����������� ������ 50 (1998), 81–89

UDK 519.633.6

���������� ������ ��� research paper

CONVERGENCE OF A FINITE DIFFERENCE METHOD FOR THE THIRD BVP FOR POISSON’S EQUATION

Boško S. Jovanović, Branislav Z. Popović

Abstract. In this paper we study the convergence of finite difference schemes to weak solutions of the third boundary value problem for Poisson’s equation on the unit square. Using the theory of interpolation of function spaces, we obtain error estimates in a discrete W_{2}^{1} Sobolev norm consistent, or “almost” consistent, with the smoothness of the data.

1. Introduction

For a class of finite difference schemes (FDS) for elliptic boundary value problems (BVP), estimates of the convergence rate consistent with the smoothness of the data:

∥u − v∥_{W_{p}^{k}(ω)} ⩽ Ch^{s−k}∥u∥_{W_{p}^{s}(Ω)},  s ⩾ k,  (1)

are of major interest (see [13], [7]). Here u = u(x) denotes the solution of the BVP, v denotes the solution of corresponding FDS, h is the discretisation parameter, W_{p}^{k}(ω) denotes the discrete Sobolev space, and C is a positive generic constant, independent of h and u.

A standard technique for the derivation of such estimates (see [7], [10], [18]) is based on the Bramble–Hilbert lemma (see [4], [6]). In particular, for the third BVP estimates of type (1) were derived in [5], [12], [15] and [16]. In this paper we, firstly, obtain better convergence then in [12], [15] or [16] and, secondly, expose an alternative technique, based on the theory of interpolation of Banach spaces (see [3], [8], [9], [11], [19]).

AMS Subject Classification: 65N15, 46B70

Keywords and phrases: Third Boundary Value Problems, Finite Difference Schemes, Sobolev Spaces, Interpolation of Function Spaces, Convergence Rate Estimates.

Supported by MST of Serbia, grant number 04M03 / C

2. Boundary value problem and its approximation

As a model problem let us consider the third boundary value problem (TBVP) for the Poisson’s equation in the domain Ω = (0, 1)^{2}:

\begin{aligned}L(u) & ≡ − ∑_{i=1}^{2} \frac{∂^{2}u}{∂x_{i}^{2}} = f in Ω, \\ l(u) & ≡ \frac{∂u}{∂n} + σu = 0 on Γ = ∂Ω , \end{aligned}  (2)

where \frac{∂u}{∂n} = ∑_{i=1}^{2} \frac{∂u}{∂x_{i}} cos (ν, x_{i}) and ν—the unit outward normal to ∂Ω.

We assume that the generalized solution of the problems (2) belongs to the Sobolev space W_{2}^{s}(Ω), 1 < s ⩽ 3, and that the right-hand side f(x) ∈ W_{2}^{s−2}(Ω). Then, the coefficient σ belongs to the corresponding space of multipliers [14], i.e. σ ∈ M (W_{2}^{s−1/2}(Γ) ↦ W_{2}^{s−3/2}(Γ)). Sufficient conditions for this are: σ ∈ L_{2}(Γ)

for 1 < s ⩽ \frac{3}{2} and σ ∈ W_{2}^{s− \frac{3}{2}} (Γ) for s > \frac{3}{2}. Also, let the condition σ(x) ≥ σ_{0} > 0 be satisfied. We shall assume that the other additional conditions which are possible and which guarantee the existence of the solution u ∈ W_{2}^{s}(Ω) are also satisfied (consistency conditions of input data at the vertices of the domain Ω).

