p¹₂₂)⁻²

2 v_x¹_¯2+ ˜α₂¹v¹i

+ ˜q¹v¹, x= (b1, d) and analogously at the other boundary nodes, x∈γ¹₂₁∪γ¹₂₂ L²_hv is defined in an analogous manner,

f˜^k=







T_k1²T_k2² f^k, x∈ω^k

T_ki^2±T_k,3−i² f^k, x∈γ_i1^k / x∈γ_i2^k T_k1^2±T_k2^2±f^k, x∈γ_?^k

˜ q^k=







T_k1²T_k2² q^k, x∈ω^k

T_ki^2±T_k,3−i² q^k, x∈γ^k_i1/ x∈γ^k_i2 T_k1^2±T_k2^2±q^k, x∈γ^k_?

˜

α^k=T_k,3−i² α^k, x∈γ_i1^k ∪γ^k_i2, i= 1,2,

˜

α^k_i =T_k,3−i^2± α^k, x∈γ_?^k, i= 1,2 and

β˜^k =

(T_k2² β^k, x∈γ^k_1,3−k, T_k2^2±β^k, x∈γ^k?_1,3−k.

The FDS (10) may be compactly presented as operator-difference scheme

Lhv= ˜f , (11)

wherev= (v¹, v²), ˜f = ( ˜f¹,f˜²) andLhv= (L¹_hv, L²_hv).

In the sequel we shall assume that the generalized solution of the problem (1)–

(5) belongs to the Sobolev spaceH^s, 2< s≤3, while the data satisfy the following smoothness conditions:

p^k_ij ∈H^s−1(Ω^k), α^k∈H^s−3/2(Γ^k_ij), α^k ∈C(Γ^k), β^k∈H^s−1(∆^k),

f^k∈H^s−2(Ω^k), q^k ∈H^s−2(Ω^k) k, i, j= 1,2. (12)

Let us consider the bilinear formAh(v, w) associated with difference operatorLhv:

Ah(v, w) = [Lhv, w] = X2

k=1

½1 2

X2

i=1

h

[p^k_iiv_x^k_i, w_x^k_i)k,i+ (p^k_iiv^k_¯xi, w^k_x¯i]k,i

+ [p^k_i,3−iv_x^k_3−i, w_x^k_i)k+ (p^k_i,3−iv_x^k_¯3−i, w^k_¯xi]k

i

+ [˜q^kv^k, w^k]k+ X2

i,j=1

X

x∈γ_ij^k

hk,3−iα˜^kv^kw^k

+1 2

X

x∈γ^k_?

(hk2α˜^k₁+hk1α˜^k₂)v^kw^k−h²_k2 X

x∈γ_1,3−k^k

X

x⁰∈γ_1,k^3−k

β^k(x, x⁰)v^3−k(x⁰)w^k(x)

−h²_k2 2

X

x∈γ_1,3−k^k?

X

x⁰∈γ_1,k^(3−k)?

β^k(x, x⁰)v^3−k(x⁰)w^k(x)

¾

Lemma 2. Let p^k_ij, α^k > 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 steph, there exist positive constantsc2 andc3 such that

c2|[v]|²_H1

h ≤Ah(v, v) = [Lhv, v]≤c3|[v]|²_H1 h.

The proof is analogous to the proof of Lemma 1.

5. Convergence of the finite difference scheme

Letu= (u¹, u²) be the solution of the BVP (1)–(5), and letv= (v¹, v²) denote the solution of the FDS (10). The errorz= (z¹, z²) =u−vsatisfies the following conditions

L^k_hz=ψ^k, x∈ω¯^k, (13) whereψ¹ is equal to:

