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 solution of the problem belongs to the Sobolev space W_{2}^{s,s/2} , s ⩽ 3. An almost second-order convergence rate estimate (with additional logarithmic factor) in the discrete W_{2}^{1,1/2} 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)  ∥u − v∥_{W_{p}^{k} (ω)} ⩽ Ch^{s−k} ∥u∥_{W_{p}^{s} (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, W_{p}^{k}(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

(1.2)  ∥u − v∥_{W_{p}^{k,k/2} (Q_{hτ} )} ⩽ C ( h^{s−k} + τ^{\frac{s−k}{2}} ) ∥u∥_{W_{p}^{s,s/2} (Q)} ,  s ⩾ k,

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(h^{3/2} )) is often observed, caused by the approximation of the boundary condition. Nevertheless, improved results are obtained in some cases, mainly for elliptic problems (see [4, 7]).

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.

In the present paper, for the FDS approximating initial-boundary value problem (IBVP) for a parabolic equation with variable coefficients and an oblique derivative boundary condition a second order error bound in the discrete W_{2}^{1,1/2} 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:

\frac{∂u}{∂t} + Lu = f, (x, t) = (x_{1}, x_{2}, t) ∈ Q, (2.1)

lu = 0, (x, t) ∈ Γ × (0, T) = ∂Ω × (0, T), (2.2)

u(x, 0) = u_{0}(x), x ∈ Ω. (2.3)

where

(2.4)  Lu := − ∑_{i,j=1}^{2} \frac{∂}{∂x_{i}} ( a_{ij} \frac{∂u}{∂x_{j}} ) , lu := ∑_{i,j=1}^{2} a_{ij} \frac{∂u}{∂x_{j}} cos(ν, x_{i}) + αu

and ν is the unit outward normal to Γ. We assume that the conditions of strong ellipticity are satisfied:

a_{ij} = a_{ij} (x) = a_{ji} , α = α(x),

c_{0} ∑_{i=1}^{2} ξ_{i}^{2} ⩽ ∑_{i,j=1}^{2} a_{ij} ξ_{i}ξ_{j} ⩽ c_{1} ∑_{i=1}^{2} ξ_{i}^{2}, ∀x ∈ Ω̄, ∀ξ ∈ ℝ^{2}, c_{i} = const. > 0, (2.5)

α_{0} ⩽ α(x) ⩽ α_{1}, α_{i} = const. > 0.

Let us denote Γ = ⋃ _{i=1}^{2} ⋃ _{k=0}^{1} Γ_{ik}, where Γ_{1k} = {k} × [0, 1], Γ_{2k} = [0, 1] × {k}, and Σ_{ik} = Γ_{ik} × [0, T].

We also assume that the generalized solution of the problem (2.1)–(2.3) belongs to the Sobolev space W_{2}^{s,s/2} (Q), 2 < s ⩽ 3, while the data satisfy the following smoothness conditions: a_{ij} ∈ W_{2}^{s−1} (Ω), α ∈ W_{2}^{s−3/2} (Γ_{ik}), α ∈ C(Γ), f ∈ W_{2}^{s−2,s/2−1} (Q) and u_{0} ∈ W_{2}^{s−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 ω̄_{τ} with mesh size τ on [0, T]. We also denote ω = ω̄ ∩Ω, ω_{τ} = ω̄_{τ} ∩(0, T), ω_{τ}^{−} = ω̄_{τ} ∩[0, T), ω_{τ}^{+} = ω̄_{τ} ∩ (0, T], γ = ω̄ ∩ Γ, γ̄_{ik} = ω̄ ∩ Γ_{ik}, γ_{ik} = {x ∈ γ̄_{ik} : 0 < x_{3−i} < 1}, γ_{ik}^{−} = {x ∈ γ̄_{ik} : 0 ⩽ x_{3−i} < 1}, γ_{ik}^{+} = {x ∈ γ̄_{ik} : 0 < x_{3−i} ⩽ 1}, γ_{ik}^{⋆} = γ̄_{ik} ∖ γ_{ik},

FINITE DIFFERENCE APPROXIMATION OF A PARABOLIC PROBLEM 51

γ^{⋆} = γ ∖ {_{⋃ i,k} γ_{ik} }, σ_{ik} = γ_{ik} × ω_{τ}^{+} , σ̄_{ik} = γ̄_{ik} × ω_{τ}^{+} , i = 1, 2, k = 0, 1, and Q̄_{hτ} = ω̄ × ω̄_{τ} .

The finite difference operators are defined in the usual manner [12]: v_{x_{i}} = (v^{+i} − v)/h, vx̄_{i} = (v − v^{−i} )/h, v_{t} = (v̂ − v)/τ, vt̄ = (v − v̌)/τ,

where v^{±i} (x, t) = v(x±he_{i}, t), e_{i} is the unit vector of the axis x_{i}, 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]:

T_{i}^{+} f(x, t) = ∫_{0}^{1} f(x + hx′e_{i}, t) dx′ = T_{i}^{−} f(x + he_{i}, t) = T_{i}f(x + \frac{h}{2}e_{i}, t),
T_{i}^{2±} f(x, t) = 2 ∫_{0}^{1}(1 − x′)f(x ± hx′e_{i}) dx′, i = 1, 2,
T_{t}^{+} f(x, t) = ∫_{0}^{1} f(x, t + τt′) dt′ = T_{t}^{−} f(x, t + τ) = T_{i}f(x, t + \frac{τ}{2}).

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

T_{i}^{+} ( \frac{∂u}{∂x_{i}} ) = u_{x_{i}} , T_{i}^{−} ( \frac{∂u}{∂x_{i}} ) = ux̄_{i} , T_{i}^{2} ( \frac{∂^{2}u}{∂x_{i}^{2}}) = u_{x̄_{i} x_{i}} , i = 1, 2,

T_{t}^{+} ( \frac{∂u}{∂t}) = u_{t}, T_{t}^{−} ( \frac{∂u}{∂t}) = ut̄.

We approximate the IBVP (2.1)–(2.3) with the following implicit FDS:

