PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 84(98) (2008), 37–48 DOI: 10.2298/PIM0898037J

FINITE DIFFERENCE APPROXIMATION OF STRONG SOLUTIONS OF A PARABOLIC INTERFACE PROBLEM ON DISCONNECTED DOMAINS

Boško S. Jovanović and Lubin G. Vulkov

Abstract. We investigate an initial boundary value problem for one dimensional parabolic equation in two disconnected intervals. A finite difference scheme for its solution is proposed and investigated. Convergence rate estimate compatible with the smoothness of input data is obtained.

1. Introduction

Interface problems occur in many applications in science and engineering. From the mathematical point of view, interface problems lead to partial differential equations whose input data and solutions have discontinuities across one or several hypersurfaces, which have lower dimension than the domain where the problem is defined. Various forms of conjugation conditions satisfied by the solution and its derivatives on the interface are known. The numerical methods designed for smooth solutions do not work efficiently for interface problems. Problems of this type we considered in [6, 7]

There exists another similar type of problems whose solutions are defined in two (or more) disconnected domains. For example, such situation occurs when the solution in the intermediate region is known or can be determined from a simpler equation. Its effect can be modelled (see [2]) by means of nonlocal jump conditions across the intermediate region (layer).

In this paper we consider the following initial-boundary-value problem (IBVP): Find functions u_{1}(x, t) and u_{2}(x, t) that satisfy the parabolic equations

\frac{∂u_{1}}{∂t} − \frac{∂}{∂x} (p_{1}(x)\frac{∂u_{1}}{∂x} ) = f_{1}(x, t), x ∈ Ω_{1} ≡ (a_{1}, b_{1}), t > 0, (1)

2000 Mathematics Subject Classification: 65M12, 65M15.

Key words and phrases: initial-boundary value problem, disconnected domains, interface, Sobolev spaces, finite differences, convergence rate.

The research of the first author was supported by Ministry of Science of Republic of Serbia under project 144005A while the research of the second author was supported by Bulgarian Fund of Science under project VU-MI-106/2005.

\frac{∂u_{2}}{∂t} − \frac{∂}{∂x} (p_{2}(x)\frac{∂u_{2}}{∂x} ) = f_{2}(x, t), x ∈ Ω_{2} ≡ (a_{2}, b_{2}), t > 0, (2)

where −∞ < a_{1} < b_{1} < a_{2} < b_{2} < +∞, the internal conjugation conditions of Robin–Dirichlet type

(3)  p_{1}(b_{1})\frac{∂u_{1}(b_{1}, t)}{∂x} + α_{1}u_{1}(b_{1}, t) = β_{1}u_{2}(a_{2}, t) + γ_{1}(t),
(4)  −p_{2}(a_{2})\frac{∂u_{2}(a_{2}, t)}{∂x} + α_{2}u_{2}(a_{2}, t) = β_{2}u_{1}(b_{1}, t) + γ_{2}(t),

the simplest external Dirichlet boundary conditions

(5)  u_{1}(a_{1}, t) = 0, u_{2}(b_{2}, t) = 0,

and initial conditions

(6)  u_{1}(x, 0) = u_{10}(x), u_{2}(x, 0) = u_{20}(x).

Throughout the paper we assume that the data satisfy the usual regularity and ellipticity conditions

(7)  p_{i}(x) ∈ L_{∞}(Ω_{i}), 0 < p_{i0} ⩽ p_{i}(x), a.e. in Ω_{i}, i = 1, 2.

We also assume that

(8)  α_{i} > 0 , β_{i} > 0, i = 1, 2 and β_{1}β_{2} ⩽ α_{1}α_{2}.

Similar problem is investigated in [8] and a finite difference scheme (FDS) for its numerical solution is proposed. The convergence of FDS in the weak discrete norm (H_{h,τ}^{1,1/2}) is proved. In this paper the properties of the strong solution of IBVP (1)–(6) are examined and the convergence of the corresponding FDS is proved in the strong discrete norm (H_{h,τ}^{2,1}). To ensure the second order of convergence in the space step-size h a special approximation for x = b_{1} and x = a_{2} was needed.

By C we shall denote a positive generic constant, independent of the solution of IBVP and the mesh-sizes, which can take different values in different formulas.

The layout of the paper is as follows. Section 2 is devoted to the analysis of the existence and the uniqueness of the strong solution of IBVP (1)–(6). In Section 3 we introduce a FDS approximating IBVP (1)–(6) and investigate its convergence. A convergence rate estimate compatible with the smoothness of input data is obtained.

2. Existence and uniqueness of the strong solution

Let the conditions (8) hold. We introduce the product space

L = L_{2}(Ω_{1}) × L_{2}(Ω_{2}) = {v = (v_{1}, v_{2}) | v_{i} ∈ L_{2}(Ω_{i})},

endowed with the inner product and associated norm

(u, v)_{L} = β_{2}(u_{1}, v_{1})_{L_{2}(Ω_{1})} + β_{1}(u_{2}, v_{2})_{L_{2}(Ω_{2})}, ∥v∥_{L} = (v, v)_{L}^{1/2},

where

(u_{i}, v_{i})_{L_{2}(Ω_{i})} = ∫_{Ω_{i}} u_{i}v_{i} dx, i = 1, 2.

We also define the spaces

H^{k} = {v = (v_{1}, v_{2}) | v_{i} ∈ H^{k}(Ω_{i})}, k = 1, 2, …,

endowed with the inner products and norms

(u, v)_{H^{k}} = β_{2}(u_{1}, v_{1})_{H^{k}(Ω_{1})} + β_{1}(u_{2}, v_{2})_{H^{k}(Ω_{2})}, ∥v∥_{H^{k}} = (v, v)_{H^{k}}^{1/2},

where

(u_{i}, v_{i})_{H^{k}(Ω_{i})} = ∑_{j=0}^{k} (\frac{d^{j}u_{i}}{dx^{j}} , \frac{d^{j}v_{i}}{dx^{j}} )_{L_{2}(Ω_{i})}, i = 1, 2, k = 1, 2, ….