Let ω̄ be a uniform mesh in Ω with the step size h = \frac{1}{n}, ω = ω̄ ∩ Ω, γ = ω̄ ∩ Γ, γ_{iα} = {x ∈ ω̄ : x_{i} = α = const, 0 < x_{3−i} < 1}, γ̃_{iα} = {x ∈ ω̄ : x_{i} = α = const, 0 ⩽ x_{3−1} < 1}, γ̄_{iα} = {x ∈ ω̄ : x_{i} = α = const, 0 ⩽ x_{3−1} ⩽ 1}, γ_{iα}^{⋆} = γ̄_{iα} ∩ γ_{iα}, ω_{i} = ω ∪ γ_{i0}, γ^{⋆} = γ \ {⋃_{i,k} γ_{ik}}, i = 1, 2; k = 0, 1. The finite difference

operators are defined in the following maner: v_{x_{i}} = (v^{+i} − v)/h, vx̄_{i} = (v − v^{−i})/h, where v^{±i}(x) = v(x ± hr_{i}), r_{i} is the unit vector of the axis x_{i}.

We define the following scalar products and discrete norms:

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

[u, v]_{i} = h^{2} ∑_{x∈ω_{i}} u(x)v(x) + \frac{h^{2}}{2} ∑_{x∈γ\(γ_{i0}∪γ̄_{i1})} u(x)v(x);

[u, v]γ̄_{iα} = h ∑_{x∈γ_{iα}} u(x)v(x) + \frac{h}{2} ∑_{x∈γ_{iα}^{⋆}} u(x)v(x); |[v]|γ̄_{iα} = [v, v]1/2 ; γ̄_{iα};

[u, v]γ̃_{iα} = h ∑_{x∈γ̃_{iα}} u(x)v(x); |[v]|γ̃_{iα} = [v, v]1/2 ; γ̃_{iα};

|[v]| = |[v]|_{L_{2}(ω̄)} = |[v]|_{W_{2}^{0}(ω̄)} = [u, v]^{1/2}; |[v]|_{i} = [v, v]_{i}^{1/2};

|[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)|.

We also define the Steklov smoothing operators with the step size h:

T_{i}^{+}f(x) = ∫_{0}^{1} f(x + htr_{i}) dt = T_{i}^{−}f(x + hr_{i}) = T_{i}f(x + \frac{h}{2}r_{i}),

T_{i}^{2}f(x) = ∫_{−1}^{1}(1 − |t|)f(x + htr_{i}) dt and
T_{i±}^{2}f(x) = 2 ∫_{0}^{1}(1 − t)f(x ± htr_{i}) dt,

where i = 1, 2. These operators commute and transform derivatives to differences:

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 .

We approximate TBVP (2) with the FDS:

L_{h}v = g(x),  (3)

where L_{h}(v) = L_{h,1}v + L_{h,2}v,

L_{h,i}v = \{ \begin{matrix}−\frac{2}{h} (v_{x_{i}} − σ̄v) , & x ∈ γ̄_{i0}, \\ −v_{x_{i}x̄_{i}}, & x ∈ ω ∪ γ_{3−i,0} ∪ γ_{3−i,1}, \\ \frac{2}{h} (vx̄_{i} + σ̄v) , & x ∈ γ̄_{i1}, \end{matrix}  (4)

g(x) is equal to:

T_{1}^{2}T_{2}^{2}f , x ∈ ω; T_{i+}^{2}T_{3−i}^{2}f , x ∈ γ_{i0}; T_{i−}^{2}T_{3−i}^{2}f , x ∈ γ_{i1};
T_{1±}^{2}T_{2±}^{2}f, x = (0.5 ∓ 0.5, 0.5 ∓ 0.5) ∈ γ^{⋆}

and

σ̄ = T_{i}^{2}σ , x ∈ γ_{3−i,0} ∪ γ_{3−i,1} ,
σ̄_{00} = T_{i+}^{2}σ(0, 0), σ̄_{11} = T_{i−}^{2}σ(1, 1) (in L_{h,i}v) i = 1, 2 ,
σ̄_{01} = T_{1+}^{2}σ(0, 1) (in L_{h,2}v), σ̄_{01} = T_{2−}^{2}σ(0, 1) (in L_{h,1}v) ,
σ̄_{10} = T_{1−}^{2}σ(1, 0) (in L_{h,2}v), σ̄_{10} = T_{2+}^{2}σ(1, 0) (in L_{h,1}v) .

