����������� ������ 49 (1997), 249–256

UDK 517.962.2

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SOME CONVERGENCE RATE ESTIMATES FOR FINITE DIFFERENCE SCHEMES

Boško S. Jovanović and Branislav Z. Popović

Abstract. In this work we use function space interpolation to prove some convergence rate estimates for finite difference schemes. We concentrate on a Dirichlet boundary value problem for a second-order linear elliptic equation with variable coefficients in the unit 3-dimensional cube. We assume that the solution to the problem and the coefficients of the equation belong to corresponding Sobolev spaces.

1. Introduction

In this work we use interpolation theory to prove some convergence rate estimates for FDS. Our model problem will be a Dirichlet BVP for a second-order linear elliptic equation with variable coefficients in the unit 3-dimensional cube Ω = (0, 1)^{3}:

− ∑_{i,j=1}^{3} D_{i}(a_{ij}D_{j}u) = f in Ω,  u = 0 on Γ = ∂Ω.  (1)

We shall assume that the generalized solution of the BVP belongs to the Sobolev spaceW_{2}^{s}(Ω), 2 ≤ s ≤ 4, with the right-hand side f(x) belonging to W_{2}^{s−2}(Ω). Initially we assume that the coefficients a_{ij}(x) belong to the space of multipliers M(W_{2}^{s−1}(Ω)); for this it is sufficient that [10]:

a_{ij} ∈ W_{2}^{s−1}(Ω), for \frac{5}{2} < s ≤ 4,

a_{ij} ∈ W_{3/(s−1)}^{s−1+δ}(Ω), δ > 0, for 2 ≤ s ≤ \frac{5}{2}.

AMS Subject Classification: 65 N 15, 46 B 70

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

Communicated at the 4th Symposium on Mathematical Analysis and Its Applications, Aran - delovac 1997.

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

We also assume that the corresponding differential operator is symmetric and strongly elliptic, i.e.

a_{ij} = a_{ji}, ∑_{i,j=1}^{3} a_{ij}y_{i}y_{j} ≥ c_{0} ∑_{i=1}^{3} y_{i}^{2}, x ∈ Ω, c_{0} = const > 0.

Let \overline{ω} be a uniform mesh in \overline{Ω} with the step size h, ω = \overline{ω} ∩ Ω, γ = \overline{ω} ∩ Γ, etc. We define finite differences v_{x_{i}} and v_{\overline{x}_{i}} in the usual manner [11]:

v_{x_{i}} = (v^{+i} − v)/h, v_{\overline{x}_{i}} = (v − v^{−i})/h,

where v^{±i}(x) = v(x ± hr_{i}), and r_{i} is the unit vector along the x_{i} axis. We approximate the BVP with following FDS:

L_{h}v = T_{1}^{2}T_{2}^{2}T_{3}^{2}f in ω, v = 0 on γ,  (2)

where T_{i} is Steklov smoothing operator on x_{i}, i.e.

T_{i}^{+}f(x) = ∫_{0}^{1} f(x + htr_{i}) dt = T_{i}^{−}f(x + hr_{i}) = T_{i}f(x + 0.5hr_{i})

(Hence, T_{i}T_{j}f = T_{j}T_{i}f and T_{i}^{+}D_{i}u = u_{x_{i}}, T_{i}^{−}D_{i}u = u_{\overline{x}_{i}}), and

L_{h}v = −\frac{1}{2} ∑_{i,j=1}^{3} [(a_{ij}v_{\overline{x}_{j}})_{x_{i}} + (a_{ij}v_{x_{j}})_{\overline{x}_{i}}] .

Let u be the solution of the BVP and v the solution of the FDS. We define the error as z = u − v. Pur aim is to show that

∥u − v∥_{W_{2}^{2}(ω)} ≤ C · h^{s−2}∥u∥_{W_{2}^{s}(Ω)}, 2 ≤ s ≤ 4,  (3)

where C is positive constant depending of the coefficients, but independent of h and u.

The finite-difference scheme (2) is the standard symmetric FDS [11] with averaged right-hand side. Note that for s ≤ 7/2 the right-hand side is a discontinuous function, so without averaging the FDS is not well-defined.

Estimates of the type

∥u − v∥_{W_{p}^{k}(ω)} ≤ C · h^{s−k}∥u∥_{W_{p}^{s}(Ω)}  (4)

are said to be consistent with the smoothness of the solution of the BVP [9].