In particular, we set

H_{0}^{1} = {v = (v_{1}, v_{2}) ∈ H^{1} | v_{1}(a_{1}) = 0, v_{2}(b_{2}) = 0}.

Finally, we define the bilinear form

(9) A(u, v) = β_{2} ∫_{Ω_{1}} p_{1} \frac{du_{1}}{dx} \frac{dv_{1}}{dx} dx + β_{1} ∫_{Ω_{2}} p_{2} \frac{du_{2}}{dx} \frac{dv_{2}}{dx} dx

+ β_{2}α_{1}v_{1}(b_{1})w_{1}(b_{1}) + β_{1}α_{2}v_{2}(a_{2})w_{2}(a_{2}) − β_{1}β_{2}[v_{1}(b_{1})w_{2}(a_{2})+v_{2}(a_{2})w_{1}(b_{1})].

Lemma 2.1. [8] Under the conditions (7) and (8) the bilinear form A, defined by (9), is symmetric and bounded on H^{1} × H^{1}. Moreover, this form is also coercive on H_{0}^{1}, i.e., there exists a constant c_{0} > 0 such that A(v, v) ⩾ c_{0}∥v∥_{H^{1}}^{2}, ∀ v ∈ H_{0}^{1}.

Let Ω be a domain in ℝ^{n} and u(t) a function mapping Ω into a Hilbert space H. In a standard manner (see [9]) we define the Sobolev space of vector valued functions H^{k}(Ω, H), endowed with the inner product

(u, v)_{H^{k}(Ω,H)} = ∫_{Ω} ∑_{|α|⩽k}(D^{α}u(t), D^{α}v(t))_{H} dt , k = 0, 1, 2, …

and with the usual modification for non-integer k. For k = 0 we set L_{2}(Ω, H) = H^{0}(Ω, H).

We define the spaces H^{k,k/2} = L_{2}((0, T), H^{k})∩H^{k/2}((0, T), L). We also denote Q_{i} = Ω_{i} × (0, T), i = 1, 2.

Theorem 2.1. Let the assumptions (7) and (8) hold and u_{0} = (u_{10}, u_{20}) ∈ H_{0}^{1}, f = (f_{1}, f_{2}) ∈ L_{2}((0, T), L), γ_{i} ∈ H^{1/4}(0, T), p′_{i} ∈ L_{∞}(Ω_{i}), i = 1, 2. Then the IBVP (1)–(6) has a unique strong solution u = (u_{1}, u_{2}) ∈ H^{2,1} and the following a priori estimate holds true

(10)  ∥u∥_{H^{2,1}}^{2} ⩽ C ∑_{i=1}^{2} β_{3−i}(∥u_{i0}∥_{H^{1}(Ω_{i})}^{2} + ∥f_{i}∥_{L_{2}(Q_{i})}^{2} + ∥γ_{i}∥_{H^{1/4}(0,T)}^{2}).

Proof. Existence and uniqueness of the weak solution of IBVP (1)–(6) is proved in [?, Theorems 2.1 and 3.1]. In such a way, it remains to prove that under our assumptions this solution is strong, i.e., satisfies the a priori estimate (10).

Multiplying the equation (1) by ∂u_{1}/∂t and integrating by parts, we obtain

\begin{aligned}∥\frac{∂u_{1}}{∂t} (·, t)∥_{L_{2}(Ω_{1})}^{2} & + ∫_{Ω_{1}} p_{1} \frac{∂u_{1}}{∂x} \frac{∂^{2}u_{1}}{∂t∂x} dx \\ & + α_{1}u_{1}(b_{1}, t) \frac{∂u_{1}}{∂t} (b_{1}, t) − β_{1}u_{2}(a_{2}, t) \frac{∂u_{1}}{∂t} (b_{1}, t) \end{aligned}

= (f_{1}(·, t), \frac{∂u_{1}}{∂t} (·, t))_{L_{2}(Ω_{1})} + γ_{1}(t) \frac{∂u_{1}}{∂t} (b_{1}, t)

⩽ \frac{1}{2} ∥f_{1}(·, t)∥_{L_{2}(Ω_{1})}^{2} + \frac{1}{2} ∥\frac{∂u_{1}}{∂t} (·, t)∥_{L_{2}(Ω_{1})}^{2} + γ_{1}(t) \frac{∂u_{1}}{∂t} (b_{1}, t)

and analogously

\frac{1}{2} ∥\frac{∂u_{2}}{∂t} (·, t)∥_{L_{2}(Ω_{2})}^{2} + ∫_{Ω_{2}} p_{2} \frac{∂u_{2}}{∂x} \frac{∂^{2}u_{2}}{∂t∂x} dx

+ α_{2}u_{2}(a_{2}, t) \frac{∂u_{2}}{∂t} (a_{2}, t) − β_{2}u_{1}(b_{1}, t) \frac{∂u_{2}}{∂t} (a_{2}, t)

⩽ \frac{1}{2} ∥f_{2}(·, t)∥_{L_{2}(Ω_{2})}^{2} + γ_{2}(t) \frac{∂u_{2}}{∂t} (a_{2}, t).

Multiplying the first of these inequalities by 2β_{2}, the second by 2β_{1} and summing up we get

∥\frac{∂u}{∂t}(·, t)∥_{L}^{2} + \frac{d}{dt}(A(u(·, t), u(·, t)))

⩽ ∥f(·, t)∥_{L}^{2} + 2(β_{2}γ_{1}(t) \frac{∂u_{1}}{∂t} (b_{1}, t) + β_{1}γ_{2}(t) \frac{∂u_{2}}{∂t} (a_{2}, t)).

Integrating this inequality on t ∈ (0, T) and using Lemma 2.1 one obtains

(11)  ∥\frac{∂u}{∂t}∥_{L_{2}((0,T),L)}^{2} ⩽ C∥u_{0}∥_{H^{1}}^{2} + ∥f∥_{L_{2}((0,T),L)}^{2}

+ 2 ∫_{0}^{T}(β_{2}γ_{1}(t) \frac{∂u_{1}}{∂t} (b_{1}, t) + β_{1}γ_{2}(t) \frac{∂u_{2}}{∂t} (a_{2}, t))dt.