Let u be the solution of the TBVP (2), and v the solution of the FDS (3). For s > 1, the error of the FDS (3) z = u − v is defined and satisfies the condition:

L_{h}z = ψ,  (5)

where

ψ = \{ \begin{matrix}∑_{i=1}^{2} η_{i,x_{i}x̄_{i}}, & x ∈ ω, \\ η̄_{3−i,x_{3−i}x̄_{3−i}} + \frac{2}{h}(η_{i,x_{i}} + ζ_{i}), & x ∈ γ_{i0}, i = 1, 2, \\ \frac{2}{h} ∑_{i=1}^{2}(η̄_{i,x_{i}} + ζ̄_{i}), & x = (0, 0), \end{matrix}  (6)

and analogously in the other boundary nodes. Here

\begin{aligned}η_{i} & = T_{3−i}^{2}u − u , x ∈ ω, \\ η̄_{i} & = T_{(3−i)±}^{2}u − u x ∈ γ_{3−i,0.5∓0.5}, \\ ζ_{i} & = (T_{3−i}^{2}σ) u − T_{3−i}^{2}(σu), x ∈ γ_{ik}, \\ ζ̄_{i} & = (T_{(3−i)+}^{2}σ) u − T_{(3−i)+}^{2}(σu), x = (0, 0), \end{aligned}  (7)

and analogously in the other nodes x ∈ γ^{⋆}.

3. Stability of the finite difference scheme

Stability of the scheme (3) for s ∈ (1, 3] is proved in [12] and [15]. However, we need a new a priori estimate specially for the case s = 3. Note that in this case σ(x) ∈ W_{2}^{3/2}(Γ) and C(Γ) ⊂ W_{2}^{3/2}(Γ) (see [1]). It is not so hard to prove the following lemma:

Lemma 1. Let v denote a mesh-function on ω̄, operator L_{h} defined by (4) and 0 ⩽ σ_{0} ⩽ σ(x) ⩽ σ_{1}. Then there exist positive constants C_{1} and C_{2} and following inequalities hold:

C_{1}|[v]|^{2} ⩽ [L_{h}v, v] ⩽ C_{2}|[v]|^{2}.

Using (6) and (7), we can transform [L_{h}z, z] into

[L_{h}z, z] = −h^{2} ∑_{x∈ω_{1}} η_{1,x_{1}} z_{x_{1}} − h^{2} ∑_{x∈ω_{2}} η_{2,x_{2}} z_{x_{2}} − \frac{h^{2}}{2} ∑_{k=0}^{1} ∑_{i=1}^{2} ∑_{x∈γ̃_{3−i,k}} η̄_{i,3−i,x_{i}} z_{x_{i}}
+ I_{10}(η̄_{22}) + I_{11}(η̄_{22}) + I_{20}(η̄_{11}) + I_{21}(η̄_{11})
+ h ∑_{k=0}^{1} ∑_{i=1}^{2} ∑_{x∈γ_{ik}} ζ_{i} z + \frac{h}{2} ∑_{x∈γ^{⋆}}(ζ̄_{1} + ζ̄_{2}) z,  (8)

where η̄_{k} = η̄_{k1} + η̄_{k2}, k = 1, 2 and, for example in the node (0, x_{2}),

\begin{matrix}η̄_{21} = \frac{2}{h} ∫_{0}^{h}∫_{0}^{x′_{1}}∫_{0}^{x′′_{1}} (1 − \frac{x′_{1}}{h} ) \frac{∂^{2}u(x′′′_{1}, x_{2})}{∂x_{1}^{2}} dx′′′_{1} dx′′_{1} dx′_{1} \\ η̄_{22} = \frac{h}{3} σ(x_{2}) u(0, x_{2}) \end{matrix}  (9)

and

I_{10}(η̄_{22}) = h ∑_{x∈γ_{10}} η̄_{22,x_{2}x̄_{2}}z + η̄_{22,x_{2}}(0, 0) z(0, 0) − η̄_{22,x̄_{2}}(0, 1) z(0, 1)  (10)

Analogously we define the other I_{ik}.