The same technique is used in papers of B.S. Jovanovic [6] (constant coeficient case) and [7], [14] (n = 2).

Estimates of type (4) have been obtained for a broad class of elliptic problems by Lazarov, Makarov, Samarski, Jovanović, Süli, Ivanović etc (see [5, 8, 9, 12]). As a rule the Bramble-Hilbert lemma [3] and results of Dupont and Scott [4] are used for proving those results.

2. Interpolation of Banach Spaces

Let A_{0} and A_{1} be two Banach spaces, linearly and continuosly embedded in a topological linear space A. Two such spaces are called interpolation pair {A_{0}, A_{1}}. Consider also the spaces A_{0} ∩ A_{1} and A_{0} + A_{1} with corresponding norms (see [2, 13]).

Let us introduce a category C_{0}, whose objects A, B, C,… are Banach spaces, and whose morphisms are bounded linear operators L ∈ L(A, B), and a category C_{1}, whose objects are interpolations pairs {A_{0}, A_{1}}, {B_{0}, B_{1}}, … and whose morphisms are L ∈ L({A_{0}, A_{1}}, {B_{0}, B_{1}}), where L({A_{0}, A_{1}}, {B_{0}, B_{1}}) denotes the set of bounded linear operators from A_{0} + A_{1} into B_{0} + B_{1}, whose restrictions on A_{i} belong to the set L(A_{i}, B_{i}), i = 1, 2.

A functor F : C_{1} → C_{0} is called an interpolation functor if A_{0} ∩ A_{1} ⊂ F({A_{0}, A_{1}}) ⊂ A_{0} + A_{1} for every interpolation pair {A_{0}, A_{1}}, while for every morphism L ∈ L({A_{0}, A_{1}}, {B_{0}, B_{1}}), F(L) is the restriction of the operator L on F({A_{0}, A_{1}}).

The corresponding Banach space A = F({A_{0}, A_{1}}) is called an interpolation space.Obviously A_{0} ∩ A_{1} and A_{0} + A_{1} are interpolation spaces.

If the inequality

∥L∥_{F({A_{0},A_{1}})→F({B_{0},B_{1}})} ≤ C∥L∥_{A_{0}→B_{0}}^{1−θ}∥L∥_{A_{1}→B_{1}}^{θ},

where 0 < θ < 1 and C = const ≥ 1, is satisfied for every morphism L of category C_{1}, the interpolation functor F is said to be of the type θ.

Let us consider the so called complex interpolation method [13]. We define the following sets of complex numbers: S = \{z ∈ ℂ : 0 < ℜz < 1\} and \overline{S} = \{z ∈ ℂ : 0 ≤ ℜz ≤ 1}. For a given interpolation pair {A_{0}, A_{1}} we introduce the set M(A_{0}, A_{1}) of continuous functions f: \overline{S} → A_{0} + A_{1}, analytic in S, which satisfy the following conditions:

(i) sup_{z∈\overline{S}} ∥f(z)∥_{A_{0}+A_{1}} < ∞,

(ii) f(j + it) ∈ A_{j}, j = 0, 1, t ∈ ℝ,

(iii) the mapings t → f(j + it), j = 0, 1, are continuous on t, and

(iv) ∥f∥_{M(A_{0},A_{1})} = max{sup_{t∈ℝ} ∥f(it)∥_{A_{0}}, sup_{t∈ℝ} ∥f(1 + it)∥_{A_{1}}} < ∞.

For 0 < θ < 1 with [A_{0}, A_{1}]_{θ} we denote the set of elements a ∈ A_{0} + A_{1} which satisfy the conditions:

(i) there exists a function f ∈ M(A_{0}, A_{1}) such that f(θ) = a, and

(ii) ∥a∥_{[A_{0},A_{1}]_{θ}} = inf_{\begin{matrix}f∈M(A_{0},A_{1}) \\ f(θ)=a \end{matrix}} ∥f∥_{M(A_{0},A_{1})} < ∞.