(3.1)  \begin{matrix}vt̄ + L_{h}v = f̃,  x ∈ ω̄, t ∈ ω_{τ}^{+} , \\ v(x, 0) = u_{0}(x),  x ∈ ω̄, \end{matrix}

where

L_{h}v = \{ \begin{matrix}−\frac{1}{2} ∑_{i,j=1}^{2} [(a_{ij} v_{x_{j}} )x̄_{i} + (a_{ij} vx̄_{j} )_{x_{i}} ], & x ∈ ω \\ \frac{2}{h}( − \frac{a_{11} +a_{11}^{+1}}{2} v_{x_{1}} − a_{12} \frac{v_{x_{2}} +vx̄_{2}}{2} + α̃v) − (a_{12} vx̄_{2} )_{x_{1}} \\ −(a_{21} v_{x_{1}} )x̄_{2} − \frac{1}{2} (a_{22} v_{x_{2}} )x̄_{2} − \frac{1}{2} (a_{22} vx̄_{2} )_{x_{2}} , & x ∈ γ_{10} \\ \frac{2}{h}[− \frac{a_{11} +a_{11}^{+1}}{2} v_{x_{1}} − a_{12}v_{x_{2}} − a_{21}v_{x_{1}} − \frac{a_{22} +a_{22}^{+2}}{2} v_{x_{2}} \\ +(α̃_{1} + α̃_{2} )v], & x = (0, 0) \\ \frac{2}{h}[− \frac{a_{11} +a_{11}^{+1}}{2} v_{x_{1}} − a_{12}vx̄_{2} + a_{21}v_{x_{1}} + \frac{a_{22} +a_{22}^{−2}}{2} vx̄_{2} \\ +(α̃_{1} + α̃_{2} )v] − 2 (a_{12} vx̄_{2} )_{x_{1}} − 2 (a_{21} v_{x_{1}} )x̄_{2} , & x = (0, 1) \\ and analogously at the other boundary nodes, & x ∈ γ ∖ γ̄_{10} \end{matrix}

f̃ = \{ \begin{matrix}T_{1}^{2}T_{2}^{2}T_{t}^{−} f, & x ∈ ω \\ T_{i}^{2±} T_{3−i}^{2} T_{t}^{−} f, & x ∈ γ_{i, 0.5∓0.5} \\ T_{1}^{2±} T_{2}^{2±} T_{t}^{−} f, & x = (0.5 ∓ 0.5, 0.5 ∓ 0.5) ∈ γ^{⋆} \end{matrix}

α̃ = T_{3−i}^{2} α ,  x ∈ γ_{i0} ∪ γ_{i1} ,  i = 1, 2

and

α̃_{i} = T_{i}^{2±} α , x ∈ γ^{⋆}, x_{i} = 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 Q̄_{hτ} and satisfies the following conditions

(4.1)  \begin{matrix}zt̄ + L_{h}z = ψ,  x ∈ ω̄, t ∈ ω_{τ}^{+} , \\ z(x, 0) = 0,  x ∈ ω̄, \end{matrix}

where

ψ = \{ \begin{matrix}ξt̄ + ∑ _{i,j=1}^{2} η_{ij, x̄_{i}} , & x ∈ ω \\ ξ̃t̄+ \frac{2}{h} η_{11} + \frac{2}{h} η_{12} + η̃_{21, x̄_{2}} + η̃_{22, x̄_{2}} + \frac{2}{h} ζ, & x ∈ γ_{10} \\ ˜ξ̃t̄ + \frac{2}{h} η̃_{11} + \frac{2}{h} η̃_{12} + \frac{2}{h} η̃_{21} + \frac{2}{h} η̃_{22} + \frac{2}{h} (ζ_{1} + ζ_{2}) , & x = (0, 0) \\ and analogously at the other boundary nodes, & x ∈ γ ∖ γ_{10}^{−} \end{matrix}

\begin{aligned}ξ & = u − T_{1}^{2}T_{2}^{2}u,  x ∈ ω \\ ξ̃ & = u − T_{i}^{2}T_{3−i}^{2±} u,  x ∈ γ_{3−i, 0.5∓0.5} \\ ˜ξ̃ & = u − T_{1}^{2±} T_{2}^{2±} u,  x = (0.5 ∓ 0.5, 0.5 ∓ 0.5) ∈ γ^{⋆} \\ η_{ij} & = T_{i}^{+} T_{3−i}^{2} T_{t}^{−} ( a_{ij} \frac{∂u}{∂x_{j}} ) − \frac{1}{2} (a_{ij} u_{x_{j}} + a_{ij}^{+i} u+i ; x̄_{j} ),  x ∈ ω \\ η̃_{ii} & = T_{i}^{+} T_{3−i}^{2±} T_{t}^{−} ( a_{ii} \frac{∂u}{∂x_{i}} ) − \frac{a_{ii} + a_{ii}^{+i}}{2} u_{x_{i}} ,  x ∈ γ_{3−i, 0.5∓0.5}^{−} \\ η̃_{i,3−i} & = \{ \begin{matrix}T_{i}^{+} T_{3−i}^{2+} T_{t}^{−} ( a_{i,3−i} \frac{∂u}{∂x_{3−i}} ) − a_{i,3−i} u_{x_{3−i}} , x ∈ γ_{3−i,0}^{−} \\ T_{i}^{+} T_{3−i}^{2−} T_{t}^{−} ( a_{i,3−i} \frac{∂u}{∂x_{3−i}} ) − a_{i,3−i}^{+i} u+i ; x̄_{3−i} , x ∈ γ_{3−i,1}^{−} \end{matrix} \end{aligned}

\begin{aligned}ζ & = (T_{i}^{2}α)u − T_{i}^{2}T_{t}^{−} (αu),  x ∈ γ_{3−i,0} ∪ γ_{3−i,1} \\ ζ_{i} & = (T_{i}^{2±} α)u − T_{i}^{2±} T_{t}^{−} (αu),  x ∈ γ^{⋆}, x_{i} = 0.5 ∓ 0.5. \end{aligned}

We define the following discrete inner products and norms:

[v, w] = h^{2} ∑_{x∈ω} v(x)w(x)+ \frac{h^{2}}{2} ∑_{x∈γ∖γ^{⋆}} v(x)w(x)+ \frac{h^{2}}{4} ∑_{x∈γ^{⋆}} v(x)w(x), |[v]|^{2} = [v, v],

[v, w)_{i} = h^{2} ∑_{x∈ω∪γ_{i0}} v(x)w(x) + \frac{h^{2}}{2}  ∑_{x∈γ_{3−i,0}^{−} ∪γ_{3−i,1}^{−}} v(x)w(x),  |[v∥_{i}^{2} = [v, v)_{i},

(v, w]_{i} = h^{2} ∑_{x∈ω∪γ_{i1}} v(x)w(x) + \frac{h^{2}}{2}  ∑_{x∈γ_{3−i,0}^{+} ∪γ_{3−i,1}^{+}} v(x)w(x),  ∥v]|_{i}^{2} = (v, v]_{i},

[v, w) = h^{2} ∑_{x∈ω∪γ_{10}^{−} ∪γ_{20}^{−}} v(x)w(x),  |[v∥^{2} = [v, v),  (v, w] = h^{2} ∑_{x∈ω∪γ_{11}^{+} ∪γ_{21}^{+}} v(x)w(x),
∥v]|^{2} = (v, v],  |[v]|_{W_{2}^{1} (ω̄)}^{2} = |[v]|^{2} + |[v_{x_{1}} ||_{1}^{2} + |[v_{x_{2}} ∥_{2}^{2},  |[v]|_{C(ω̄)} = max_{x∈ω̄}|v(x)|,

[v, w]γ̄_{ik} = h ∑_{x∈γ_{ik}} v(x)w(x) + \frac{h}{2} ∑_{x∈γ_{ik}^{⋆}} v(x)w(x), |[v]|2 ; γ̄_{ik} = [v, v]γ̄_{ik} ,