Analogously

(12)  ∥Au∥_{L_{2}((0,T),L)}^{2} ⩽ C∥u_{0}∥_{H^{1}}^{2} + ∥f∥_{L_{2}((0,T),L)}^{2}

+ 2 ∫_{0}^{T} (β_{2}γ_{1}(t) \frac{∂u_{1}}{∂t} (b_{1}, t) + β_{1}γ_{2}(t) \frac{∂u_{2}}{∂t} (a_{2}, t))dt,

where we denoted Au = (A_{1}u_{1}, A_{2}u_{2}) and

A_{i}u_{i} = −\frac{∂}{∂x}(p_{i}(x)\frac{∂u_{i}}{∂x} ), i = 1, 2.

Let us estimate ∥Au∥_{L} = ∥Au(·, t)∥_{L}. From the inequalities

∥\frac{∂^{2}u}{∂x^{2}} ∥_{L} ⩽ C ∥p \frac{∂^{2}u}{∂x^{2}} ∥_{L} = C ∥Au + \frac{dp}{dx} \frac{∂u}{∂x}∥_{L} ⩽ C(∥Au∥_{L} + ∥u∥_{H^{1}})

and

c_{0} ∥u∥_{H^{1}}^{2} ⩽ A(u, u) = (Au, u)_{L} + β_{2}γ_{1}(t) u_{1}(b_{1}, t) + β_{1}γ_{2}(t)u_{2}(a_{2}, t)

⩽ ε(∥u∥_{L}^{2} + β_{2}u_{1}^{2}(b_{1}, t) + β_{1}u_{2}^{2}(a_{2}, t)) + \frac{1}{4ε}(∥Au∥_{L}^{2} + β_{2}γ_{1}^{2}(t) + β_{1}γ_{2}^{2}(t))

⩽ Cε ∥u∥_{H^{1}}^{2} + \frac{1}{4ε}(∥Au∥_{L}^{2} + β_{2}γ_{1}^{2}(t) + β_{1}γ_{2}^{2}(t))

after taking ε = c_{0}/(2C) one obtains

(13)  ∥u∥_{H^{2}}^{2} = ∥\frac{∂^{2}u}{∂x^{2}} ∥_{L}^{2} + ∥u∥_{H^{1}}^{2} ⩽ C(∥Au∥_{L}^{2} + β_{2}γ_{1}^{2}(t) + β_{1}γ_{2}^{2}(t)).

Let us now estimate ∫_{0}^{T} γ_{1}(t) \frac{∂u_{1}}{∂t} (b_{1}, t) dt. Using Fourier sine and cosine expansions

γ_{1}(t) = ∑_{j=1}^{∞} b_{j}[γ_{1}] sin \frac{jπt}{T}, u_{1}(b_{1}, t) = \frac{a_{0}[u_{1}(b_{1}, ·)]}{2} + ∑_{j=1}^{∞} a_{j}[u_{1}(b_{1}, ·)] cos \frac{jπt}{T}

(where b_{j}[γ_{1}] and a_{j}[u_{1}(b_{1}, ·)] are the corresponding Fourier coefficients) and the orthogonality of sines, we obtain

∫_{0}^{T} γ_{1}(t) \frac{∂u_{1}}{∂t} (b_{1}, t) dt = −\frac{π}{2} ∑_{k=1}^{∞} k b_{k}[γ_{1}]a_{k}[u_{1}(b_{1}, ·)]

⩽ \frac{π}{2} ( ∑_{k=1}^{∞} k^{1/2} b_{k}^{2}[γ_{1}])^{1/2}( ∑_{k=1}^{∞} k^{3/2} a_{k}^{2}[u_{1}(b_{1}, ·)])^{1/2}.

From this inequality, using Lemma 3.1 from [8] and the trace theorem for anisotropic Sobolev spaces [9], one obtains

(14) | ∫_{0}^{T} γ_{1}(t) \frac{∂u_{1}}{∂t} (b_{1}, t) dt| ⩽ C∥γ_{1}∥_{H^{1/4}(0,T)}∥u_{1}(b_{1}, ·)∥_{H^{3/4}(0,T)}

⩽ C∥γ_{1}∥_{H^{1/4}(0,T)}∥u_{1}∥_{H^{2,1}(Q_{1})} ⩽ ε ∥u_{1}∥_{H^{2,1}(Q_{1})}^{2} + \frac{C}{ε} ∥γ_{1}∥_{H^{1/4}(0,T)}^{2}

and analogously

(15)  | ∫_{0}^{T} γ_{2}(t) \frac{∂u_{2}}{∂t} (a_{2}, t) dt| ⩽ ε ∥u_{2}∥_{H^{2,1}(Q_{2})}^{2} + \frac{C}{ε} ∥γ_{2}∥_{H^{1/4}(0,T)}^{2}.

Finally, for sufficiently small ε > 0, from (11)–(15) we get the a priori estimate (10). □

3. Finite difference approximation

3.1. Meshes, finite differences and discrete norms. Let ω̄_{1,h_{1}} be an uniform mesh in Ω̄_{1} with the step-size h_{1} = (b_{1} − a_{1})/n_{1}, ω_{1,h_{1}} = ω̄_{1,h_{1}} ∩ Ω_{1}, ω_{1,h_{1}}^{−} = ω_{1,h_{1}} ∪{a_{1}} and ω_{1,h_{1}}^{+} = ω_{1,h_{1}} ∪{b_{1}}. Analogously, in Ω̄_{2} we define uniform mesh ω̄_{2,h_{2}} with the step-size h_{2} = (b_{2}−a_{2})/n_{2} and its submeshes ω_{2,h_{2}} = ω̄_{2,h_{2}}∩Ω_{2}, ω_{2,h_{2}}^{−} = ω_{2,h_{2}} ∪{a_{2}} and ω_{2,h_{2}}^{+} = ω_{2,h_{2}} ∪{b_{2}}. Finally, in [0, T] we introduce uniform mesh ω̄_{τ} with the step-size τ = T/n and set ω_{τ} = ω̄_{τ} ∩ (0, T), ω_{τ}^{−} = ω_{τ} ∪ {0} and ω_{τ}^{+} = ω_{τ} ∪ {T }. We will consider vector-functions of the form v = (v_{1}, v_{2}) where v_{i} is mesh function defined on ω̄_{i,h_{i}} × ω̄_{τ}, i = 1, 2. We define finite differences in the usual way [10]