X2

i,j=1

η_ij,¹ _x¯i+µ¹, x∈ω¹,

2

h1η₁₁¹ + 2

h1η₁₂¹ + ˜η_21,¹ _x¯2+ ˜η_22,¹ _x¯2+ 2

h1ζ¹+ ˜µ¹, x∈γ₁₁¹ , 2

h₁

¡η˜₁₁¹ + ˜η¹₁₂+ζ₁¹+ζ₂¹¢ + 2

h₃

¡η˜₂₁³ + ˜η¹₂₂+ζ₂¹¢

+ ˜˜µ¹, x= (a1, c),

− 2 h1

(η₁₁¹ )⁻¹− 2 h1

(η₁₂¹ )⁻¹+ ˜η_21,¹ _¯x2+ ˜η_22,¹ _¯x2+ 2 h1

ζ¹+ 2 h1

χ¹+ ˜µ¹, x∈γ₁₂¹ , 2

h1

£−(˜η¹₁₁)⁻¹−(˜η₁₂¹ )⁻¹+ζ₁¹+χ¹¤ + 2

h3

¡η˜₂₁¹ + ˜η₂₂¹ +ζ₂¹¢

+ ˜˜µ¹, x= (b1, c), and analogously at the other boundary nodes.

ψ² is defined in an analogous manner, η_ij^k =T_ki⁺T_k,3−i²

µ p^k_ij ∂u^k

∂xj

¶

−1 2

h

p^k_ijuxj+ (p^k_ij)⁺ⁱ(u^k_¯xj)⁺ⁱi

, x∈ω^k,

˜

η_ii^k =T_ki⁺T_k,3−i^2±

µ p^k_ii∂u^k

∂xi

¶

−p^k_ii+ (p^k_ii)⁺ⁱ

2 u^k_xi, x∈γ^k−_3−i,1/ x∈γ_3−i,2^k− ,

˜ η_i,3−i^k =









T_ki⁺T_k,3−i²⁺

µ

p^k_i,3−i∂x^∂uk

3−i

¶

−p^k_i,3−iu^k_x3−i, x∈γ_3−i,1^k− ,

T_ki⁺T_k,3−i²⁻

µ

p^k_i,3−i∂x^∂uk

3−i

¶

−(p^k_i,3−i)⁺ⁱ(u^k_x¯3−i)⁺ⁱ, x∈γ_3−i,2^k− ,

ζ^k = (T_ki²α^k)u^k−T_ki²(α^ku^k), x∈γ_3−i,1^k ∪γ_3−i,2^k , ζ_i^k = (T_ki^2±α^k)u^k−T_ki^2±(α^ku^k), x∈γ_?^k,

µ^k = (T_ki²T_k,3−i² q^k)u^k−T_ki²T_k,3−i² (q^ku^k), x∈ω^k,

˜

µ^k = (T_k,3−i^2± T_ki²q^k)u^k−T_k,3−i^2± T_ki²(q^ku^k), x∈γ_3−i,1^k− / x∈γ_3−i,2^k− , µ˜˜^k = (T_k,3−i^2± T_ki^2±q^k)u^k−T_k,3−i^2± T_ki^2±(q^ku^k), x∈γ_?^k,

χ^k = Z

Γ^3−k_1k

T_k2²β^k(x, x⁰)u^3−k(x⁰) dΓ^3−k_1k −h3

X

x⁰∈¯γ_1k^3−k

T_k2²β^k(x, x⁰)u^3−k(x⁰)

−h3

2 X

x⁰∈¯γ_1k^3−k

T_k2²β^k(x, x⁰)u^3−k(x⁰), x∈γ_1,3−k^k ,

χ^k = Z

Γ^3−k_1k

T_k2^2±β^k(x, x⁰)u^3−k(x⁰) dΓ^3−k_1k −h3

X

x⁰∈¯γ^3−k_1k

T_k2^2±β^k(x, x⁰)u^3−k(x⁰)

−h3

2 X

x⁰∈¯γ_1k^3−k

T_k2^2±β^k(x, x⁰)u^3−k(x⁰), x∈γ_1,3−k^k? .

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:

¯¯

¯[v^k, w^k_x3−i)_γk−

ij

¯¯

¯≤C|[v^k]|H¨^1/2(γ_ij^k−)|[w^k]|_H1(¯ω^k).

Lemma 4. [6]Let v^k be a mesh function on ω¯^k, then

|[v^k]|_C(¯ωk)≤C r

log1

h|[v^k]|_H1(¯ω^k).