[v, w)_{γ_{ik}^{−}} = h ∑_{x∈γ_{ik}^{−}} v(x)w(x),  |[v||_{γ_{ik}^{−}}^{2} = [v, v)_{γ_{ik}^{−}} ,  ∥v∥_{γ_{ik}}^{2} = h ∑_{x∈γ_{ik}} v^{2}(x),

|v|_{W_{2}^{1/2} (γ_{ik}^{−} )}^{2} = h^{2} ∑_{x, x′∈γ_{ik}^{−} , x′≠x} [ \frac{v(x) − v(x′)}{x_{3−i} − x′_{3−i}} ]^{2}, |[v∥_{W_{2}^{1/2} (γ_{ik}^{−} )}^{2} = |v|_{W_{2}^{1/2} (γ_{ik}^{−} )}^{2}+|[v∥_{γ_{ik}^{−}}^{2} ,

|[v∥_{Ẅ_{2}^{1/2} (γ_{ik}^{−} )}^{2} = |v|_{W_{2}^{1/2} (γ_{ik}^{−} )}^{2} + h ∑_{x∈γ_{ik}^{−}} ( \frac{1}{x_{3−i} + h/2} + \frac{1}{1 − x_{3−i} − h/2} ) v^{2}(x),

∥v∥_{τ}^{2} = τ ∑_{t∈ω_{τ}^{+}} v^{2}(t), |[v∥_{i,hτ}^{2} = τ ∑_{t∈ω_{τ}^{+}} |[v(·, t)∥_{i}^{2},

∥v∥_{σ_{ik}}^{2} = τ ∑_{t∈ω_{τ}^{+}} ∥v(·, t)∥_{γ_{ik}}^{2} , |[v]|2 ; σ̄_{ik} = τ ∑_{t∈ω_{τ}^{+}} |[v(·, t)]|2 ; γ̄_{ik} ,

|[v∥_{L_{2} (ω_{τ}^{+} , Ẅ_{2}^{1/2} (γ_{ik}^{−} ))}^{2} = τ ∑_{t∈ω_{τ}^{+}} |[v(·, t)∥_{Ẅ_{2}^{1/2} (γ_{ik}^{−} )}^{2} ,

|v|_{W_{2}^{1/2} (ω̄_{τ} ,L_{2} (ω̄))}^{2} = τ^{2} ∑_{t, t′∈ω̄_{τ} , t′≠t} \frac{|[v(·, t) − v(·, t′)]|^{2}}{(t − t′)^{2}},

|[v]|_{Ẅ_{2}^{1/2} (ω̄_{τ} ,L_{2} (ω̄))}^{2} = |v|_{W_{2}^{1/2} (ω̄_{τ} ,L_{2} (ω̄))}^{2} + τ ∑_{t∈ω̄_{τ}} ( \frac{1}{t + τ/2} + \frac{1}{T − t + τ/2} ) |[v(·, t)]|^{2},

|[v]|_{L_{2} (ω_{τ}^{+} , W_{2}^{1} (ω̄))}^{2} = τ ∑_{t∈ω_{τ}^{+}} |[v(·, t)]|_{W_{2}^{1} (ω̄)}^{2},

|[v]|_{W_{2}^{1, 1/2} (Q_{hτ} )}^{2} = |[v]|_{L_{2} (ω_{τ}^{+} , W_{2}^{1} (ω̄))}^{2} + |v|_{W_{2}^{1/2} (ω̄_{τ} ,L_{2} (ω̄))}^{2} .

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 a_{ij} and α satisfy the assumptions from Section 2. Then, for a sufficiently small mesh step h, there exist positive constants C_{1} and C_{2} such that

C_{1}|[v]|_{W_{2}^{1} (ω̄)}^{2} ⩽ [L_{h}v, v] ⩽ C_{2}|[v]|_{W_{2}^{1} (ω̄)}^{2}.

Proof. The proof immediately follows from

[L_{h}v, v] = \frac{1}{2} ∑_{i=1}^{2} \{[a_{ii} v_{x_{i}} , v_{x_{i}} )_{i} + (a_{ii} vx̄_{i} , vx̄_{i} ]_{i} + [a_{i,3−i} v_{x_{3−i}} , v_{x_{i}} )

+ (a_{i,3−i} vx̄_{3−i} , vx̄_{i} ]\} + h ∑_{x∈γ∖γ^{⋆}} α̃ v^{2} + \frac{h}{2} ∑_{x∈γ^{⋆}} (α̃_{1} + α̃_{2} ) v^{2}

= \frac{1}{2} ∑_{i,j=1}^{2} \{[a_{ij} v_{x_{j}} , v_{x_{i}} ) + (a_{ij} vx̄_{j} , vx̄_{i} ]\} + \frac{h}{4} ∑_{i=1}^{2} ∑_{k=0}^{1} (−1)^{k}[a_{ii}^{+i} − a_{ii} , v_{x_{i}}^{2} )_{γ_{3−i, k}^{−}}

+ ∑_{x∈γ∖γ^{⋆}} α̃ v^{2} + \frac{h}{2} ∑_{x∈γ^{⋆}} (α̃_{1} + α̃_{2} ) v^{2}

and a discrete imbedding theorem [12]. □

Lemma 4.2. [2, 7] The following inequality holds true:

|[v, w_{x_{3−i}} )_{γ_{ik}^{−}} | ⩽ C|[v∥_{Ẅ_{2}^{1/2} (γ_{ik}^{−} )} |[w]|_{W_{2}^{1} (ω̄)}.

Lemma 4.3. [7] Let v be a mesh function on ω̄, then

|[v]|_{C(ω̄)} ⩽ C \sqrt{log \frac{1}{h}} |[v]|_{W_{2}^{1} (ω̄)}.

Lemma 4.4. [8] The solution of the FDS

vt̄ + L_{h}v = φ, (x, t) ∈ ω̄ × ω_{τ}^{+} ; v(x, 0) = 0, x ∈ ω̄.

satisfies the a priori estimate

|[v]|_{W_{2}^{1, 1/2} (Q_{hτ} )} ⩽ C( τ ∑_{t∈ω_{τ}^{+}} |[φ(·, t)]|_{−1}^{2} ) ^{1/2}

where

|[φ(·, t)]|_{−1} := sup_{w} \frac{[φ(·, t), w]}{|[w]|_{W_{2}^{1} (ω̄)}}.

Lemma 4.5. [8] The solution of the FDS

vt̄ + L_{h}v = ϕt̄, (x, t) ∈ ω̄ × ω_{τ}^{+} ; v(x, 0) = 0, x ∈ ω̄.

satisfies the a priori estimate

|[v]|_{W_{2}^{1, 1/2} (Q_{hτ} )} ⩽ C|[ϕ]|_{Ẅ_{2}^{1/2} (ω̄_{τ} , L_{2} (ω̄))} .

FINITE DIFFERENCE APPROXIMATION OF A PARABOLIC PROBLEM 55

Lemma 4.6. Let w ∈ W_{2}^{r}(Γ_{ik}), 0 < r ⩽ 0.5. Then

|T_{3−i}^{+} w|_{W_{2}^{1/2} (γ_{ik}^{−} )} ⩽ C(r)h^{r−1/2} |w|_{W_{2}^{r} (Γ_{ik} )} .