The space [A_{0}, A_{1}]_{θ} defined in that way is an interpolation space. The corresponding interpolation functor F({A_{0}, A_{1}}) = [A_{0}, A_{1}]_{θ} is of the type θ, with constant C = 1. Analogous assertion holds true for bilinear operators [13]:

Lemma 1. Let A_{0} ⊂ A_{1}, B_{0} ⊂ B_{1}, C_{0} ⊂ C_{1} and let L: A_{1} × B_{1} → C_{1} be a continuous bilinear form whose restriction on A_{0} × B_{0} is a continuous maping with values in C_{0}. Then L is continuous maping from [A_{0}, A_{1}]_{θ} × [B_{0}, B_{1}]_{θ} into [C_{0}, C_{1}]_{θ}, and

∥L∥_{[A_{0},A_{1}]_{θ}×[B_{0},B_{1}]_{θ}→[C_{0},C_{1}]_{θ}} ≤ ∥L∥_{A_{0}×B_{0}→C_{0}}^{1−θ}∥L∥_{A_{1}×B_{1}→C_{1}}^{θ}.

3. Spaces H_{p}^{s}, B_{pq}^{s} and W_{p}^{s}

As examples of interpolation function spaces we consider the spaces of Bessel potentials H_{p}^{s}, the Besov spaces B_{pq}^{s} and the Sobolev spaces W_{p}^{s} (see [1], [2] and [13]). The spaces H_{p}^{s} and B_{pq}^{s} are spaces of distributions. We know that D(ℝ^{n}) ⊂ H_{p}^{s}(ℝ^{n}) and B_{pq}^{s} ⊂ D′(ℝ^{n}) where D(ℝ^{n}) = C_{0}^{∞}(ℝ^{n}) is the set of infinitely differentiable functions with compact support, and D′(ℝ^{n}) is the set of distributions. For s = 0, H_{p}^{0}(ℝ^{n}) = L_{p}(ℝ^{n}), where L_{p} is the Lebesgue space of integrable functions. For 1 < p < ∞ the Sobolev spaces W_{p}^{s} are defined in the following manner:

W_{p}^{s}(ℝ^{n}) = \{ \begin{matrix}H_{p}^{s}(ℝ^{n}), & s = 0, 1, 2,… \\ pp^{s}(ℝ^{n}), & 0 < s ≠ integer \end{matrix}  (5)

with the norm defined as

∥f∥_{W_{p}^{s}} = ( ∑_{k<s} |f|_{W_{p}^{k}}^{p} + |f|_{W_{p}^{s}}^{p})^{1/p},

where

|f|_{W_{p}^{r}} = \{ \begin{matrix}( ∑_{|α|=r} ∫_{ℝ^{n}} |D^{α}f(x)|^{p}dx)^{1/p}, & r = 0, 1, 2,… \\ ( ∑_{|α|=[r]} ∫_{ℝ^{n}} ∫_{ℝ^{n}} \frac{|D^{α}f(x)−D^{α}f(y)|^{p}}{|x−y|^{n+p(r−[r])}} dxdy)^{1/p}, & 0 < r ≠ integer. \end{matrix}

Here α = (α_{1},…, α_{n}) is a multi-index, |α| = α_{1} + ⋯ + α_{n}, x = (x_{1},…, x_{n}) ∈ ℝ^{n}, |x| = (x_{1} + ⋯ + x_{n})^{1/2}, D^{α} = D_{1}^{α_{1}} ⋯ D_{n}^{α_{n}} = (∂/∂x_{1})^{α_{1}} ⋯ (∂/∂x_{n})^{α_{n}} and [r] is the integer part of r. Obviously, W_{p}^{s}(ℝ^{n}) ⊂ L_{p}(ℝ^{n}), s ≥ 0.

For −∞ < s < ∞, 1 < p < ∞, ε > 0 and 1 ≤ q_{0} ≤ q_{1} ≤ ∞ the folloving imbeddings hold true [13]:

B_{p,∞}^{s+ε}(ℝ^{n}) ⊂ B_{p1}^{s}(ℝ^{n}) ⊂ B_{pq_{0}}^{s}(ℝ^{n}) ⊂ B_{pq_{1}}^{s}(ℝ^{n}) ⊂ B_{p,∞}^{s}(ℝ^{n}) ⊂ B_{p1}^{s−ε}(ℝ^{n}),
H_{p}^{s+ε}(ℝ^{n}) ⊂ H_{p}^{s}(ℝ^{n}) and
B_{p,min{p,2}}^{s}(ℝ^{n}) ⊂ H_{p}^{s}(ℝ^{n}) ⊂ B_{p,max{p,2}}^{s}(ℝ^{n}).  (6)

For −∞ < t ≤ s < ∞, 1 < p ≤ q < ∞, 1 ≤ r ≤ ∞ and s − n/p ≥ t − n/q we also have

B_{pr}^{s}(ℝ^{n}) ⊂ B_{qr}^{t}(ℝ^{n}) and H_{p}^{s}(ℝ^{n}) ⊂ H_{q}^{t}(ℝ^{n}).