v_{i,x}(x, t) = \frac{v_{i}(x + h_{i}, t) − v_{i}(x, t)}{h_{i}} = v_{i,x̄}(x + h_{i}, t),

v_{i,t}(x, t) = \frac{v_{i}(x, t + τ) − v_{i}(x, t)}{τ} = v_{i,t̄}(x, t + τ),

v_{x} = (v_{1,x}, v_{2,x}), vx̄ = (v_{1,x̄}, v_{2,x̄}), v_{t} = (v_{1,t}, v_{2,t}), vt̄ = (v_{1,t̄}, v_{2,t̄}).

We also introduce the discrete inner products and norms

(v, w)_{L_{h}} = β_{2} ∑_{x∈ω_{1,h_{1}}^{+}} v_{1}w_{1}ℏ_{1} + β_{1} ∑_{x∈ω_{2,h_{2}}^{−}} v_{2}w_{2}ℏ_{2},

(v, w)_{L_{h′}} = β_{2} h_{1} ∑_{x∈ω_{1,h_{1}}^{+}} v_{1}w_{1} + β_{1} h_{2} ∑_{x∈ω_{2,h_{2}}^{+}} v_{2}w_{2},

∥v∥_{L_{h}}^{2} = (v, v)_{L_{h}}, ∥v∥_{L_{h′}}^{2} = (v, v)_{L_{h′}} , ∥v_{i}∥_{L_{2}(ω_{τ}^{+})}^{2} = τ ∑_{t∈ω_{τ}^{+}} v_{i}^{2},

∥v∥_{L_{2}(ω_{τ}^{+},L_{h})}^{2} = τ ∑_{t∈ω_{τ}^{+}} ∥v(·, t)∥_{L_{h}}^{2}, ∥v∥_{L_{2}(ω_{τ}^{+},L_{h′})}^{2} = τ ∑_{t∈ω_{τ}^{+}} ∥v(·, t)∥_{L_{h′}}^{2} ,

|v|_{H^{1}(ω_{τ}^{+},L_{h})}^{2} = τ ∑_{t∈ω_{τ}^{+}} ∥vt̄(·, t)∥_{L_{h}}^{2},

∥v∥_{H_{h,τ}^{2,1}}^{2} = ∥v_{x̄x}∥_{L_{2}(ω_{τ}^{+},L_{h})}^{2} + ∥vx̄∥_{L_{2}(ω_{τ}^{+},L_{h′})}^{2} + ∥v∥_{L_{2}(ω_{τ}^{+},L_{h})}^{2} + |v|_{H^{1}(ω_{τ}^{+},L_{h})}^{2},

where

ℏ_{i} = h_{i}, x ∈ ω_{i,h_{i}}, i = 1, 2, ℏ_{1}(b_{1}) = h_{1}/2, ℏ_{2}(a_{2}) = h_{2}/2,

v_{1,x̄x}(b_{1}, t) = \frac{−2}{h_{1}}[v_{1,x̄}(b_{1}, t) + α_{1}^{0}v_{1}(b_{1}, t) − β_{1}^{0}v_{2}(a_{2}, t)],

v_{2,x̄x}(a_{2}, t) = \frac{2}{h_{2}} [v_{2,x}(a_{2}, t) − α_{2}^{0}v_{2}(a_{2}, t) + β_{2}^{0}v_{1}(b_{1}, t)],

and

α_{1}^{0} = \frac{α_{1}}{p_{1}(b_{1})}, β_{1}^{0} = \frac{β_{1}}{p_{1}(b_{1})}, α_{2}^{0} = \frac{α_{2}}{p_{2}(a_{2})}, β_{2}^{0} = \frac{β_{2}}{p_{2}(a_{2})}.

3.2. Finite difference scheme. In this and subsequent sections we shall assume that u_{i} belongs to H^{4,2}(Q_{i}), while p_{i} ∈ H^{3}(Ω_{i}). Consequently, f_{i} ∈ H^{2,1}(Q_{i}) is continuous function. We approximate the equations (1) and (2) in the following manner:

v_{1,t̄} − (p̄_{1}v_{1,x̄})_{x} = f_{1}, x ∈ ω_{1,h_{1}}, t ∈ ω_{τ}^{+}, (16)

v_{2,t̄} − (p̄_{2}v_{2,x̄})_{x} = f_{2}, x ∈ ω_{2,h_{2}}, t ∈ ω_{τ}^{+}, (17)

where p̄_{i}(x) = \frac{1}{2} [p_{i}(x) + p_{i}(x − h_{i})], i = 1, 2. To ensure the same order of approximation for x = b_{1} and x = a_{2} we set:

(18) v_{1,t̄}(b_{1}, t) − \frac{h_{1}}{3} v_{1,t̄x̄}(b_{1}, t) + \frac{h_{1}}{6} \frac{p′_{1}(b_{1})}{p_{1}(b_{1})} v_{1,t̄}(b_{1}, t) + \frac{2}{h_{1}} \{p̄_{1}(b_{1})v_{1,x̄}(b_{1}, t)

+ [1 + \frac{h_{1}^{2}}{12} (\frac{p′_{1,x̄}(b_{1})}{p_{1}(b_{1})} + \frac{(p′_{1}(b_{1}))^{2}}{p_{1}^{2}(b_{1})} )](α_{1}v_{1}(b_{1}, t) − β_{1}v_{2}(a_{2}, t))\}

