����������� ������ 47 (1995), 23–30

UDC 519.632

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CONVERGENCE OF FINITE-DIFFERENCE SCHEMES FOR POISSON’S EQUATION WITH BOUNDARY CONDITION OF THE THIRD KIND

Branislav Popović

Abstract. In this paper we study the convergence of finite-difference schemes to generalized solutions of the third boundary-value problem for Poisson’s equation on the unit square. Using the generalized Bramble-Hilbert lemma, we obtain error estimates in discrete H^{1} Sobolev norm compatible, in some cases, with the smoothness of the data.

The outline of the paper is as follows. In section 1 notational conventions are presented. The stability theorem is proved in section 2. In section 3 we prove estimates of the energy of the operator Δ_{h}. Finally, in section 4, we derive our main results.

1. Preliminaries and notation

Consider the third boundary-value problem for Poisson’s equation on the unit square Ω = (0, 1)^{2}:

Δu = f in Ω,  \frac{∂u}{∂n} + σu = 0 on ∂Ω  (1)

where \frac{∂u}{∂n} = \frac{∂u}{∂x} cos(x, n) + \frac{∂u}{∂y} cos(y, n), n the unit outward normal to ∂Ω, and σ(x, y) is continuous function on ∂Ω such that σ(x, y) ≥ σ_{0} > 0, σ_{0} = const.

We suppose that, for f ∈ H^{0}(Ω), our problem (1) has a unique solution in H^{2}(Ω) and, provided f ∈ H^{s−2}(Ω), u ∈ H^{s}(Ω) for 2 ≤ s ≤ 4 (see [1,4]).

Problem (1) is discretised on the uniform mesh with step-size h : \overline{Ω}_{h} = \{(ih, jh) : i, j = 0, 1, 2, ..., N; Nh = 1\}. We define Ω_{h} = Ω ∩ \overline{Ω}_{h} and ∂Ω_{h} = ∂Ω ∩ \overline{Ω}_{h}. In ∂Ω_{h} we distinguish between two kinds of meshpoints: ∂Ω_{h}^{2} = ∂Ω_{h} \ ∂Ω_{h}^{1} and ∂Ω_{h}^{1} = {(0, 0), (1, 0), (0, 1), (1, 1)}.

For a function U defined on \overline{Ω}_{h}, the following notation will be used: U_{ij} = U(x_{i}, y_{j}), x_{i} = ih, y_{j} = jh, i, j = 1, 2, ..., N and

Δ_{x}^{−}U_{ij} = \frac{U_{ij} − U_{i−1,j}}{h} , Δ_{x}^{+}U_{ij} = Δ_{x}^{−}U_{i+1,j},

Δ_{y}^{−}U_{ij} = \frac{U_{ij} − U_{i,j−1}}{h} , Δ_{y}^{+}U_{ij} = Δ_{y}^{−}U_{i,j+1}.

1991 AMS Subject Classification 65N12, 65N15

24 B. Popović

In the linear space of functions defined on \overline{Ω}_{h} let

[U, V] = h^{2} ∑_{i,j=1}^{N−1} U_{ij}V_{ij} + \frac{h^{2}}{2} ∑_{i=1}^{N−1} (U_{i0}V_{i0} + U_{0i}V_{0i} + U_{Ni}V_{Ni} + U_{iN}V_{iN})+

+ \frac{h^{2}}{4} (U_{00}V_{00} + U_{N0}V_{N0} + U_{0N}V_{0N} + U_{NN}V_{NN})

be the scalar product and | [U] | = \sqrt{[U, U]} the corresponding norm. The discrete H^{1} norm | [ · ] |_{1,h} is defined by | [U] |_{1,h} = \sqrt{| [U] |^{2} + |U|_{1,h}^{2}}, where | · |_{1,h} is the discrete H^{1} seminorm:

|U|_{1,h} = \sqrt{||Δ_{x}^{−}U]|_{x}^{2} + ||Δ_{y}^{−}U]|_{y}^{2}},  ||U]|_{x} = \sqrt{(U, U]_{x}},  ||U]|_{y} = \sqrt{(U, U]_{y}},
(U, V]_{x} = h^{2} ∑_{i=1}^{N} ∑_{j=0}^{N} U_{ij}V_{ij}  and  (U, V]_{y} = h^{2} ∑_{i=0}^{N} ∑_{j=1}^{N} U_{ij}V_{ij}.