Let us define the operator Λ_{i} on γ̄_{3−i,α} as

Λ_{i}v = \{ \begin{matrix}−\frac{2}{h}v_{x_{i}}, & x_{i} = 0, \\ −v_{x_{i}x̄_{i}}, & 0 < x_{i} < 1, \\ \frac{2}{h}vx̄_{i}, & x_{i} = 1. \end{matrix}  (11)

The operator Λ_{i} is self-adjoint with respect to the inner product [u, v]γ̄_{3−i,α}. The eigenvalues of the operator Λ_{i} are (see [7])

λ_{k} = \frac{4}{h^{2}} sin^{2} \frac{kπh}{2} , k = 0, 1,…, n,

and its corresponding eigenfunction are ψ_{k} = ψ_{k}(x_{i}) = cos kπx_{i}. Since λ_{0} = 0, the operator Λ_{i} is non-negative. Using (9), (10) and (11) we obtain

I_{10}(η̄_{22}) = [Λ_{2}η̄_{22}, z]γ̄_{10} = [Λ_{2}^{3/4}η̄_{22}, Λ_{2}^{1/4}z]γ̄_{10} |[Λ_{2}^{3/4}η̄_{22}]|γ̄_{10}|[Λ_{2}^{1/4}z]|γ̄_{10}

and analogously for the other I_{ik} at γ̄_{ik}.

Convergence of a finite difference method … 85

Let us prove the following assertion:

Lemma 2. Let v be a mesh-function on ω̄ and Λ_{i} defined by (11). Then |[Λ_{i}^{1/4}v]|γ̄_{3−i,α} ⩽ C |[v]|_{W_{2}^{1}(ω̄)}.

Proof. If we take i = 2 (proof in the other case is the same), the function v can be presented in the form

v(x_{1}, x_{2}) = ∑_{l=0}^{n} ∑_{k=0}^{n} a_{kl}ψ_{k}(x_{1})ψ_{l}(x_{2}) = ∑_{l=0}^{n} C_{l}(x_{1})ψ_{l}(x_{2})

where ψ_{k}(x_{1}) = cos kπx_{1}, ψ_{l}(x_{2}) = cos lπx_{2} and then (see [7])

\begin{matrix}|[v]|2 ; γ̄_{1,x_{1}} = \frac{1}{2} \begin{matrix}n \\ ∑ & ′ \\ l=0 \end{matrix} C_{l}^{2}(x_{1}) ≡ \frac{1}{2} ∑_{l=1}^{n−1} C_{l}^{2}(x_{1}) + \frac{1}{4} (C_{0}^{2}(x_{1}) + C_{n}^{2}(x_{1})) \\ |[Λ_{2}^{1/2}v]|2 ; γ̄_{1,x_{1}} = \frac{1}{2} \begin{matrix}n \\ ∑ & ′ \\ l=1 \end{matrix} λ_{l} C_{l}^{2}(x_{1}) ≡ \frac{1}{2} ∑_{l=1}^{n−1} λ_{l} C_{l}^{2}(x_{1}) + \frac{λ_{n}}{4} C_{n}^{2}(x_{1}). \\ |[Λ_{2}^{1/4}v]|2 ; γ̄_{1,x_{1}} = \frac{1}{2} \begin{matrix}n \\ ∑ & ′ \\ l=1 \end{matrix} \sqrt{λ_{l}} C_{l}^{2}(x_{1}) ≡ \frac{1}{2} ∑_{l=1}^{n−1} \sqrt{λ_{l}} C_{l}^{2}(x_{1}) + \frac{\sqrt{λ_{n}}}{4} C_{n}^{2}(x_{1}). \end{matrix} (12)