= f_{1}(b_{1}, t) − \frac{h_{1}}{3} f_{1,x̄}(b_{1}, t) + \frac{h_{1}}{6} \frac{p′_{1}(b_{1})}{p_{1}(b_{1})} f_{1}(b_{1}, t)

+ \frac{2}{h_{1}} [1 + \frac{h_{1}^{2}}{12} (\frac{p′_{1,x̄}(b_{1})}{p_{1}(b_{1})} + \frac{(p′_{1}(b_{1}))^{2}}{p_{1}^{2}(b_{1})} )]γ_{1}(t), t ∈ ω_{τ}^{+},

(19) v_{2,t̄}(a_{2}, t)+ \frac{h_{2}}{3} v_{2,t̄x}(a_{2}, t)− \frac{h_{2}}{6} \frac{p′_{2}(a_{2})}{p_{2}(a_{2})}v_{2,t̄}(a_{2}, t)− \frac{2}{h_{2}} \{p̄_{2}(a_{2} + h_{2})v_{2,x}(a_{2}, t)

− [1 + \frac{h_{2}^{2}}{12} (\frac{p′_{2,x}(a_{2})}{p_{2}(a_{2})} + \frac{(p′_{2}(a_{2}))^{2}}{p_{2}^{2}(a_{2})} )](α_{2}v_{2}(a_{2}, t) − β_{2}v_{1}(b_{1}, t))\}

= f_{2}(a_{2}, t) + \frac{h_{2}}{3} f_{2,x}(a_{2}, t) − \frac{h_{2}}{6} \frac{p′_{2}(a_{2})}{p_{2}(a_{2})} f_{2}(a_{2}, t)

+ \frac{2}{h_{2}} [1 + \frac{h_{2}^{2}}{12} (\frac{p′_{2,x}(a_{2})}{p_{2}(a_{2})} + \frac{(p′_{2}(a_{2}))^{2}}{p_{2}^{2}(a_{2})} )]γ_{2}(t), t ∈ ω_{τ}^{+}.

The Dirichlet boundary conditions (5) and initial conditions (6) can be satisfied exactly:

v_{1}(a_{1}, t) = 0,  v_{2}(b_{2}, t) = 0,  t ∈ ω_{τ}^{+}, (20)

v_{i}(x, 0) = u_{i0}(x), x ∈ ω̄_{i,h_{i}}, i = 1, 2. (21)

In each time level t = jτ FDS (16)–(21) reduces to a tridiagonal linear system with n_{1} + n_{2} unknowns. In such a way, FDS (16)–(21) is computationally efficient. From the general theory of difference schemes [10] it follows that FDS (16)–(21) is unconditionally stable.

3.3. Convergence of the finite difference scheme. Let u = (u_{1}, u_{2}) be the solution of the IBVP (1)–(6) and v = (v_{1}, v_{2}) the solution of the FDS (16)–(21). Then the error z = u − v satisfies the following FDS:

z_{1,t̄} − (p̄_{1}z_{1,x̄})_{x} = φ_{1}, x ∈ ω_{1,h_{1}}, t ∈ ω_{τ}^{+}, (22)

z_{2,t̄} − (p̄_{2}z_{2,x̄})_{x} = φ_{2}, x ∈ ω_{2,h_{2}}, t ∈ ω_{τ}^{+}, (23)

(24) z_{1,t̄}(b_{1}, t) − \frac{h_{1}}{3} z_{1,t̄x̄}(b_{1}, t) + \frac{h_{1}}{6} \frac{p′_{1}(b_{1})}{p_{1}(b_{1})} z_{1,t̄}(b_{1}, t) + \frac{2}{h_{1}} \{p̄_{1}(b_{1})z_{1,x̄}(b_{1}, t)

+[1+\frac{h_{1}^{2}}{12} (\frac{p′_{1,x̄}(b_{1})}{p_{1}(b_{1})}+\frac{(p′_{1}(b_{1}))^{2}}{p_{1}^{2}(b_{1})} )](α_{1}z_{1}(b_{1}, t)−β_{1}z_{2}(a_{2}, t))\} = φ_{1}(b_{1}, t), t ∈ ω_{τ}^{+},

(25) z_{2,t̄}(a_{2}, t)+ \frac{h_{2}}{3} z_{2,t̄x}(a_{2}, t)− \frac{h_{2}}{6} \frac{p′_{2}(a_{2})}{p_{2}(a_{2})}z_{2,t̄}(a_{2}, t)− \frac{2}{h_{2}} \{p̄_{2}(a_{2} + h_{2})z_{2,x}(a_{2}, t)

−[1+\frac{h_{2}^{2}}{12} (\frac{p′_{2,x}(a_{2})}{p_{2}(a_{2})}+\frac{(p′_{2}(a_{2}))^{2}}{p_{2}^{2}(a_{2})} )](α_{2}z_{2}(a_{2}, t)−β_{2}z_{1}(b_{1}, t))\} = φ_{2}(a_{2}, t), t ∈ ω_{τ}^{+},

z_{1}(a_{1}, t) = 0,  z_{2}(b_{2}, t) = 0,  t ∈ ω_{τ}^{+}, (26)
z_{i}(x, 0) = 0,  x ∈ ω̄_{i,h_{i}},  i = 1, 2, (27)

where

φ_{i} = ψ_{i} + χ_{i} , x ∈ ω_{i,h_{i}}, t ∈ ω_{τ}^{+}, i = 1, 2,

φ_{1}(b_{1}, t) = ψ_{1}(b_{1}, t) − \frac{h_{1}}{3} ψ_{1,x̄}(b_{1}, t) + \frac{h_{1}}{6} \frac{p′_{1}(b_{1})}{p_{1}(b_{1})} ψ_{1}(b_{1}, t) + χ̃_{1}(b_{1}, t) ,

φ_{2}(a_{2}, t) = ψ_{2}(a_{2}, t) + \frac{h_{2}}{3} ψ_{2,x}(a_{2}, t) − \frac{h_{2}}{6} \frac{p′_{2}(a_{2})}{p_{2}(a_{2})} ψ_{2}(a_{2}, t) + χ̃_{2}(a_{2}, t) ,