Let T_{i}, \overline{T}_{i} and \overline{\overline{T}}_{i} (i = 1, 2) denote the mollifiers defined by

T_{1}f(x, y) = ∫− \frac{1}{2}^{\frac{1}{2}} f(x + th, y) dt , T_{2}f(x, y) = ∫− \frac{1}{2}^{\frac{1}{2}} f(x, y + th) dt,

\overline{T}_{1}f(0, y) = 2 ∫0^{\frac{1}{2}} f(th, y) dt  ,  \overline{T}_{2}f(x, 0) = 2 ∫0^{\frac{1}{2}} f(x, th) dt,

\overline{\overline{T}}_{1}f(1, y) = 2 ∫_{− \frac{1}{2}}^{0} f(1 + th, y) dt , \overline{\overline{T}}_{2}f(x, 1) = 2 ∫_{− \frac{1}{2}}^{0} f(x, 1 + th) dt.

We approximate problem (1) by the finite-difference scheme

Δ_{h}U = F in \overline{Ω}_{h},  (2)

where Δ_{h}U = Δ_{h,x}U + Δ_{h,y}U,

Δ_{h,x}U = \{ \begin{matrix}\frac{2}{h}(Δ_{x}^{+}U − σU) , & i = 0 ,  j = 0, 1, 2, ..., N \\ Δ_{x}^{+}Δ_{x}^{−}U , & i = 1, 2, ..., N − 1, j = 0, 1, 2, ..., N \\ −\frac{2}{h}(Δ_{x}^{−}U + σU), & i = N ,  j = 0, 1, 2, ..., N \end{matrix} ,
F_{ij} = T_{1}T_{2}f_{ij}, F_{0j} = \overline{T}_{1}T_{2}f_{0j}, (i, j = 1, 2, ..., N − 1), F_{00} = \overline{T}_{1}\overline{T}_{2}f_{00},

and Δ_{h,y}U, F_{Nj}, F_{i0}, F_{iN}, F_{N0}, F_{0N}, F_{NN} defined analogously.

2. Stability of the scheme

To begin, let us prove two lemmas.

Lemma 1. Let U, V denote mesh-functions on \overline{Ω}_{h}. Then [Δ_{h}U, V] = [U, Δ_{h}V].

Proof. Using summation by parts it is easy to prove that

[Δ_{h,x}U, V] = −h^{2} ∑_{j=1}^{N−1} ∑_{i=1}^{N} Δ_{x}^{−}U_{ij}Δ_{x}^{−}V_{ij}−

−\frac{h^{2}}{2} ∑_{i=1}^{N} (Δ_{x}^{−}U_{i0}Δ_{x}^{−}V_{i0} + Δ_{x}^{−}U_{iN}Δ_{x}^{−}V_{iN}) − h ∑_{i=1}^{N−1} (σ_{Ni}U_{Ni}V_{Ni} + σ_{0i}U_{0i}V_{0i})

−\frac{h}{2} (σ_{00}U_{00}V_{00} + σ_{0N}U_{0N}V_{0N} + σ_{N0}U_{N0}V_{N0} + σ_{NN}U_{NN}V_{NN}) = [U, Δ_{h,x}V] .

The operator Δ_{h,y} has the same property. Therefore,

[Δ_{h}U, V] = [Δ_{h,x}U, V] + [Δ_{h,y}U, V] = [U, Δ_{h,x}V] + [U, Δ_{h,y}V] = [U, Δ_{h}V] .

Lemma 2. Let U denote mesh-function on \overline{Ω}_{h}. Then [Δ_{h}U, U] ≤ −C|[U]|^{2}, where C = min {1, 2σ_{0}}.

Proof. For fixed j = 0, 1, 2, ..., N , using an inequality from [8] we get

max {U_{ij}^{2} : 0 ≤ i ≤ N} ≤ 2h ∑_{i=1}^{N} (Δ_{x}^{−}U_{ij})^{2} + U_{0j}^{2} + U_{Nj}^{2}.