The following assertion holds true [13]:

Lemma 2. For −∞ < s_{0}, s_{1} < ∞, 1 < p_{0}, p_{1} < ∞, 1 ≤ q_{0} < ∞, 1 ≤ q_{1} ≤ ∞ and 0 < θ < 1 we have

[H_{p_{0}}^{s_{0}}(ℝ^{n}), H_{p_{1}}^{s_{1}}(ℝ^{n})]_{θ} = H_{p}^{s}(ℝ^{n}) and  (7)
[B_{p_{0}q_{0}}^{s_{0}}(ℝ^{n}), B_{p_{1}q_{1}}^{s_{1}}(ℝ^{n})]_{θ} = B_{pq}^{s}(ℝ^{n}),  (8)

where

s = (1 − θ)s_{0} + s_{1}, \frac{1}{p} = \frac{1 − θ}{p_{0}} + \frac{θ}{p_{1}} , \frac{1}{q} = \frac{1 − θ}{q_{0}} + \frac{θ}{q_{1}} .

From (7), (8) and (5), for s_{0}, s_{1} ≥ 0, it follows that
[W_{p}^{s_{0}}(ℝ^{n}), W_{p}^{s_{1}}(ℝ^{n})]_{θ} = W_{p}^{s}(ℝ^{n}), s = (1 − θ)s_{0} + s_{1},  (9)

if s_{0}, s_{1} and s are all integer, or fractional numbers. For p = 2 from (6) it follows that W_{2}^{s}(ℝ^{n}) = H_{2}^{s}(ℝ^{n}) = B_{22}^{s}(ℝ^{n}) and (9) holds without the restriction that s_{0}, s_{1} and s are of the same kind.

The previous results hold for the spaces H_{p}^{s}, B_{pq}^{s} and W_{p}^{s} in a bounded domain Ω ⊂ ℝ^{n} which satisfies the cone condition. Here we assume that s ≥ 0 for H_{p}^{s} spaces, and s > 0 for B_{pq}^{s} spaces.

4. Convergence of Finite Difference Schemes

Let u be the solution of the BVP (1) and let v be the solution of the FDS (2). The error z = u − v satisfies the conditions

L_{h}z = ∑_{i,j=1}^{3} ψ_{ij}, in ω, z = 0 on γ,  (10)

where

ψ_{ij} = T D_{i}(a_{ij}D_{j}u)− \frac{1}{2} [(a_{ij}u_{\overline{x}_{j}})_{x_{i}} + (a_{ij}u_{x_{j}})_{\overline{x}_{i}}] , T u = T_{1}^{2}T_{2}^{2}T_{3}^{2}u, i, j = 1, 2, 3.

Let (v, w)_{ω} = (v, w)_{L_{2}(ω)} = h^{3} ∑_{x∈ω} v(x)w(x) and ∥v∥_{ω}^{2} = (v, v)_{ω} denote the discrete inner product and the discrete L_{2}-norm on ω. We also define the discrete Sobolev norm

∥v∥_{W_{2}^{2}(ω)}^{2} = ∥v∥_{ω}^{2} + ∑_{i=1}^{3} ∥v_{x_{i}}∥_{ω_{i}}^{2} + ∑_{i=1}^{3} ∥v_{x_{i}\overline{x}_{i}}∥_{ω}^{2} + ∑_{i<j}^{3} ∥v_{x_{i}x_{j}}∥_{ω_{ij}}^{2},

where ω_{i} and ω_{ij} are subsets of \overline{ω} where corresponding finite differences are well defined.

The following assertion holds true [5]:

Lemma 3. The FDS (10) satisfies the a priori estimate

∥z∥_{W_{2}^{2}(ω)} ≤ C · ∑_{i,j=1}^{3} ∥ψ_{ij}∥_{ω}.  (11)

254 B. S. Jovanović and B. Z. Popović

The problem of deriving the convergence rate estimates for thr FDS (2) is reduced to estimating the right-hand side terms in (11). Let us decompose ψ_{ij} in the following manner: ψ_{ij} = ∑_{k=1}^{7} ψ_{ijk}, where