Proof. Without loss of generality let us set i = 1 and k = 0. Hence

|T_{2}^{+} w|_{W_{2}^{1/2} (γ_{10}^{−} )}^{2} = h^{2} ∑_{i=1}^{n−1} ∑_{j=1, j≠i}^{n−1} \frac{[T_{2}^{+} w(0, ih) − T_{2}^{+} w(0, jh)]^{2}}{(ih − jh)^{2}}

= 2h^{2} ∑_{i=1}^{n−1} ∑_{j=1}^{i−1} \frac{[T_{2}^{+} w(0, ih) − T_{2}^{+} w(0, jh)]^{2}}{(ih − jh)^{2}}

= 2h^{2} ∑_{i=1}^{n−1} ∑_{j=1}^{i−1} \{ \frac{1}{h^{2}} ∫_{ih}^{ih+h} ∫_{jh}^{jh+h} [w(0, x) − w(0, x′)] dx′dx\} ^{2} (ih − jh)^{−2}

= 2h^{2} ∑_{i=1}^{n−1} ∑_{j=1}^{i−1} \frac{1}{h^{4}} \{ ∫_{ih}^{ih+h} ∫_{jh}^{jh+h} \frac{[w(0, x) − w(0, x′)]^{2}}{(x − x′)^{1+2r}} dx′dx\}

× \{ ∫_{ih}^{ih+h} ∫_{jh}^{jh+h} \frac{(x − x′)^{1+2r}}{(ih − jh)^{2}} dx′dx\}

⩽ 2h^{−2} 2^{1+2r} h^{2}h^{2r−1} ∑_{i=1}^{n−1} ∑_{j=1}^{i−1} \{ ∫_{ih}^{ih+h} ∫_{jh}^{jh+h} \frac{[w(0, x) − w(0, x′)]^{2}}{(x − x′)^{1+2r}} dx′dx\}

= 2^{1+2r} h^{2r−1} 2 ∫_{0}^{1} ∫_{0}^{x} \frac{[w(0, x) − w(0, x′)]^{2}}{(x − x′)^{1+2r}} dx′dx = 2^{1+2r} h^{2r−1} |w|_{W_{2}^{r} (Γ_{10} )}^{2} .

□

Let us rearrange the summands in truncation error ψ in the following manner:

η̃_{ij} = η_{ij} + η′_{ij} , ξ̃ = ξ + ξ′, ˜ ξ̃ = ξ + ξ^{⋆},

where

η′_{ii} = ±\frac{h}{3} T_{i}^{+} T_{t}^{−} ( \frac{∂}{∂x_{3−i}} ( a_{ii} \frac{∂u}{∂x_{i}} ) ) , x ∈ γ_{3−i, 0.5∓0.5} ,

\begin{matrix}η′_{i,3−i} = ±\frac{h}{3} T_{i}^{+} T_{t}^{−} ( \frac{∂}{∂x_{3−i}} ( a_{i,3−i} \frac{∂u}{∂x_{3−i}} ) ) ∓ \frac{h}{2} T_{i}^{+} T_{t}^{−} ( a_{i,3−i} \frac{∂^{2}u}{∂x_{3−i}^{2}} ) \\ +\frac{h}{2} T_{i}^{+} T_{t}^{−} ( \frac{∂}{∂x_{i}} ( a_{i,3−i} \frac{∂u}{∂x_{3−i}} ) ) , x ∈ γ_{3−i, 0.5∓0.5} , \end{matrix}

ξ′ = ∓\frac{h}{3} T_{3−i}^{2} ( \frac{∂u}{∂x_{i}} ) , x ∈ γ_{i, 0.5∓0.5} ,

ξ^{⋆} = ∓\frac{h}{3} T_{2}^{2±} ( \frac{∂u}{∂x_{1}} ) ∓ \frac{h}{3} T_{1}^{2±} ( \frac{∂u}{∂x_{2}} ) , x = (0.5 ∓ 0.5, 0.5 ∓ 0.5) ∈ γ^{⋆}.

Using the boundary condition (2.2) we further obtain

ξ′ ; t̄ = λ_{i, x̄_{3−i}} + μ_{i} + ν_{i}, x ∈ γ_{i, 0.5∓0.5} ,

where

λ_{i} = ±\frac{h}{3} T_{3−i}^{+} T_{t}^{−} ( \frac{a_{i,3−i}}{a_{ii}} \frac{∂u}{∂t}) ,

μ_{i} = ∓\frac{h}{3} T_{3−i}^{2} T_{t}^{−} ( \frac{∂}{∂x_{3−i}} ( \frac{a_{i,3−i}}{a_{ii}} ) \frac{∂u}{∂t}) ,

ν_{i} = −\frac{h}{3} T_{3−i}^{2} T_{t}^{−} ( \frac{α}{a_{ii}} \frac{∂u}{∂t}) .

Similarly, for x = (0, 0), we obtain

ξ⋆ ; t̄ = \frac{2}{h} λ_{1} − \frac{2}{h} λ_{1}^{⋆} + μ_{1} + ν_{1} + \frac{2}{h} λ_{2} − \frac{2}{h} λ_{2}^{⋆} + μ_{2} + ν_{2},

where λ_{1} and λ_{2} are the same as before and

λ_{i}^{⋆} = ±\frac{h}{3} T_{t}^{−} ( \frac{a_{i,3−i}}{a_{ii}} \frac{∂u}{∂t}) ,

μ_{i} = −\frac{h}{3} T_{3−i}^{2+} T_{t}^{−} ( \frac{∂}{∂x_{3−i}} ( \frac{a_{i,3−i}}{a_{ii}} ) \frac{∂u}{∂t}) ,

ν_{i} = −\frac{h}{3} T_{3−i}^{2+} T_{t}^{−} ( \frac{α}{a_{ii}} \frac{∂u}{∂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

|[z]|_{W_{2}^{1, 1/2} (Q_{hτ} )} ⩽ C{ |[ξ]|_{Ẅ_{2}^{1/2} (ω̄_{τ} , L_{2} (ω̄))} + ∑_{i,j=1}^{2} |[η_{ij} ∥_{i,hτ} + ∑_{k=0}^{1} ∑_{i=1}^{2} ∥ζ∥_{σ_{ik}}

(4.2)  + h ∑_{k=0}^{1} ∑_{i,j=1}^{2} |[η′_{ij} ∥_{L_{2} (ω_{τ}^{+} , Ẅ_{2}^{1/2} (γ_{3−i,k}^{−} ))} + h ∑_{k=0}^{1} ∑_{i=1}^{2} |[λ_{i}∥_{L_{2} (ω_{τ}^{+} , Ẅ_{2}^{1/2} (γ_{ik}^{−} ))}

+h ∑_{k=0}^{1} ∑_{i=1}^{2} ( |[μ_{i}]|σ̄_{ik} +|[ν_{i}]|σ̄_{ik} ) + h\sqrt{log \frac{1}{h}} ∑_{i=1}^{2} ∑_{x∈γ^{⋆}} ( ∥ζ_{i}(x, ·)∥_{τ}+∥λ_{i}^{⋆}(x, ·)∥_{τ})\} .