This yields the following inequality:

h ∑_{i=1}^{N−1} U_{ij}^{2} + \frac{h}{2} (U_{0j}^{2} + U_{Nj}^{2}) ≤ 2h ∑_{i=1}^{N} (Δ_{x}^{−}U_{ij})^{2} + U_{0j}^{2} + U_{Nj}^{2}.

Now let us prove that [Δ_{h,x}U, U] ≤ −\frac{C}{2} |[U]|^{2}. Summing by parts and using last inequality, we obtain:

[Δ_{h,x}U, U] = −h^{2} ∑_{j=1}^{N−1} ∑_{i=1}^{N} (Δ_{x}^{−}U_{ij})^{2} − \frac{h^{2}}{2} ∑_{i=1}^{N} [(Δ_{x}^{−}U_{i0})^{2} + (Δ_{x}^{−}U_{iN})^{2}]−

−h ∑_{j=1}^{N−1} (σ_{Nj}U_{Nj}^{2} + σ_{0j}U_{0j}^{2}) − \frac{h}{2} (σ_{00}U_{00}^{2} + σ_{0N}U_{0N}^{2} + σ_{N0}U_{N0}^{2} + σ_{NN}U_{NN}^{2}) ≤

−h ∑_{j=1}^{N−1} [h ∑_{i=1}^{N} (Δ_{x}^{−}U_{ij})^{2} + σ_{0}U_{Nj}^{2} + σ_{0}U_{0j}^{2}] − \frac{h}{2} [h ∑_{i=1}^{N} (Δ_{x}^{−}U_{i0})^{2} + σ_{0}U_{00}^{2} + σ_{0}U_{N0}^{2}]

−\frac{h}{2} [h ∑_{i=1}^{N} (Δ_{x}^{−}U_{iN})^{2} + σ_{0}U_{0N}^{2} + σ_{0}U_{NN}^{2}] ≤ −\frac{C}{2} |[U]|^{2}.

The inequality [Δ_{h,y}U, U] ≤ −\frac{C}{2} |[U]|^{2} can be proved analogously and we easily obtain Lemma 2.

Theorem 1. For any f ∈ H^{s}(Ω), s ≥ 0, finite-difference scheme (2) has unique solution U. Moreover,

|[U]|_{1,h} ≤ \sqrt{2 + \frac{1}{C}} |[F]| ,  (3)

where C = min {1, 2σ_{0}}.

Proof. The existence and uniqueness of solutions follow from the fact that Δ_{h} is a self-adjoint and negative definite operator (Lemmas 1. and 2.). Further, (using Lemma 2.) we can prove stability in the norm | [ · ] | : |[U]|^{2} ≤ \frac{1}{C} [−Δ_{h}U, U] = \frac{1}{C} [−F, U] ≤ \frac{1}{C} |[F]| |[U]| and thus |[U]| ≤ \frac{1}{C} |[F]|. Summing by parts we can also prove that [Δ_{h,x}U, U] ≤ −\frac{1}{2}||Δ_{x}^{−}U]|_{x}^{2}, [Δ_{h,y}U, U] ≤ −\frac{1}{2} ||Δ_{y}^{−}U]|_{y}^{2}, and |U|_{1,h}^{2} ≤ 2 [−Δ_{h}U, U] . Thence and using Lemma 2. we get (3).

3. The estimates of energy norm [−Δ_{h}U, U]

In this section we present three lemmas. Each of them will be used to obtain an appropriate error estimate for scheme (2).

Lemma 3. Let U denote a mesh-function on \overline{Ω}_{h} which is a solution of finitedifference scheme (2). Then

[−Δ_{h}U, U] ≤ C_{3} \{h^{2} ∑_{i,j=1}^{N−1} F_{ij}^{2} + \frac{h^{3}}{4} [∑_{i=0}^{N} (F_{i0}^{2} + F_{iN}^{2}) + ∑_{i=1}^{N−1} (F_{0i}^{2} + F_{Ni}^{2})]\} , (4)

where C_{3} is a positive constant.