ψ_{ij1} = T (a_{ij}D_{i}D_{j}u) − (T a_{ij})(T D_{i}D_{j}u),
ψ_{ij2} = (T a_{ij} − a_{ij})(T D_{i}D_{j}u),
ψ_{ij3} = a_{ij} [T D_{i}D_{j}u − 0.5(u_{\overline{x}_{i}x_{j}} + u_{x_{i}\overline{x}_{j}})] ,
ψ_{ij4} = T (D_{i}a_{ij}D_{j}u) − (T D_{i}a_{ij})(T D_{j}u),
ψ_{ij5} = [T D_{i}a_{ij} − 0.5(a_{ij,x_{i}} + a_{ij,\overline{x}_{i}})] (T D_{j}u),
ψ_{ij6} = 0.5(a_{ij,x_{i}} + a_{ij,\overline{x}_{i}}) [T D_{j}u − 0.5(u_{x_{j}}^{−i} + u_{\overline{x}_{j}}^{+i})] ,
ψ_{ij7} = 0.25(a_{ij,x_{i}} + a_{ij,\overline{x}_{i}})(u_{x_{j}}^{−i} + u_{\overline{x}_{j}}^{+i}).

The value ψ_{ij1} in the node x ∈ ω can be represented in the form

\begin{aligned}ψ_{ij1} = & \frac{1}{h^{6}} ∫ ⋯ ∫_{e×e} Φ(ξ_{1}, ξ_{2}, ξ_{3})Φ(σ_{1}, σ_{2}, σ_{3})[a_{ij}(ξ_{1}, ξ_{2}, ξ_{3}) − a_{ij}(σ_{1}, σ_{2}, σ_{3})]× \\ & × D_{i}D_{j}u(ξ_{1}, ξ_{2}, ξ_{3}) dξ_{1}dξ_{2}dξ_{3}dσ_{1}dσ_{2}dσ_{3}, \end{aligned} (12)

where e = (x_{1} − h, x_{1} + h) × (x_{2} − h, x_{2} + h) × (x_{3} − h, x_{3} + h) and

Φ(ξ_{1}, ξ_{2}, ξ_{3}) = (1 − \frac{|ξ_{1} − x_{1}|}{h}) (1 − \frac{|ξ_{2} − x_{2}|}{h}) (1 − \frac{|ξ_{3} − x_{3}|}{h}) .

Now, from (12) it follows that:

|ψ_{ij1}| ≤ \frac{C}{h^{3/2}} ∥a_{ij}∥_{C(ē)}∥D_{i}D_{j}u∥_{L_{2}(e)} ≤ \frac{C}{h^{3/2}} ∥a_{ij}∥_{C(\overline{Ω})}∥u∥_{W_{2}^{2}(e)}.

From here, summing over the mesh ω we obtain

∥ψ_{ij1}∥_{ω} ≤ C · ∥a_{ij}∥_{C(\overline{Ω})}∥u∥_{W_{2}^{2}(Ω)} ≤ C · ∥a_{ij}∥_{W_{p}^{1+ε}(Ω)}∥u∥_{W_{2}^{2}(Ω)}, ε > 0, p ≥ 3. (13)

Transforming a_{ij}(ξ_{1}, ξ_{2}, ξ_{3}) − a_{ij}(σ_{1}, σ_{2}, σ_{3}) in (12) to integral form

\begin{aligned}a_{ij}(ξ_{1}, ξ_{2}, ξ_{3}) & − a_{ij}(σ_{1}, σ_{2}, σ_{3}) = ∫_{σ_{1}}^{ξ_{1}} D_{1}a_{ij}(τ_{1}, σ_{2}, σ_{3}) dτ_{1}+ \\ & + ∫_{σ_{2}}^{ξ_{2}} D_{2}a_{ij}(ξ_{1}, τ_{2}, σ_{3}) dτ_{2} + ∫_{σ_{3}}^{ξ_{3}} D_{3}a_{ij}(ξ_{1}, ξ_{2}, τ_{3}) dτ_{3} \end{aligned}  (14)

and exchanging ξ_{i} and σ_{i}, we obtain

\begin{aligned}ψ_{ij1} = & \frac{1}{2h^{6}} ∫ ⋯ ∫_{e×e} Φ(ξ_{1}, ξ_{2}, ξ_{3})Φ(σ_{1}, σ_{2}, σ_{3}) × [∫_{σ_{1}}^{ξ_{1}} D_{1}a_{ij}(τ_{1}, ξ_{2}, ξ_{3}) dτ_{1}+ \\ & + ∫_{σ_{2}}^{ξ_{2}} D_{2}a_{ij}(σ_{1}, τ_{2}, ξ_{3}) dτ_{2} + ∫_{σ_{3}}^{ξ_{3}} D_{3}a_{ij}(σ_{1}, σ_{2}, τ_{3}) dτ_{3}] × \\ & × [D_{i}D_{j}u(ξ_{1}, ξ_{2}, ξ_{3}) − D_{i}D_{j}u(σ_{1}, σ_{2}, σ_{3})] dξ_{1}dξ_{2}dξ_{3}dσ_{1}dσ_{2}dσ_{3}. \end{aligned} (15)