ψ_{i} = u_{i,t̄} − \frac{∂u_{i}}{∂t} , χ_{i} = \frac{∂}{∂x}(p_{i} \frac{∂u_{i}}{∂x} ) − (p̄_{i} u_{i,x̄})_{x},

χ̃_{1} = \frac{∂}{∂x}(p_{1} \frac{∂u_{1}}{∂x} ) + \frac{2}{h_{1}} (p̄_{1}u_{1,x̄} − p_{1} \frac{∂u_{1}}{∂x} )

+ \frac{h_{1}}{6} (p′_{1} \frac{∂^{2}u_{1}}{∂x^{2}} − p′_{1,x̄} \frac{∂u_{1}}{∂x} ) − \frac{h_{1}}{3} [\frac{∂}{∂x}(p_{1} \frac{∂u_{1}}{∂x} )]x̄,

χ̃_{2} = \frac{∂}{∂x}(p_{2} \frac{∂u_{2}}{∂x} ) − \frac{2}{h_{2}} (p̄_{2}(x + h_{2})u_{2,x} − p_{2} \frac{∂u_{2}}{∂x} )

− \frac{h_{2}}{6} (p′_{2} \frac{∂^{2}u_{2}}{∂x^{2}} − p′_{2,x} \frac{∂u_{2}}{∂x} ) + \frac{h_{2}}{3} [\frac{∂}{∂x}(p_{2} \frac{∂u_{2}}{∂x} )]_{x}.

Analogously as in the continuous case one obtains the next assertion.

Theorem 3.1. Let the assumptions (7) and (8) hold and let p_{i} ∈ C^{1}(Ω̄_{i}), for i = 1, 2. Then, for sufficiently small h_{1} and h_{2}, the solution z of FDS (22)–(27) satisfies the a priori estimate

(28)  ∥z∥_{H_{h,τ}^{2,1}} ⩽ C∥φ∥_{L_{2}(ω_{τ}^{+},L_{h})}.

Therefore, in order to determine the convergence rate of the FDS (16)–(21), it is enough to estimate the right hand side term in the inequality (28).

From the integral representation

ψ_{1}(x, t) = ψ_{10}(x, t) + ψ_{11}(x, t) = −\frac{1}{τh_{1}} ∫_{t−τ}^{t} ∫_{t′}^{t} ∫_{x−h_{1}}^{x} \frac{∂^{2}u_{1}}{∂t^{2}} (x′, t′′) dx′ dt′′ dt′

FINITE DIFFERENCE APPROXIMATION OF THE STRONG SOLUTIONS 45

+ \frac{1}{τh_{1}} ∫_{t−τ}^{t} ∫_{x−h_{1}}^{x} [\frac{∂u_{1}}{∂t} (x, t′) − \frac{∂u_{1}}{∂t} (x, t) − \frac{∂u_{1}}{∂t} (x′, t′) + \frac{∂u_{1}}{∂t} (x′, t)]dx′ dt′

we immediately obtain

|ψ_{10}(x, t)| ⩽ \frac{τ}{τh_{1}} ∫_{t−τ}^{t} ∫_{x−h_{1}}^{x} |\frac{∂^{2}u_{1}}{∂t^{2}} (x′, t′)|dx′ dt′ ⩽ \frac{τ}{\sqrt{τh_{1}}} ∥\frac{∂^{2}u_{1}}{∂t^{2}} ∥_{L_{2}(e(x,t))},

where e(x, t) = (x − h_{1}, x) × (t − τ, t). Summation over the mesh yields

(29)  τ ∑_{t∈ω_{τ}^{+}} ∑_{x∈ω_{1,h_{1}}^{+}} ℏ_{1}|ψ_{10}|^{2} ⩽ Cτ^{2} ∥\frac{∂^{2}u_{1}}{∂t^{2}} ∥_{L_{2}(Q_{1})}^{2} ⩽ Cτ^{2} ∥u_{1}∥_{H^{4,2}(Q_{1})}^{2}.

The term of the same form as ψ_{11} is estimated in [5] wherefrom follows

(30)  τ ∑_{t∈ω_{τ}^{+}} ∑_{x∈ω_{1,h_{1}}^{+}} ℏ_{1}|ψ_{11}|^{2} ⩽ C(h_{1}^{4} + τ^{2}) ∥u_{1}∥_{H^{4,2}(Q_{1})}^{2}.
For x ∈ ω_{1,h_{1}}, t ∈ ω_{τ}^{+} term χ_{1} can be represented in the following way
χ_{1}(x, t) = χ_{10}(x, t) + χ_{11}(x, t) + χ_{12}(x, t) + χ_{13}(x, t) + χ_{14}(x, t)

= −\frac{p_{1}(x)}{τh_{1}} ∫_{x−h_{1}}^{x+h_{1}}∫_{x′}^{x} ∫_{x′′}^{x} ∫_{t−τ}^{t} (1 − \frac{|x′ − x|}{h_{1}})\frac{∂^{4}u_{1}}{∂x^{4}} (x′′′, t′) dt′ dx′′′ dx′′ dx′

+ \frac{p_{1}(x)}{τh_{1}} ∫_{x−h_{1}}^{x+h_{1}} ∫_{t−τ}^{t} (1 − \frac{|x′ − x|}{h_{1}})[\frac{∂^{2}u_{1}}{∂x^{2}} (x, t) − \frac{∂^{2}u_{1}}{∂x^{2}} (x′, t)

− \frac{∂^{2}u_{1}}{∂x^{2}} (x, t′) + \frac{∂^{2}u_{1}}{∂x^{2}} (x′, t′)]dt′ dx′

− \frac{p′_{1}(x)}{2h_{1}} ∫_{x−h_{1}}^{x+h_{1}}∫_{x′}^{x} ∫_{x′′}^{x} \frac{∂^{3}u_{1}}{∂x^{3}} (x′′′, t) dx′′′ dx′′ dx′

− \frac{h_{1}}{4} p′′_{1}(x) ∫_{x−h_{1}}^{x+h_{1}}(1 − \frac{|x′ − x|}{h_{1}})\frac{∂^{2}u_{1}}{∂x^{2}} (x′, t) dx′

− \frac{1}{2h_{1}^{2}} [( ∫_{x}^{x+h_{1}}∫_{x}^{x′} ∫_{x}^{x′′}p′′′_{1}(x′′′) dx′′′ dx′′ dx′)( ∫_{x}^{x+h_{1}} \frac{∂u_{1}}{∂x} (x′, t) dx′)

+ ( ∫_{x−h_{1}}^{x} ∫_{x′}^{x} ∫_{x′′}^{x} p′′′_{1}(x′′′) dx′′′ dx′′ dx′)( ∫_{x−h_{1}}^{x} \frac{∂u_{1}}{∂x} (x′, t) dx′)].