Proof. Using the ε-inequality: |ab| ≤ εa^{2} + \frac{1}{4ε}b^{2}; a, b ∈ R, ε > 0; in the identity [−Δ_{h}U, U] = [−F, U] as follows:

−h^{2} ∑_{i,j=1}^{N−1} F_{ij}U_{ij} ≤ εh^{2} ∑_{i,j=1}^{N−1} U_{ij}^{2} + \frac{h^{2}}{4ε} ∑_{i,j=1}^{N−1} F_{ij}^{2},

−\frac{h^{2}}{2} ∑_{i=1}^{N−1} F_{i0}U_{i0} ≤ εh ∑_{i=1}^{N−1} U_{i0}^{2} + \frac{h^{3}}{16ε} ∑_{i=1}^{N−1} F_{i0}^{2} and − \frac{h^{2}}{4} F_{00}U_{00} ≤ \frac{εh}{4}U_{00}^{2} + \frac{h^{3}}{16ε}F_{00}^{2}

we obtain the following inequality:

[−Δ_{h}U, U] ≤ ε𝕌 + \frac{1}{4ε} \{h^{2} ∑_{i,j=1}^{N−1} F_{ij}^{2} + \frac{h^{3}}{4} ∑_{i=0}^{N} (F_{i0}^{2} + F_{iN}^{2}) + \frac{h^{3}}{4} ∑_{i=1}^{N−1} (F_{0i}^{2} + F_{Ni}^{2})\}

where 𝕌 = 𝕌(U, h), more precisely

𝕌 = h^{2} ∑_{i,j=1}^{N−1} U_{ij}^{2} + h ∑_{i=1}^{N−1} (U_{0i}^{2} + U_{i0}^{2} + U_{Ni}^{2} + U_{iN}^{2}) + \frac{h}{4} (U_{00}^{2} + U_{0N}^{2} + U_{N0}^{2} + U_{NN}^{2}) .

It is easy to prove that 𝕌 (U, h) ≤ (\frac{1}{C} + \frac{1}{σ_{0}} ) [−Δ_{h}U, U] , where C = min \{1, 2σ_{0}\}

and σ ≥ σ_{0} > 0. Thus we get (4) where C_{3} = [4ε − 4ε^{2} (C^{−1} + σ_{0}^{−1})]^{−1} and the value of ε can be chosen so that 1 > ε (C^{−1} + σ_{0}^{−1}), the optimal choice being

ε = Cσ_{0}(2σ_{0} + 2C)^{−1} = \{ \begin{matrix}1 + σ_{0}^{−1}, & σ_{0} ≥ \frac{1}{2}, \\ \frac{3}{2}σ_{0}^{−1}, & 0 < σ_{0} < \frac{1}{2}. \end{matrix}

Using the same technique, we can prove the following two lemmas. Their proofs are omitted.

Lemma 4. Let U denote a mesh-function on \overline{Ω}_{h} which is the solution of finitedifference scheme (2). If we substitute F_{ij} by F_{ij} = Δ_{x}^{+}ξ_{1,ij} + Δ_{y}^{+}ξ_{2,ij}, (i, j = 1, 2, ..., N − 1), F_{i0} = Δ_{x}^{+}ξ_{1,i0} + η_{2,i0}, F_{0i} = η_{1,0i} + Δ_{y}^{+}ξ_{2,0i}, F_{Ni} = η_{1,Ni} + Δ_{y}^{+}ξ_{2,Ni}, F_{iN} = Δ_{x}^{+}ξ_{1,iN} + η_{2,iN}, (i = 1, 2, ..., N − 1) , F_{00} = η_{1,00} + η_{2,00}, F_{0N} = η_{1,0N} + η_{2,0N}, F_{N0} = η_{1,N0} + η_{2,N0} and F_{NN} = η_{1,NN} + η_{2,NN} then

[−Δ_{h}U, U] ≤ C_{4} {h^{2} ∑_{j=0}^{N} ∑_{i=1}^{N} (ξ_{1,ij}^{2} + ξ_{2,ji}^{2}) + h ∑_{i=1}^{N} (ξ_{1,Ni}^{2} + ξ_{1,1i}^{2} + ξ_{2,iN}^{2} + ξ_{2,i1}^{2})

+h^{3} ∑_{i=0}^{N} (η_{1,i0}^{2} + η_{1,iN}^{2} + η_{2,0i}^{2} + η_{2,Ni}^{2})} .