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 ≍ h^{2}, i.e., c_{2}h^{2} ⩽ τ ⩽ c_{3}h^{2} for some positive constants c_{2} and c_{3}.

The term η_{ij} at the internal mesh nodes can be estimated in the same manner as in the case of the Dirichlet IBVP (see [6]):

(4.3) τ ∑_{t∈ω_{τ}^{+}} h^{2} ∑_{x∈ω∪γ_{i0}} η_{ij}^{2} ⩽ Ch^{2s−2} ∥a_{ij} ∥_{W_{2}^{s−1} (Ω)}^{2} ∥u∥_{W_{2}^{s,s/2} (Q)}^{2} , 2 < s ⩽ 3.

In boundary nodes η_{ii} can be decomposed in the following manner

η_{ii} = η_{ii,1} + η_{ii,2} + η_{ii,3} + η_{ii,4} , x ∈ γ_{3−i, 0.5∓0.5}^{−} ,

where

η_{ii,1} = T_{i}^{+} T_{t}^{−} ( a_{ii} \frac{∂u}{∂x_{i}} ) − (T_{i}^{+} a_{ii} )( T_{i}^{+} T_{t}^{−} \frac{∂u}{∂x_{i}} ) ,

η_{ii,2} = [ (T_{i}^{+} a_{ii} )− \frac{a_{ii} + a_{ii}^{+i}}{2} ]( T_{i}^{+} T_{t}^{−} \frac{∂u}{∂x_{i}} ) ,

η_{ii,3} = \frac{a_{ii} + a_{ii}^{+i}}{2} [( T_{i}^{+} T_{t}^{−} \frac{∂u}{∂x_{i}} ) − ( T_{i}^{+} \frac{∂u}{∂x_{i}} ) ],

η_{ii,4} = T_{i}^{+} T_{3−i}^{2±} T_{t}^{−} ( a_{ii} \frac{∂u}{∂x_{i}} ) − T_{i}^{+} T_{t}^{−} ( a_{ii} \frac{∂u}{∂x_{i}} ) ∓ \frac{h}{3} T_{i}^{+} T_{t}^{−} ( \frac{∂}{∂x_{3−i}} ( a_{ii} \frac{∂u}{∂x_{i}} ) ) .

The terms η_{ii,l} for l = 1, 2, 3 satisfy the same conditions as analogous terms in [6] whereby it follows that

(4.4) τ ∑_{t∈ω_{τ}^{+}} h^{2} ∑_{x∈γ_{3−i,0}^{−} ∪γ_{3−i,1}^{−}} η_{ii,l}^{2} ⩽ Ch^{2s−2} ∥a_{ii} ∥_{W_{2}^{s−1} (Ω)}^{2} ∥u∥_{W_{2}^{s,s/2} (Q)}^{2} , 2 < s ⩽ 3.

For s > 2.5 the term η_{ii,4} is a bounded linear functional of w = a_{ii} \frac{∂u}{∂x_{i}} ∈W_{2}^{s−1,(s−1)/2} which vanishes when w = 1, x_{1}, x_{2}, t. Using the Bramble–Hilbert lemma [3, 5] and properties of multipliers in Sobolev spaces [10] we obtain the following result

(4.5) τ ∑_{t∈ω_{τ}^{+}} h^{2} ∑_{x∈γ_{3−i,0}^{−} ∪γ_{3−i,1}^{−}} η_{ii,4}^{2} ⩽ Ch^{2s−2} ∥a_{ii} \frac{∂u}{∂x_{i}} ∥_{W_{2}^{s−1,(s−1)/2} (Q)}^{2}

⩽ Ch^{2s−2} ∥a_{ii} ∥_{W_{2}^{s−1} (Ω)}^{2} ∥u∥_{W_{2}^{s,s/2} (Q)}^{2} , 2.5 < s ⩽ 3.

Similarly, at the boundary nodes η_{i,3−i} can be decomposed in the following manner

η_{i,3−i} = η_{i,3−i,1} + η_{i,3−i,2} + η_{i,3−i,3} + η_{i,3−i,4} , x ∈ γ_{3−i, 0.5∓0.5}^{−} ,

where

η_{i,3−i,1} = T_{i}^{+} T_{3−i}^{2±} T_{t}^{−} ( a_{i,3−i} \frac{∂u}{∂x_{3−i}} ) − T_{i}^{+} T_{t}^{−} ( a_{i,3−i} \frac{∂u}{∂x_{3−i}} )

∓ \frac{h}{3} T_{i}^{+} T_{t}^{−} ( \frac{∂}{∂x_{3−i}} ( a_{i,3−i} \frac{∂u}{∂x_{3−i}} ) ) ,

η_{i,3−i,2} = T_{i}^{+} T_{t}^{−} ( a_{i,3−i} \frac{∂u}{∂x_{3−i}} ) − T_{t}^{−} ( a_{i,3−i} \frac{∂u}{∂x_{3−i}} )

− \frac{h}{2} T_{i}^{+} T_{t}^{−} ( \frac{∂}{∂x_{i}} ( a_{i,3−i} \frac{∂u}{∂x_{3−i}} ) ) ,

η_{i,3−i,3} = a_{i,3−i} [T_{t}^{−} ( \frac{∂u}{∂x_{3−i}} ) − T_{3−i}^{±} ( \frac{∂u}{∂x_{3−i}} ) ± \frac{h}{2} T_{i}^{+} T_{t}^{−} ( \frac{∂^{2}u}{∂x_{3−i}^{2}} ) ],
η_{i,3−i,4} = ±\frac{h}{2}[T_{i}^{+} T_{t}^{−} ( a_{i,3−i} \frac{∂^{2}u}{∂x_{3−i}^{2}} ) − a_{i,3−i} T_{i}^{+} T_{t}^{−} ( \frac{∂^{2}u}{∂x_{3−i}^{2}} ) ].

The terms η_{i,3−i,1} and η_{i,3−i,2} satisfy estimates analogous to (4.5). For s > 2.5 and a_{i,3−i} ∈ C(Ω̄) the term η_{i,3−i,3} is a bounded linear functional of u ∈ W_{2}^{s,s/2} which vanishes when w = 1, x_{1}, x_{2}, t, x_{1}^{2}, x_{1}x_{2}, x_{2}^{2}. Using the Bramble–Hilbert lemma and the Sobolev imbedding theorem [1] we obtain the following result

(4.6) τ ∑_{t∈ω_{τ}^{+}} h^{2} ∑_{x∈γ_{3−i,0}^{−} ∪γ_{3−i,1}^{−}} η_{i,3−i,3}^{2} ⩽ Ch^{2s−2} ∥a_{i,3−i} ∥_{C(Ω̄ )}^{2}∥u∥_{W_{2}^{s,s/2} (Q)}^{2}

⩽ Ch^{2s−2} ∥a_{i,3−i} ∥_{W_{2}^{s−1} (Ω)}^{2} ∥u∥_{W_{2}^{s,s/2} (Q)}^{2} , 2.5 < s ⩽ 3.