Finally, transforming D_{i}D_{j}u(ξ_{1}, ξ_{2}, ξ_{3}) − D_{i}D_{j}u(σ_{1}, σ_{2}, σ_{3}) in (15) to integral form (like in (14)) and applying Hölder’s inequality, we obtain

|ψ_{ij1}| ≤ C · h^{1/2}∥a_{ij}∥_{W_{p}^{1}(e)}∥u∥_{W_{2p/(p−2)}^{3}(e)}, p > 2.

From here, summing over the mesh ω, and using the imbeddings W_{p}^{3} ⊂ W_{p}^{1} and W_{2}^{4} ⊂ W_{2p/(p−2)}^{3} for p ≥ 3, we obtain

\begin{aligned}∥ψ_{ij1}∥_{ω} & ≤ C · h^{2}∥a_{ij}∥_{W_{p}^{1}(Ω)}∥u∥_{W_{2p/(p−2)}^{3}(Ω)} \\ & ≤ C · h^{2}∥a_{ij}∥_{W_{p}^{3}(Ω)}∥u∥_{W_{2}^{4}(Ω)}, p ≥ 3. \end{aligned}  (16)

Estimates analogous to (13) and (16) hold true for the other terms ψ_{ijk} [14] and so we obtain

∥ψ_{ij}∥_{ω} ≤ C · ∥a_{ij}∥_{W_{p}^{1+ε}(Ω)}∥u∥_{W_{2}^{2}(Ω)}, ε > 0, p ≥ 3  (17)
∥ψ_{ij}∥_{ω} ≤ C · h^{2}∥a_{ij}∥_{W_{p}^{3}(Ω)}∥u∥_{W_{2}^{4}(Ω)}, p ≥ 3.  (18)

The mapping (a_{ij}, u) → ψ_{ij} is bilinear. From (17) and (18) it follows that it is a bounded bilinear operator from W_{p}^{1+ε}(Ω) × W_{2}^{2}(Ω) to L_{2}(ω) and from W_{p}^{3}(Ω) × W_{2}^{4}(Ω) to L_{2}(ω). Applying Lemma 1, from (17) and (18) it follows that ψ_{ij} is a bounded bilinear operator from [W_{p}^{1+ε}(Ω), W_{p}^{3}(Ω)]_{θ} × [W_{2}^{2}(Ω), W_{2}^{4}(Ω)]_{θ} to L_{2}(ω), with the norm M ≤ C · h^{2θ}. According to Lemma 2, (8) and (9)

[W_{p}^{1+ε}(Ω), W_{p}^{3}(Ω)]_{θ} = B_{pp}^{1+ε+θ(2−ε)}(Ω) and [W_{2}^{2}(Ω), W_{2}^{4}(Ω)]_{θ} = W_{2}^{2+2θ}(Ω).

Setting 2 + 2θ = s, we obtain

∥ψ_{ij}∥_{ω} ≤ C · h^{s−2}∥a_{ij}∥_{B_{pp}^{s−1+ε(2−s/2)}(Ω)}∥u∥_{W_{2}^{s}(Ω)}, p ≥ 3, 2 < s < 4. (19)

Combining (11) and (17)–(19) we have thus proved the following result:

Theorem. The FDS (2) converges in the norm of the space W_{2}^{2}(ω) and following estimate, which is consistent with the smoothness of the data, holds true

∥u − v∥_{W_{2}^{2}(ω)} ≤ C · h^{s−2} · max_{ij} ∥a_{ij}∥_{B_{pp}^{s−1+ε(2−s/2)}(Ω)} · ∥u∥_{W_{2}^{s}(Ω)}, p ≥ 3, 2 ≤ s ≤ 4.

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(received 20.08.1997.)

University of Belgrade, Faculty of Mathematics, Studentski trg 16, 11000 Belgrade, Yugoslavia

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