Lemma 5. Under the same assumptions as in Lemma 4, and defining α_{0i} = ξ_{1,1i} − \frac{h}{2}η_{1,0i}, α_{Ni} = −ξ_{1,Ni} − \frac{h}{2}η_{1,Ni}, β_{i0} = ξ_{2,i1} − \frac{h}{2}η_{2,i0} and β_{iN} = −ξ_{2,iN} − \frac{h}{2}η_{2,iN}, (i = 0, 1, 2, ..., N), the following inequality holds:

[−Δ_{h}U, U] ≤ C_{5} {h^{2} ∑_{j=0}^{N} ∑_{i=1}^{N} (ξ_{1,ij}^{2} + ξ_{2,ji}^{2}) + h ∑_{i=0}^{N} (α_{0i}^{2} + α_{Ni}^{2} + β_{i0}^{2} + β_{iN}^{2})} .

4. Convergence of the finite-difference scheme

Before stating our main results we quote the following theorem which is a variant of the well-known Dupont-Scott approximation theorem (see [2]).

Theorem 2. Let E be a bounded connected domain in R^{2} satisfying the cone condition and A(u) a bounded linear functional on H^{s}(E) (s = {s} + α, {s} ≥ 0 is integer and {s} < s, 0 < α ≤ 1) such that P_{{s}} ⊂ Kernel (A(u)) , where P_{{s}} denotes the set of polynomials of degree ≤ {s}. Then, for any u ∈ H^{s}(E), |A(u)| ≤ C|u|_{H^{s}(E)}, where C = C(E, s) is a positive constant independent of u and | · |_{H^{s}(E)} is the highest seminorm of H^{s}(E).

The derivations of all error estimates below are based on the above theorem.

Theorem 3. Suppose that u ∈ H^{s}(Ω), 2 ≤ s ≤ 4, is the solution of problem (1) and U is the solution of the finite-difference scheme (2). Then

|[U − u]|_{1,h} ≤ Ch^{s−2}∥u∥_{H^{s}(Ω)} = O(h^{s−2}).

Proof. Let us define the global error as z = U − u. Then Δ_{h}z_{ij} = Δ_{h}U_{ij} − Δ_{h}u_{ij} = F_{ij} − Δ_{h}u_{ij} = φ_{ij}. We shall consider three cases:

i) If (ih, jh) ∈ Ω_{h}, then

φ_{ij} = T_{2}Δ_{x}^{−} \frac{∂u}{∂x} (ih + \frac{h}{2}, jh) + T_{1}Δ_{y}^{−} \frac{∂u}{∂x} (ih, jh + \frac{h}{2}) − Δ_{h}u_{ij}.

Using Theorem 2 and standard technique based on Theorem 2 as in [3], [5] or [9], we obtain |φ_{ij}| ≤ Ch^{s−3}|u|_{H^{s}(e_{ij})}, 2 < s ≤ 4, where e_{ij} = {(x, y) : ih − h ≤ x ≤ ih + h; jh − h ≤ y ≤ jh + h}.

ii) If (ih, jh) ∈ ∂Ω_{h}^{2}, for example (0, jh), then

φ_{0j} = \frac{2}{h}T_{2} [\frac{∂u}{∂x} (\frac{h}{2}, jh) − \frac{∂u}{∂x} (0, jh)] + \overline{T}_{1}Δ_{y}^{−} \frac{∂u}{∂y} (0, jh + \frac{h}{2}) −

− \frac{1}{h^{2}} [u_{0,j+1} + u_{0,j−1} + 2u_{1,j} − 2h\frac{∂u}{∂x} (0, jh) − 4u_{0j}] .