Term η_{i,3−i,4} can be estimated directly. Let us set i = 2 and x = (0, x_{2}) ∈ γ_{10}^{−}. Then

η_{2,1,4} (0, x_{2}, t) = \frac{h}{2} \frac{2}{hτ} ∫_{x_{2}}^{x_{2} +h} ( 1 − \frac{x′_{2}}{h} ) ∫_{t−τ}^{t} ∫_{x_{2}}^{x′_{2}} \frac{∂a_{21}}{∂x_{2}} (0, x′′_{2})\frac{∂^{2}u}{∂x_{1}^{2}}(0, x′_{2}, t′) dx′′_{2}dt′dx′_{2},

while for other boundary nodes, and also for i = 1, we have analogous integral representations. Hence

(4.7) τ ∑_{t∈ω_{τ}^{+}} h^{2} ∑_{x∈γ_{3−i,k}^{−}} η_{i,3−i,4}^{2} ⩽ Ch^{4} ∥ \frac{∂a_{i,3−i}}{∂x_{i}} ∥_{L_{2} (Γ_{ik} )}^{2} ∥ \frac{∂^{2}u}{∂x_{1}^{2}} ∥_{L_{2} (Σ_{ik} )}^{2}

⩽ Ch^{4} ∥a_{i,3−i} ∥_{W_{2}^{s−1} (Ω)}^{2} ∥u∥_{W_{2}^{s,s/2} (Q)}^{2} , s > 2.5 .

In such a way, from (4.3)-(4.7) one obtains

(4.8)  |[η_{ij} ∥_{i,hτ} ⩽ Ch^{s−1} ∥a_{ij} ∥_{W_{2}^{s−1} (Ω)} ∥u∥_{W_{2}^{s,s/2} (Q)} , 2.5 < s ⩽ 3.

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

(4.9) |[ξ]|_{Ẅ_{2}^{1/2} (ω̄_{τ} , L_{2} (ω̄))} ⩽ Ch^{s−1} \sqrt{log \frac{1}{h}} ∥a_{ij} ∥_{W_{2}^{s−1} (Ω)} ∥u∥_{W_{2}^{s,s/2} (Q)} , 2 < s ⩽ 3.

The term ζ on σ_{3−i,k} can be represented in the following manner:

ζ = (T_{i}^{2}α) (u − T_{i}^{2}T_{t}^{−} u) + [(T_{i}^{2}α) (T_{i}^{2}T_{t}^{−} u)− T_{i}^{2} (α T_{t}^{−} u)] = ζ_{01} + ζ_{02}.

For s > 2.5 and α ∈ C(Γ_{3−i,k} ) the term ζ_{01} is a bounded linear functional of u ∈ W_{2}^{s−1,(s−1)/2} (Σ_{3−i,k} ) which vanishes when u = 1 and u = x_{3−i} . Using the Bramble– Hilbert lemma and the Sobolev imbedding theorem we obtain the following result

(4.10)  ∥ζ_{01}∥_{σ_{3−i,k}} ⩽ Ch^{s−1} ∥α∥_{C(Γ_{3−i,k} )} ∥u∥_{W_{2}^{s−1,(s−1)/2}  (Σ_{3−i,k} )}
⩽ Ch^{s−1} ∥α∥_{W_{2}^{s−3/2} (Γ_{3−i,k} )} ∥u∥_{W_{2}^{s,s/2} (Q)} , 2.5 < s ⩽ 3.

The term ζ_{02} is a bounded linear functional of (α, T_{t}^{−} u)∈W_{q}^{r}(Γ_{3−i,k} )×W p_{\frac{2q}{q−2}} (Γ_{3−i,k} ),

q > 2, which vanishes when u = 1 or T_{t}^{−} u = 1. Using the bilinear version of the Bramble–Hilbert lemma, the Sobolev imbedding theorem and the Hölder inequality, after summation over the mesh γ_{3−i,k} we obtain the following result

∥ζ_{02}∥_{γ_{3−i,k}} ⩽ Ch^{r+p} ∥α∥_{W_{q}^{r} (Γ_{3−i,k} )} ∥T_{t}^{−} u∥_{W_{2q/(q−2)}^{p} (Γ_{3−i,k} )}

⩽ Ch^{r+p} ∥α∥_{W_{2}^{r+ \frac{1}{2} − \frac{1}{q}} (Γ_{3−i,k} )} ∥T_{t}^{−} u∥_{W_{2}^{p+ \frac{1}{q}} (Γ_{3−i,k} )} , 0 ⩽ r ⩽ 1, 1 − \frac{1}{q} < p ⩽ 1.

Summing over the mesh ω_{τ}^{+} and using imbedding and trace theorems we obtain

∥ζ_{02}∥_{σ_{3−i,k}} ⩽ Ch^{r+p} ∥α∥_{W_{2}^{r+ \frac{1}{2} − \frac{1}{q}} (Γ_{3−i,k} )} ∥u∥_{W_{2}^{p+ \frac{1}{q} , \frac{1}{2} (p+ \frac{1}{q} )} (Σ_{3−i,k} )}
⩽ Ch^{r+p} ∥α∥_{W_{2}^{r+ \frac{1}{2} − \frac{1}{q}} (Γ_{3−i,k} )} ∥u∥_{W_{2}^{p+ \frac{1}{2} + \frac{1}{q} , \frac{1}{2} (p+ \frac{1}{2} + \frac{1}{q} )} (Q)}
⩽ Ch^{r+p} ∥α∥_{W_{2}^{r+p− \frac{1}{2}} (Γ_{3−i,k} )} ∥u∥_{W_{2}^{r+p+1, \frac{1}{2} (r+p+1)} (Q)} .

Finally, setting r + p = s − 1, we get

(4.11)  ∥ζ_{02}∥_{σ_{3−i,k}} ⩽ Ch^{s−1} ∥α∥_{W_{2}^{s−3/2} (Γ_{3−i,k} )} ∥u∥_{W_{2}^{s,s/2} (Q)} ,  2 < s ⩽ 3.

From (4.10) and (4.11) it follows that

(4.12)  ∥ζ∥_{σ_{3−i,k}} ⩽ Ch^{s−1} ∥α∥_{W_{2}^{s−3/2} (Γ_{3−i,k} )} ∥u∥_{W_{2}^{s,s/2} (Q)} ,  2.5 < s ⩽ 3.

For x ∈ γ_{3−i,k}^{⋆} we set

ζ_{i} = (T_{i}^{2±} α) (u − T_{t}^{−} u) + [(T_{i}^{2±} α) (T_{t}^{−} u)− T_{i}^{2±} (α T_{t}^{−} u)] = ζ_{i1} + ζ_{i2}.