46 JOVANOVIĆ AND VULKOV

Summands χ_{10} and χ_{11} can be estimated in the same manner as ψ_{10} and ψ_{11}:

(31)  h_{1}τ ∑_{t∈ω_{τ}^{+}} ∑_{x∈ω_{1,h_{1}}} |χ_{10}|^{2} ⩽ Ch_{1}^{4} ∥p_{1}∥_{C(Ω̄_{1})}^{2} ∥\frac{∂^{4}u_{1}}{∂x^{4}} ∥_{L_{2}(Q_{1})}^{2}
⩽ Ch_{1}^{4} ∥p_{1}∥_{H^{3}(Ω_{1})}^{2} ∥u_{1}∥_{H^{4,2}(Q_{1})}^{2},
(32)  h_{1}τ ∑_{t∈ω_{τ}^{+}} ∑_{x∈ω_{1,h_{1}}} |χ_{11}|^{2} ⩽ C(h_{1}^{4} + τ^{2}) ∥p_{1}∥_{H^{3}(Ω_{1})}^{2} ∥u_{1}∥_{H^{4,2}(Q_{1})}^{2}.

Remaining terms can be estimated directly using Cauchy–Schwartz inequality and Sobolev imbedding theorem:

(33)  h_{1}τ ∑_{t∈ω_{τ}^{+}} ∑_{x∈ω_{1,h_{1}}} |χ_{12}|^{2} ⩽ Ch_{1}^{4} ∥p_{1}∥_{C^{1}(Ω̄_{1})}^{2} max_{t∈[0,T]} ∥u_{1}(·, t)∥_{H^{3}(Ω_{1})}^{2}
⩽ Ch_{1}^{4} ∥p_{1}∥_{H^{3}(Ω_{1})}^{2} ∥u_{1}∥_{H^{4,2}(Q_{1})}^{2},
(34)  h_{1}τ ∑_{t∈ω_{τ}^{+}} ∑_{x∈ω_{1,h_{1}}} |χ_{13}|^{2} ⩽ Ch_{1}^{4} ∥p_{1}∥_{C^{2}(Ω̄_{1})}^{2} ∥u_{1}∥_{C^{2}(Q̄_{1})}^{2}
⩽ Ch_{1}^{4} ∥p_{1}∥_{H^{3}(Ω_{1})}^{2} ∥u_{1}∥_{H^{4,2}(Q_{1})}^{2},
(35)  h_{1}τ ∑_{t∈ω_{τ}^{+}} ∑_{x∈ω_{1,h_{1}}} |χ_{14}|^{2} ⩽ Ch_{1}^{4} ∥p_{1}∥_{H^{3}(Ω_{1})}^{2} ∥u_{1}∥_{C^{1}(Q̄_{1})}^{2}
⩽ Ch_{1}^{4} ∥p_{1}∥_{H^{3}(Ω_{1})}^{2} ∥u_{1}∥_{H^{4,2}(Q_{1})}^{2}.

Similarly, term χ̃_{1} can be represented in the following manner χ̃_{1}(b_{1}, t) = χ̃_{10}(b_{1}, t) + χ̃_{11}(b_{1}, t) + χ̃_{12}(b_{1}, t) + χ̃_{13}(b_{1}, t) + χ_{14}(b_{1}, t)

= \frac{2p_{1}(b_{1})}{τh_{1}^{3}} ∫_{b_{1}−h_{1}}^{b_{1}} ∫_{x′}^{b_{1}}∫_{x′′}^{b_{1}} ∫_{b_{1}−h_{1}}^{b_{1}} ∫_{x′′′′}^{x′′′} ∫_{t−τ}^{t} \frac{∂^{4}u_{1}}{∂x^{4}} (x′′′′′, t′) dt′ dx′′′′′dx′′′′dx′′′ dx′′ dx′

+ \frac{2p_{1}(b_{1})}{τh_{1}^{2}}\{\frac{2}{3} ∫_{b_{1}−h_{1}}^{b_{1}} ∫_{x′}^{b_{1}} ∫_{t−τ}^{t} [\frac{∂^{2}u_{1}}{∂x^{2}} (b_{1}, t) − \frac{∂^{2}u_{1}}{∂x^{2}} (x′′, t)

− \frac{∂^{2}u_{1}}{∂x^{2}} (b_{1}, t′) + \frac{∂^{2}u_{1}}{∂x^{2}} (x′′, t′)]dt′dx′′dx′

+ \frac{1}{3} ∫_{b_{1}−h_{1}}^{b_{1}} ∫_{x′}^{b_{1}} ∫_{t−τ}^{t} [\frac{∂^{2}u_{1}}{∂x^{2}} (b_{1} − h_{1}, t) − \frac{∂^{2}u_{1}}{∂x^{2}} (x′′, t)

− \frac{∂^{2}u_{1}}{∂x^{2}} (b_{1} − h_{1}, t′) + \frac{∂^{2}u_{1}}{∂x^{2}} (x′′, t′)]dt′dx′′dx′\}

+ \{\frac{1}{3}( ∫_{b_{1}−h_{1}}^{b_{1}} p′_{1}(x′) dx′)( ∫_{b_{1}−h_{1}}^{b_{1}} \frac{∂^{3}u_{1}}{∂x^{3}} (x′, t) dx′)