In the same way, except that 2 < s ≤ 3, we obtain |φ_{0j}| ≤ Ch^{s−3}|u|_{H^{s}(e_{0j})}, where e_{0j} = {(x, y) : 0 ≤ x ≤ h; jh − h ≤ y ≤ jh + h} .

iii) If (ih, jh) ∈ ∂Ω_{h}^{1}, for example (0, 0), then

φ_{00} = \frac{2}{h}\overline{T}_{2} [\frac{∂u}{∂x} (\frac{h}{2}, 0) − \frac{∂u}{∂x}(0, 0)] + \frac{2}{h}\overline{T}_{1} [\frac{∂u}{∂y} (0, \frac{h}{2}) − \frac{∂u}{∂y}(0, 0)] −

− \frac{2}{h^{2}} [u_{10} − u_{00} − h\frac{∂u}{∂x}(0, 0) + u_{01} − u_{00} − h\frac{∂u}{∂y}(0, 0)]

and we obtain, provided 2 < s ≤ 3, |φ_{00}| ≤ Ch^{s−3}|u|_{H^{s}(e_{00})}, where e_{00} = {(x, y) : 0 ≤ x, y ≤ h} .

However, we can obtain |[U]|_{1,h}^{2} ≤ C [Δ_{h}U, U] . Thence, using (4), we get

|[z]|_{1,h}^{2} ≤ C \{h^{2} ∑_{i,j=1}^{N−1} φ_{ij}^{2} + \frac{h^{3}}{4} ∑_{i=0}^{N} (φ_{i0}^{2} + φ_{iN}^{2}) + \frac{h^{3}}{4} ∑_{i=1}^{N−1} (φ_{0i}^{2} + φ_{Ni}^{2})\} .

Now, for 2 < s ≤ 3, it is easy to prove that |[z]|_{1,h} ≤ Ch^{s−2}|u|_{H^{s}(Ω)}. If 3 < s ≤ 4, then

h^{2} ∑_{i,j=1}^{N−1} φ_{ij}^{2} ≤ Ch^{2s−4}|u|_{H^{s}(Ω)}^{2}.  (5)

On the other hand, if u ∈ H^{s}(Ω), then u ∈ H^{3}(Ω) and

\frac{h^{3}}{4} [∑_{i=0}^{N} (φ_{i0}^{2} + φ_{iN}^{2}) + ∑_{i=1}^{N−1} (φ_{0i}^{2} + φ_{Ni}^{2})] ≤ Ch^{3}|u|_{H^{3}(∂_{h}Ω)},  (6)

where ∂_{h}Ω is the boundary strip of width h. But, according to [7]

|u|_{H^{3}(∂_{h}Ω)} ≤ C∥u∥_{H^{s}(Ω)} · \{ \begin{matrix}h^{s−3}, & 3 < s < \frac{7}{2} \\ \sqrt{h}| ln h|, & s = \frac{7}{2} \\ \sqrt{h} , & \frac{7}{2} < s ≤ 4. \end{matrix}  (7)

Using (5), (6) and (7) we obtain |[z]|_{1,h} ≤ Ch^{s−2}∥u∥_{H^{s}(Ω)} and that completes the proof.

Theorem 4. Suppose that u ∈ H^{s}(Ω), \frac{3}{2} < s ≤ 3, is the solution of problem (1) where σ ∈ M (H^{s−1}(0, 1)) (see [6]) and U is the solution of (2). Then

|[U − u]|_{1,h} = \{ \begin{matrix}O(h^{s−1}) , & \frac{3}{2} < s < \frac{5}{2} \\ O(h\sqrt{h}|lnh|), & s = \frac{5}{2} \\ O(h\sqrt{h}) , & \frac{5}{2} < s ≤ 3. \end{matrix}

Proof. This theorem is similar to the previous one. Therefore we begin the proof as before. Naturally, this time we shall use Lemma 5 and thus we have to derive the following:

i) If i = 1, 2, ..., N; j = 1, 2, ...N − 1 and \frac{3}{2} < s ≤ 3, then

ξ_{1,ij} = T_{2} [\frac{∂u}{∂x} (ih − \frac{h}{2}, jh)] − Δ_{x}^{−}u_{ij} and |ξ_{1,ij}| ≤ Ch^{s−2}|u|_{H^{s}(e_{ij})},

where e_{ij} = \{(x, y) : ih − h ≤ x ≤ ih, jh − \frac{h}{2} ≤ y ≤ jh + \frac{h}{2}\} .