Using the inequality (see [2]):

max_{x∈γ̄_{2,x_{2}}} |v(x)|^{2} ⩽ ε |[v_{x_{1}}]|2 ; γ̃_{2,x_{2}} + (\frac{1}{ε} + 1) |[v]|2 ; γ̄_{2,x_{2}} , ε > 0,

from (12) we obtain

|[Λ_{2}^{1/4}v]|2 ; γ̄_{1,x_{1}} ⩽ \frac{1}{2} \begin{matrix}n \\ ∑ & ′ \\ l=1 \end{matrix} \sqrt{λ_{l}}\{ε_{l}h ∑_{m=0}^{n−1} C_{l,x_{1}}^{2}(mh) + (\frac{1}{ε_{l}} + 1) h\begin{matrix}n \\ ∑ & ′ \\ m=0 \end{matrix} C_{l}^{2}(mh)\}

If we choose ε_{l} = \frac{1 + \sqrt{1 + 4λ_{l}}}{2λ_{l}}, then we have

|[Λ_{2}^{1/4}v]|2 ; γ̄_{1,x_{1}} ⩽ max_{1⩽l⩽n} \frac{1 + \sqrt{1 + 4λ_{l}}}{2 \sqrt{λ_{l}}} · \frac{1}{2} \begin{matrix}n \\ ∑ & ′ \\ l=1 \end{matrix}\{h ∑_{m=0}^{n−1} C_{l,x_{1}}^{2}(mh) + λ_{l}h\begin{matrix}n \\ ∑ & ′ \\ m=0 \end{matrix} C_{l}^{2}(mh)\}.

However, we know that λ_{l} ⩾ λ_{1} ⩾ 8, and then

|[Λ_{2}^{1/4}v]|2 ; γ̄_{1,x_{1}} ⩽ \frac{5}{4} h ∑_{m=0}^{n−1}(\frac{1}{2} \begin{matrix}n \\ ∑ & ′ \\ l=1 \end{matrix} C_{l,x_{1}}^{2}(mh)) + \frac{5}{4} h\begin{matrix}n \\ ∑ & ′ \\ m=0 \end{matrix}(\frac{1}{2} \begin{matrix}n \\ ∑ & ′ \\ l=1 \end{matrix} λ_{l} C_{l}^{2}(mh))

⩽ \frac{5}{4} h ∑_{m=0}^{n−1} |[v_{x_{1}}]|2 ; γ̄_{1,mh} + \frac{5}{4} h\begin{matrix}n \\ ∑ & ′ \\ m=0 \end{matrix} |[v_{x_{2}}^{2}]|2 ; γ̃_{1,mh}

⩽ \frac{5}{4} (|[v_{x_{1}}]|_{1}^{2} + |[v_{x_{2}}]|_{2}^{2}) ⩽ C |[v]|_{W_{2}^{1}(ω̄)}^{2}.

In the similar way, we can prove following two lemmas (see [2], [7]):

86 B. S. Jovanović, B. Z. Popović

Lemma 3. Let v be a mesh-function on ω̄ and Λ_{3−i} defined by (11). Then

|[Λ_{3−i}^{3/4}v]|2 ; γ̄_{1,x_{i}} ⩽ C h^{2} ∑_{x,x′∈γ̃_{1,x_{i}} , x≠x′} \frac{[v_{x_{3−i}}(x) − v_{x_{3−i}}(x′)]^{2}}{(x_{3−i} − x′_{3−i})^{2}} .

Lemma 4. Let v be a mesh-function on ω̄, then

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

From (8), using lemmas 1, 2 and 4, the following assertion holds true: Lemma 5. The finite difference scheme (5) is stable in the sense of the a priori

estimate:

|[z]|_{W_{2}^{1}(ω̄)} ⩽ C · {h^{2} ∑_{x∈ω_{1}} η_{1,x_{1}}^{2} + h^{2} ∑_{x∈ω_{2}} η_{2,x_{2}}^{2} + h^{2} ∑_{k=0}^{1} ( ∑_{x∈γ̃_{2k}} η̄_{12,x_{1}}^{2} + ∑_{x∈γ̃_{1k}} η̄_{21,x_{2}}^{2})