+ \frac{p′_{1}(b_{1} − h_{1})}{3}( ∫_{b_{1}−h_{1}}^{b_{1}} ∫_{x′}^{b_{1}} \frac{∂^{3}u_{1}}{∂x^{3}} (x′′, t) dx′′dx′)

− \frac{p′_{1}(b_{1})}{h_{1}}( ∫_{b_{1}−h_{1}}^{b_{1}} ∫_{x′}^{b_{1}}∫_{x′′}^{b_{1}} \frac{∂^{3}u_{1}}{∂x^{3}} (x′′′, t) dx′′′ dx′′ dx′)\}

+ \{(\frac{h_{1}}{3} ∫_{b_{1}−h_{1}}^{b_{1}} p′′_{1}(x′) dx′ + \frac{1}{3} ∫_{b_{1}−h_{1}}^{b_{1}} ∫_{x′}^{b_{1}}p′′_{1}(x′′) dx′′dx′)\frac{∂^{2}u_{1}}{∂x^{2}} (b_{1}, t)

− \frac{1}{h_{1}^{2}} ( ∫_{b_{1}−h_{1}}^{b_{1}} ∫_{x′}^{b_{1}}p′′_{1}(x′′) dx′′dx′)( ∫_{b_{1}−h_{1}}^{b_{1}} ∫_{x′}^{b_{1}} \frac{∂^{2}u_{1}}{∂x^{2}} (x′′, t) dx′′ dx′)\}

+ \frac{1}{h_{1}^{2}} ( ∫_{b_{1}−h_{1}}^{b_{1}} ∫_{x′}^{b_{1}} ∫_{b_{1}−h_{1}}^{b_{1}} ∫_{x′′′}^{x′′}p′′′_{1}(x′′′′) dx′′′′ dx′′′ dx′′ dx′)\frac{∂u_{1}}{∂x} (b_{1}, t).

Analogously as in the previous case one obtains:

(36)  h_{1}τ ∑_{t∈ω_{τ}^{+}} |χ̃_{11}(b_{1}, t)|^{2} ⩽ C(h_{1}^{4} + τ^{2}) ∥p_{1}∥_{H^{3}(Ω_{1})}^{2} ∥u_{1}∥_{H^{4,2}(Q_{1})}^{2},

(37)  h_{1}τ ∑_{t∈ω_{τ}^{+}} |χ̃_{1j}(b_{1}, t)|^{2} ⩽ Ch_{1}^{4} ∥p_{1}∥_{H^{3}(Ω_{1})}^{2} ∥u_{1}∥_{H^{4,2}(Q_{1})}^{2}, j = 0, 2, 3, 4.

From (29)–(37), analogous inequalities for ψ_{2}, χ_{2} and χ̃_{2} and the a priori estimate (28) one obtains the next result.

Theorem 3.2. Let p_{i} ∈ H^{3}(Ω_{i}), i = 1, 2, and let assumptions (7) and (8) hold. Let the functions γ_{i} and f_{i} be continuous and sufficiently smooth to ensure that the solution of IBVP (1)–(6) belongs to the space H^{4,2}. Let also the step sizes h_{1} and h_{2} be sufficiently small so that the a priori estimate (28) holds. Then the solution v of FDS (16)–(21) converges to the solution u of IBVP (1)–(6) in H_{h,τ}^{2,1} and the following convergence rate estimate holds true:

∥u − v∥_{H_{h,τ}^{2,1}} ⩽ C(h^{2} + τ)(1 + max_{i} ∥p_{i}∥_{H^{3}(Ω_{i})})∥u∥_{H^{4,2}}, h = max{h_{1}, h_{2}}.

Remark 3.1. For u ∈ H^{s,s/2}, 3.5 < s < 4, using Bramble–Hilbert lemma [1, 3] and methodology proposed in [4], one can obtain convergence rate O(h^{s−2}) assuming τ ≍ h^{2}. The same result holds for 2 < s < 3.5, but in this case a FDS with averaged data must be used, because the right-hand sides f_{i} may be discontinuous functions.

References

[1] J. H. Bramble, S. R. Hilbert, Bounds for a class of linear functionals with application to Hermite interpolation, Numer. Math. 16 (1971), 362–369.

[2] A. K. Datta, Biological and Bioenvironmental Heat and Mass Transfer, Marcel Dekker, 2002.

[3] T. Dupont, R. Scott, Polynomial approximation of functions in Sobolev spaces, Math. Comput. 34 (1980), 441–463.

[4] B. S. Jovanović, Finite Difference Method for Boundary Value Problems with Weak Solutions, Posebna izdanja Mat. Instituta, Vol. 16, Mat. Institut, Belgrade, 1993.

[5] B. S. Jovanović, P. P. Matus, V. S. Shcheglik, Finite difference schemes on nonuniform meshes for parabolic equation with variable coeffcients and generalized solutions, Dokl. Nats. Akad. Nauk Belarusi 42(6) (1998), 38-44 (Russian).

[6] B. S. Jovanović, L. G. Vulkov, On the convergence of finite difference schemes for the heat equation with concentrated capacity, Numer. Math. 89(4) (2001), 715–734.

[7] B. S. Jovanović, L. G. Vulkov, On the convergence of difference schemes for hyperbolic problems with concentrated data, SIAM J. Numer. Anal. 41(2) (2003), 516–538.

[8] B. S. Jovanović, L. G. Vulkov, Numerical solution of a parabolic transmission problem, IMA J. Numer. Anal. (2009), to appear

[9] J. L. Lions, E. Magenes, Non Homogeneous Boundary Value Problems and Applications, Springer-Verlag, Berlin and New York, 1972.

[10] A. A. Samarskiĭ, Theory of Difference Schemes, Marcel Dekker, New York – Basel, 2001.

University of Belgrade Faculty of Mathematics 11000 Belgrade Serbia

bosko@matf.bg.ac.yu

University of Rousse Department of Mathematics 7017 Rousse Bulgaria

vulkov@ami.ru.acad.bg