ii) If i = 1, 2, ..., N; j = 0 or j = N and \frac{3}{2} < s ≤ 2, then

ξ_{1,i0} = \overline{T}_{2} [\frac{∂u}{∂x} (ih − \frac{h}{2}, 0)] − Δ_{x}^{−}u_{i0}, ξ_{1,iN} = \overline{\overline{T}}_{2} [\frac{∂u}{∂x} (ih − \frac{h}{2}, 1)] − Δ_{x}^{−}u_{iN}

and |ξ_{1,ij}| ≤ Ch^{s−2}|u|_{H^{s}(e_{ij})}

where e_{i0} = \{(x, y) : ih − h ≤ x ≤ ih, 0 ≤ y ≤ \frac{h}{2}\} or e_{iN} = \{(x, y) : ih − h ≤ x ≤ ih, 1 − \frac{h}{2} ≤ y ≤ 1\} . (Analogous results can be obtained for ξ_{2}.)

iii) If j = 1, 2, ..., N − 1, then α_{0j} = T_{2}(σ_{0j}u_{0j}) − σ_{0j}u_{0j} and

|α_{0j}| ≤ Ch^{s− \frac{3}{2}} |σu|_{H^{s−1}(d_{0j})}, 1 ≤ s ≤ 3,

where d_{0j} = [jh − \frac{h}{2}, jh + \frac{h}{2}] . (The same results can be obtained for α_{Nj}, β_{jN} and β_{j0}.)

iv) If i = 0 and j = 0, then α_{00} = \overline{T}_{2}(σ_{00}u_{00}) − σ_{00}u_{00} and
|α_{00}| ≤ Ch^{s− \frac{3}{2}} |σu|_{H^{s−1}(d_{00})}, 1 ≤ s ≤ 2,

where d_{00} = [0, \frac{h}{2}] . (The same results can be obtained for α_{ij} and β_{ij} where (ih, jh) ∈ ∂Ω_{h}^{1}.)

Thence:

h^{2} ∑_{i=1}^{N} ∑_{j=1}^{N−1} (ξ_{1,ij}^{2} + ξ_{2,ji}^{2}) ≤ Ch^{2s−2}|u|_{H^{s}(Ω)}^{2}, \frac{3}{2} < s ≤ 3,

h^{2} ∑_{i=1}^{N} (ξ_{1,i0}^{2} + ξ_{1,iN}^{2} + ξ_{2,0i}^{2} + ξ_{2,Ni}^{2}) ≤ C∥u∥_{H^{s}(Ω)}^{2} · \{ \begin{matrix}h^{s} , & \frac{3}{2} < s < \frac{5}{2} \\ h^{3} ln^{2} h, & s = \frac{5}{2} , \\ h^{3} , & \frac{5}{2} < s ≤ 3 \end{matrix}

h ∑_{j=1}^{N−1} α_{0j}^{2} ≤ Ch^{2s−2}|σu|_{H^{s−1}(0,1)}^{2} , 1 ≤ s ≤ 3,

hα_{00}^{2} ≤ C∥σu∥_{H^{s−1}(0,1)}^{2} · \{ \begin{matrix}h^{2s−2} , & 1 ≤ s ≤ \frac{5}{2} \\ h^{3} ln^{2} h, & s = \frac{5}{2} \\ h^{3} , & \frac{5}{2} < s ≤ 3 \end{matrix}

and |σu|_{H^{s−1}(0,1)} ≤ ∥σu∥_{H^{s−1}(0,1)} ≤ C∥σ∥_{M(H^{s−1}(0,1))}∥u∥_{H^{s−1}(0,1)} ≤

≤ C∥σ∥_{M(H^{s−1}(0,1))}∥u∥_{H^{s}(Ω)}.

Now using Lemma 5, we easily complete the proof of the theorem.

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(received 08.05.1995.)

University of Kragujevac, Faculty of Sciences Radoja Domanovića 12, 34000 Kragujevac, Yugoslavia