+ h^{2} (|[Λ_{2}^{3/4}η̄_{22}]|2 ; γ̄_{10} + |[Λ_{2}^{3/4}η̄_{22}]|2 ; γ̄_{11} + |[Λ_{1}^{3/4}η̄_{11}]|2 ; γ̄_{20} + |[Λ_{1}^{3/4}η̄_{11}]|2 ; γ̄_{21}) +

+ h ∑_{i=1}^{2} ∑_{k=0}^{1} ∑_{x∈γ_{ik}} ζ_{i}^{2} + h^{2} ln^{2} \frac{1}{h} ∑_{x∈γ^{⋆}} (ζ̄_{1}^{2} + ζ̄_{2}^{2}) \}^{1/2}.

4. Convergence of the finite difference scheme

The problem of deriving the convergence rate estimate for the FDS (3) is reduced to estimating the right-hand side of the inequality in Lemma 5.

Note that in [16] the following estimates are proved:

|[u − v]|_{W_{2}^{1}(ω̄)} ⩽ Ch^{s−1}∥σ∥_{L_{2}(Γ)}∥u∥_{W_{2}^{s+ε(2−s)}(Ω)}, 1 ⩽ s < 3/2,
|[u − v]|_{W_{2}^{1}(ω̄)} ⩽ Ch^{s−1}∥σ∥_{W_{2}^{s−3/2}(Γ)}∥u∥_{W_{2}^{s+ε(2−s)}(Ω)}, 3/2 < s ⩽ 2. (13)

(i) The estimate for η_{i} in ω_{i}, i = 1, 2 was obtained in [9]:

{h^{2} ∑_{x∈ω_{i}} η_{i,x_{i}}^{2}}^{1/2} ⩽ Ch^{s−1} ∥u∥_{W_{2}^{s}(Ω)}, 2 < s ⩽ 3.  (14)

(ii) The value η̄_{21,x_{2}} in the nodes x ∈ γ̃_{10} may be represented in the form:

η̄_{21,x_{2}} = \frac{2}{h^{2}} ∫_{0}^{h}∫_{0}^{x′_{1}}∫_{0}^{x′′_{1}}∫_{x_{2}}^{x_{2}+h} (1 − \frac{x′_{1}}{h} ) \frac{∂^{3}u(x′′′_{1}, x′_{2})}{∂x_{1}^{2}∂x_{2}} dx′_{2} dx′′′_{1} dx′′_{1} dx′_{1}.

Now, from above it follows that:

|η̄_{21,x_{2}}| ⩽ Ch ∥\frac{∂^{3}u}{∂x_{1}^{2}∂x_{2}} ∥_{L_{2}(e)}, e = (0, h) × (x_{2}, x_{2} + h),

Convergence of a finite difference method … 87

and summing over the nodes x ∈ γ̃_{10} we obtain:

{h^{2} ∑_{x∈γ̃_{10}} η̄_{21,x_{2}}^{2}}^{1/2} ⩽ Ch^{2}∥u∥_{W_{2}^{3}(Ω)}.  (15)

Analogous estimates hold true for the other sum with η̄_{21,x_{2}} and η̄_{12,x_{1}}.

(iii) Let us estimate, for example, h^{2}|[Λ_{2}^{3/4}η̄_{22}|]_{γ_{10}}^{2}. Firstly, setting

S(x_{2}) = σ(x_{2}) u(0, x_{2}) = \frac{∂u(0, x_{2})}{∂x_{1}}

from (9) it follows that

η̄_{22,x_{2}} = \frac{h}{3} S_{x_{2}} = \frac{1}{3} ∫_{x_{2}}^{x_{2}+h} S′(y_{2}) dy_{2}.