For r > 0.5 and α ∈ C(Γ_{3−i,k} ) the term ζ_{i1} is a bounded linear functional of u ∈ W_{2}^{r}(0, T) which vanishes when u = 1. Using the Bramble–Hilbert lemma and the Sobolev imbedding theorem, after summation over the mesh ω_{τ}^{+} we obtain

∥ζ_{i1}∥_{τ} ⩽ Cτ^{r}∥α∥_{C(Γ_{3−i,k} )} ∥u(x, ·)∥_{W_{2}^{r} (0,T)}
⩽ Ch^{2r} ∥α∥_{W_{2}^{2r−1/2} (Γ_{3−i,k} )} ∥u∥_{W_{2}^{2r+1,r+1/2}  (Q)} ,  0.5 < r ⩽ 1,

whereby, setting 2r + 1 = s,

(4.13)  ∥ζ_{i1}∥_{τ} ⩽ Ch^{s−1} ∥α∥_{W_{2}^{s−3/2} (Γ_{3−i,k} )} ∥u∥_{W_{2}^{s,s/2} (Q)} ,  2 < s ⩽ 3.

For i = 1 and x = (0, 0) we have

ζ_{12}(0, 0, t) = \frac{2}{h} ∫_{0}^{h} ( 1 − \frac{x′_{1}}{h} ) α(x′_{1}, 0) [T_{t}^{−} u(0, 0, t) − T_{t}^{−} u(x′_{1}, 0, t)]dx′_{1}

= \frac{2}{h} ∫_{0}^{h} ∫_{0}^{x′_{1}} ( 1 − \frac{x′_{1}}{h} ) α(x′_{1}, 0) \frac{∂(T_{t}^{−} u)}{∂x_{1}}(x′′_{1}, 0, t) dx′′_{1}dx′_{1}.

Analogous representations hold for other nodes x ∈ γ^{⋆} and also for i = 2. Hence

|ζ_{i2}| ⩽ Ch ∥α∥_{C(Γ_{3−i,k} )} ∥ \frac{∂(T_{t}^{−} u)}{∂x_{i}}∥_{C(Ω̄ )} ⩽ Ch ∥α∥_{W_{2}^{s−3/2} (Γ_{3−i,k} )} ∥T_{t}^{−} u∥_{W_{2}^{s} (Ω)} , s > 2,

whereby, after summation over the mesh ω_{τ}^{+} , one obtains

(4.14)  ∥ζ_{i2}∥_{τ} ⩽ Ch ∥α∥_{W_{2}^{s−3/2} (Γ_{3−i,k} )} ∥u∥_{W_{2}^{s,s/2} (Q)} ,  s > 2.

In such a way, for x ∈ γ_{3−i,k}^{⋆} , from (4.13) and (4.14) we get

(4.15)  ∥ζ_{i}∥_{τ} ⩽ Ch ∥α∥_{W_{2}^{s−3/2} (Γ_{3−i,k} )} ∥u∥_{W_{2}^{s,s/2} (Q)} ,  s > 2.

Let us now estimate |[η′_{ij} ∥_{L_{2} (ω_{τ}^{+} , Ẅ_{2}^{1/2} (γ_{3−i,k}^{−} ))} . Using Lemma 4.6 we immediately obtain

|η′_{ii} |_{W_{2}^{1/2} (γ_{3−i,k}^{−} )} ⩽ Ch^{r+1/2} |T_{t}^{−} ( \frac{∂}{∂x_{3−i}} ( a_{ii} \frac{∂u}{∂x_{i}} ) ) |_{W_{2}^{r} (Γ_{3−i,k} )}

⩽ Ch^{r+1/2} ∥T_{t}^{−} ( \frac{∂}{∂x_{3−i}} ( a_{ii} \frac{∂u}{∂x_{i}} ) ) ∥_{W_{2}^{r+1/2} (Ω)} , 0 < r ⩽ 0.5.

Using the inequality [11]

∥F ∥_{L_{2} (0,ε)} ⩽ C \{ \begin{matrix}ε^{r} ∥F ∥_{W_{2}^{r} (0,1)} , & 0 < r < 0.5, \\ ε^{1/2} log \frac{1}{ε} ∥F ∥_{W_{2}^{1/2} (0,1)} , & r = 0.5, \\ ε^{1/2} ∥F ∥_{W_{2}^{1/2} (0,1)} , & r > 0.5, \end{matrix}

where 0 < ε < 1, we obtain

h ∑_{x∈γ_{3−i,k}^{−}} ( \frac{1}{x_{i} + h/2} + \frac{1}{1 − x_{i} − h/2} ) (η′_{ii})^{2}

⩽ Ch^{2r+1} log \frac{1}{h} ∥T_{t}^{−} ( \frac{∂}{∂x_{3−i}} ( a_{ii} \frac{∂u}{∂x_{i}} ) ) ∥_{W_{2}^{r} (Γ_{3−i,k}^{−} )}^{2}

⩽ Ch^{2r+1} log \frac{1}{h} ∥T_{t}^{−} ( \frac{∂}{∂x_{3−i}} ( a_{ii} \frac{∂u}{∂x_{i}} ) ) ∥_{W_{2}^{r+1/2} (Ω)}^{2} , 0 < r < 0.5

and

h ∑_{x∈γ_{3−i,k}^{−}} ( \frac{1}{x_{i} + h/2} + \frac{1}{1 − x_{i} − h/2} ) (η′_{ii})^{2}

⩽ Ch^{2} log^{3} \frac{1}{h} ∥T_{t}^{−} ( \frac{∂}{∂x_{3−i}} ( a_{ii} \frac{∂u}{∂x_{i}} ) ) ∥_{W_{2}^{1} (Ω)}^{2} .

From the obtained inequalities, summing over the mesh ω_{τ}^{+} , denoting r + 2.5 = s and using properties of multipliers in Sobolev spaces, we immediately obtain

(4.16)  ∥η′_{ii} ∥_{L_{2} (ω_{τ}^{+} ,Ẅ_{2}^{1/2} (γ_{3−i,k}^{−} ))} ⩽ Ch^{s−2} \sqrt{log \frac{1}{h}} ∥a_{ii} ∥_{W_{2}^{s−1} (Ω)} ∥u∥_{W_{2}^{s,s/2} (Q)} ,

for 2.5 < s < 3 and

(4.17)  ∥η′_{ii} ∥_{L_{2} (ω_{τ}^{+} ,Ẅ_{2}^{1/2} (γ_{3−i,k}^{−} ))} ⩽ Ch (log \frac{1}{h})^{3/2} ∥a_{ii} ∥_{W_{2}^{2} (Ω)} ∥u∥_{W_{2}^{3,3/2} (Q)} .

Analogous estimates hold for η′_{i,3−i} substituting a_{ii} with a_{i,3−i} .