Now, let us derive an estimate for the following expression:

η̄_{22,x_{2}}(0, x_{2}) − η̄_{22,x_{2}}(0, x′_{2}) = \frac{1}{3h} ∫_{x_{2}}^{x_{2}+h}∫_{x′_{2}}^{x′_{2}+h} \frac{S′(y_{2}) − S′(y′_{2})}{y_{2} − y′_{2}}(y_{2} − y′_{2}) dy_{2} dy′_{2}

⩽ \frac{1}{3} \{\frac{7}{6} (x_{2} − x′_{2})^{2}\}^{1/2} \{∫_{x_{2}}^{x_{2}+h}∫_{x′_{2}}^{x′_{2}+h} \frac{[S′(y_{2}) − S′(y′_{2})]^{2}}{|y_{2} − y′_{2}|^{2}} dy_{2}dy′_{2}\}^{1/2}. (16)

Now, using (16) and Lemma 3, we obtain

h^{2}|[Λ_{2}^{3/4}η̄_{22}|]_{γ_{10}}^{2} ⩽ h^{2} · C h^{2} ∑_{x,x′∈γ̃_{10}, x≠x′} \frac{[η̄_{22,x_{2}}(x) − η̄_{22,x_{2}}(x′)]^{2}}{(x_{2} − x′_{2})^{2}}

⩽ C h^{4} ∑_{x,x′∈γ̃_{10}, x≠x′} ∫_{x_{2}}^{x_{2}+h}∫_{x′_{2}}^{x′_{2}+h} \frac{[S′(y_{2}) − S′(y′_{2})]^{2}}{|y_{2} − y′_{2}|^{2}} dy_{2}dy′_{2}

⩽ C h^{4} ∫_{0}^{1}∫_{0}^{1} \frac{[S′(y_{2}) − S′(y′_{2})]^{2}}{|y_{2} − y′_{2}|^{2}} dy_{2}dy′_{2} = C h^{4} |S|_{W_{2}^{3/2}(0,1)}^{2}

= C h^{4} |\frac{∂u(0, ·)}{∂x_{1}} |_{W_{2}^{3/2}(0,1)}^{2} ⩽ C h^{4} ∥u∥_{W_{2}^{3}(Ω)}^{2}.  (17)
(iv) Let us derive the estimate for h ∑_{i=1}^{2} ∑_{k=0}^{1} ∑_{x∈γ_{ik}} ζ_{i}^{2}.

For example, consider h ∑_{x∈γ_{10}} ζ_{1}^{2}. The functional ζ_{1} = (T_{2}^{2}σ)u − T_{2}^{2}(σu) may be decomposed in the following manner:

\begin{aligned}ζ_{1} & = ζ_{11} + ζ_{12}, where \\ ζ_{11} & = \frac{1}{h} ∫_{x_{2}−h}^{x_{2}+h} (1 − \frac{|x′_{2} − x_{2}|}{h}) [σ(x′_{2}) − σ(x_{2})][u(0, x_{2}) − u(0, x′_{2})] dx′_{2}, \\ ζ_{12} & = σ(x_{2})\frac{1}{h} ∫_{x_{2}−h}^{x_{2}+h} (1 − \frac{|x′_{2} − x_{2}|}{h}) [u(0, x_{2}) − u(0, x′_{2})] dx′_{2}. \end{aligned}

Transforming ζ_{11} and ζ_{12} to the following forms

\begin{aligned}ζ_{11} & = \frac{1}{h} ∫_{x_{2}−h}^{x_{2}+h} (1 − \frac{|x′_{2} − x_{2}|}{h}) [∫_{x_{2}}^{x′′_{2}} σ′(x′′′_{2})dx′′′_{2} ∫_{x′′_{2}}^{x_{2}} \frac{∂u(0, x′′′_{2})}{∂x_{2}}dx′′′_{2}] dx′′_{2}, \\ ζ_{12} & = \frac{σ(x_{2})}{h} ∫_{x_{2}−h}^{x_{2}+h} (1 − \frac{|x′_{2} − x_{2}|}{h}) ∫_{x′_{2}}^{x_{2}}∫_{x_{2}}^{x′′_{2}} \frac{∂^{2}u(0, x′′′_{2})}{∂x_{2}^{2}} dx′′′_{2}dx′′_{2}dx′_{2}, \end{aligned}