The term λ_{i} has a similar structure as η′_{ij} . Handling it in the same manner we obtain for 2.5 < s < 3

∥λ_{i}∥_{L_{2} (ω_{τ}^{+} ,Ẅ_{2}^{1/2} (γ_{ik}^{−} ))} ⩽ Ch^{s−2} \sqrt{log \frac{1}{h}} ∥ \frac{a_{i,3−i}}{a_{ii}} ∥_{M(W_{2}^{s−2} (Ω))} ∥u∥_{W_{2}^{s,s/2} (Q)} ,

where ∥ · ∥_{M(W_{2}^{s−2} (Ω))} is the norm in the space of multipliers in W_{2}^{s−2} (Ω) [10]. Further, for 2.5 < s < 3 and q > 2, using the properties of multipliers in Sobolev spaces and the assumptions (2.4), we obtain

∥ \frac{a_{i,3−i}}{a_{ii}} ∥_{M(W_{2}^{s−2} (Ω))} ⩽ C ∥ \frac{a_{i,3−i}}{a_{ii}} ∥_{W_{q}^{1} (Ω)} ⩽ C( ∥a_{i,3−i} ∥_{W_{q}^{1} (Ω)} + ∥a_{ii} ∥_{W_{q}^{1} (Ω)} )
⩽ C( ∥a_{i,3−i} ∥_{W_{2}^{s−1} (Ω)} + ∥a_{ii} ∥_{W_{2}^{s−1} (Ω)} ) .

In such a way, we finally obtain

(4.18) ∥λ_{i}∥_{L_{2} (ω_{τ}^{+} ,Ẅ_{2}^{1/2} (γ_{ik}^{−} ))} ⩽ Ch^{s−2} \sqrt{log \frac{1}{h}}

× ( ∥a_{ii} ∥_{W_{2}^{s−1} (Ω)} + ∥a_{i,3−i} ∥_{W_{2}^{s−1} (Ω)} ) ∥u∥_{W_{2}^{s,s/2} (Q)} , 2.5 < s < 3

and analogously

(4.19) ∥λ_{i}∥_{L_{2} (ω_{τ}^{+} ,Ẅ_{2}^{1/2} (γ_{ik}^{−} ))} ⩽ Ch (log \frac{1}{h})^{3/2}

× ( ∥a_{ii} ∥_{W_{2}^{2} (Ω)} + ∥a_{i,3−i} ∥_{W_{2}^{2} (Ω)} ) ∥u∥_{W_{2}^{3,3/2} (Q)} , s = 3.

For r > 0.5 and a_{ii} , a_{i,3−i} ∈ C(Ω̄) the term λ_{i}^{⋆} is a bounded linear functional of u ∈ W_{2}^{r}(0, T) which vanishes when u = 1. Using the Bramble–Hilbert lemma and handling it in the same manner as in the case of ζ_{i1} we obtain

(4.20)  ∥λ_{i}^{⋆}∥_{τ} ⩽ Ch^{s−2} ∥a_{i,3−i} ∥_{W_{2}^{s−1} (Ω)} ∥u∥_{W_{2}^{s,s/2} (Q)} ,  2 < s ⩽ 3.

The terms ν_{i} and μ_{i} can be estimated directly:

(4.21)  \begin{aligned}|[ν_{i}]|σ̄_{ik} ⩽ Ch ∥ \frac{α}{a_{ii}} \frac{∂u}{∂t} ∥_{L_{2} (Σ_{ik} )} & ⩽ Ch ∥ \frac{α}{a_{ii}} ∥_{C(Γ_{ik} )} ∥u∥_{W_{2}^{s,s/2} (Q)} \\ & ⩽ Ch ∥u∥_{W_{2}^{s,s/2} (Q)} ,  s > 2.5 \end{aligned}

and

(4.22) |[μ_{i}]|σ̄_{ik} ⩽ Ch ( ∥a_{ii} ∥_{W_{2}^{s−1} (Ω)} + ∥a_{i,3−i} ∥_{W_{2}^{s−1} (Ω)} ) ∥u∥_{W_{2}^{s,s/2} (Q)} ,  s > 2.5

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 ≍ h^{2}. Then the FDS (3.1) converges in the norm W_{2}^{1,1/2} (Q_{hτ}) and the convergence rate estimates

|[z]|_{W_{2}^{1,1/2} (Q_{hτ} )} ⩽ Ch^{s−1} \sqrt{log \frac{1}{h}}

× ( 1 + max_{i,j}∥a_{ij} ∥_{W_{2}^{s−1} (Ω)} + max_{i,k}∥α∥_{W_{2}^{s−3/2} (Γ_{ik} )} ) ∥u∥_{W_{2}^{s,s/2} (Q)} , 2.5 < s < 3

and

|[z]|_{W_{2}^{1,1/2} (Q_{hτ} )} ⩽Ch^{2} (log \frac{1}{h})^{\frac{3}{2}} ( 1+max_{i,j}∥a_{ij} ∥_{W_{2}^{2} (Ω)} +max_{i,k}∥α∥_{W_{2}^{3/2} (Γ_{ik} )} ) ∥u∥_{W_{2}^{3,3/2} (Q)}

hold.

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 ≍ h^{2}, 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]).

References

[1] R. A. Adams, J. J. F. Fournier, Sobolev Spaces, 2nd ed., Pure Appl. Math. Ser. 140, Elsevier/Academic Press, Amsterdam, 2003.

[2] V. B. Andreev, Stability of difference schemes for elliptic equations with respect to the Dirichlet boundary conditions, Zh. Vychisl. Mat. Mat. Fiz. 12 (1972), 598–611 (in Russian).

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

[4] M. D. Chkhartishvili, G. K. Berikelashvili, On the convergence in W_{2}^{1} of difference solution of elliptic equation with mixed boundary conditions, Bull. Acad. Sci. Georgia 148:2 (1993), 180–184 (in Russian).

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

[6] B. S. Jovanović, The Finite Difference Method for Boundary-Value Problems with Weak Solutions, Posebna izdanja 16, Matematički institut, Beograd, 1993.

[7] B. S. Jovanović, B. Z. Popović, Convergence of a finite difference scheme for the third boundary value problem for elliptic equation with variable coefficients, Comput. Methods Appl. Math. 1:4 (2001), 356–366.

[8] 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.

[9] , Finite difference approximation of strong solutions of a parabolic interface problem on disconnected domains, Publ. Inst. Math. 84(98) (2008), 37–48.

[10] V. G. Maz’ya, T. O. Shaposhnikova, Theory of Multipliers in Spaces of Differentiable Functions, Monographs and Studies in Mathematics 23, Pitman, Boston, Mass., 1985.

[11] L. A. Oganesyan, L. A. Rukhovets, Variational-Difference Methods for Solution of Elliptic Equations, AS Arm., Erevan, 1979 (in Russian).

[12] A. A. Samarskiĭ, The Theory of Difference Schemes, Marcel Dekker, New York, 2001.

[13] A. A. Samarskiĭ, R. D. Lazarov, V. L. Makarov, Difference Schemes for Differential Equations with Generalized Solutions, Vyshaya Shkola, Moscow, 1987 (in Russian).

[14] E. E. Süli, B. S. Jovanović, L. D. Ivanović, On the construction of finite difference schemes approximating generalized solutions, Publ. Inst. Math. 37(51) (1985), 123–128.

University of Belgrade (Received 23 03 2013) Faculty of Mathematics (Revised 01 04 2013) 11000 Belgrade, Serbia

bosko@matf.bg.ac.rs
zorica.milovanovic@gmail.com