we simply obtain

\begin{matrix}|ζ_{11}| & ⩽ 4\sqrt{2} h^{3/2} ∥\frac{∂u}{∂x_{2}} ∥_{C(Ω̄)}(∫_{x_{2}−h}^{x_{2}+h} |σ′(x′_{2})|^{2}dx′_{2})^{1/2}, \\ |ζ_{12}| & ⩽ 4\sqrt{2} h^{3/2} ∥σ∥_{C[0,1]}(∫_{x_{2}−h}^{x_{2}+h} |\frac{∂^{2}u(0, x′_{2})}{∂x_{2}^{2}} |^{2}dx′_{2})^{1/2}. \end{matrix}

Summing over the nodes x ∈ γ_{10} we obtain

h ∑_{x∈γ_{10}} ζ_{1}^{2} ⩽ Ch^{4} ∥σ∥_{W_{2}^{3/2}(0,1)}^{2}∥u∥_{W_{2}^{3}(Ω)}^{2}

and thus we can prove the estimate

h ∑_{i=1}^{2} ∑_{k=0}^{1} ∑_{x∈γ_{ik}} ζ_{i}^{2} ⩽ Ch^{4} ∥σ∥_{W_{2}^{3/2}(0,1)}^{2}∥u∥_{W_{2}^{3}(Ω)}^{2}.  (18)

(v) Let us derive the estimate for ζ̄_{1}(0, 0) as an example of estimates for the functionals ζ̄_{i} in the nodes x ∈ γ^{⋆}. The functional ζ̄_{1}(0, 0) can be represented in the form

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

From this representation we simply obtain:

|ζ̄_{1}(0, 0)| ⩽ h∥σ∥_{C[0,1]}∥\frac{∂u}{∂x_{2}} ∥_{C(Ω̄)} and

|h^{2} ln^{2} \frac{1}{h} ζ̄_{1}(0, 0)^{2}| ⩽ Ch^{4} ln^{2} \frac{1}{h} ∥σ∥_{W_{2}^{3/2}(0,1)}^{2}∥u∥_{W_{2}^{3}(Ω)}^{2}.

(19)

Finally, from lemma 5 and estimates (14)–(19) we obtain

|[z]|_{W_{2}^{1}(ω̄)} ⩽ Ch^{2} ln\frac{1}{h} ∥σ∥_{W_{2}^{3/2}(Γ)}∥u∥_{W_{2}^{3}(Ω)}.

Now, applying interpolation theory (see [3], [19]) and using (13) we can prove the main result of this paper:

Theorem. The finite difference scheme (3) converges in the norm W_{2}^{1}(ω̄) and the following estimates hold true:

|[u − v]|_{W_{2}^{1}(ω̄)} ⩽ Ch^{s−1}∥σ∥_{L_{2}(Γ)}∥u∥_{W_{2}^{s+ε(2−s)}(Ω)}, 1 ⩽ s < 3/2,
|[u − v]|_{W_{2}^{1}(ω̄)} ⩽ Ch^{s−1}∥σ∥_{W_{2}^{s−3/2}(Γ)}∥u∥_{W_{2}^{s+ε(2−s)}(Ω)}, 3/2 < s ⩽ 2,
|[u − v]|_{W_{2}^{1}(ω̄)} ⩽ Ch^{s−1} (ln\frac{1}{h})^{s−2} ∥σ∥_{W_{2}^{s−3/2}(Γ)}∥u∥_{W_{2}^{s}(Ω)}, 2 < s ⩽ 3.

Note that the error estimates obtained are “almost” consistent with the smoothness of data.

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(received 18.12.1999.)

Boško S. Jovanović, University of Belgrade, Faculty of Mathematics, Studentski trg 16, 11000 Belgrade, Yugoslavia Branislav Z. Popović, University of Kragujevac, Faculty of Science, Radoja Domanovića 12, 34000 Kragujevac, Yugoslavia