����������� ������ 49 (1997), 257–264

UDK 517.962.2

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CONVERGENCE OF A FINITE DIFFERENCE METHOD FOR THE HEAT EQUATION — INTERPOLATION TECHNIQUE

Dejan Bojović and Boško S. Jovanović

Abstract. In this paper we show how the theory of interpolation of function spaces can be used to establish convergence rate estimates for finite difference schemes. As a model problem we consider the first initial-boundary value problem for the heat equation with variable coefficients in a domain (0, 1)^{2} × (0, T]. We assume that the solution of the problem and the coefficients of equation belong to corresponding Sobolev spaces. Using interpolation theory we construct a fractional–order convergence rate estimate which is consistent with the smoothness of the data.

1. Introduction

For a class of finite difference schemes for parabolic initial-boundary value problem, estimates of the convergence rate consistent with the smoothness of data, are of major interest, i.e.

∥u − v∥_{W_{2}^{r,r/2}(Q_{hτ})} ≤ C(h + \sqrt{τ})^{s−r}∥u∥_{W_{2}^{s,s/2}(Q)} , s ≥ r.  (1)

Here u = u(x, t) denotes the solution of the original initial-boundary value problem, v denotes the solution of corresponding finite difference scheme, h and τ are discretisation parameters, W_{2}^{s,s/2}(Q) denotes a Sobolev space, W_{2}^{s,s/2}(Q_{hτ}) denotes a discrete Sobolev space, and C is a positive generic constant, independent of h, τ and u. If parameters h and τ satisfy the condition k_{1}h^{2} ≤ τ ≤ k_{2}h^{2}, k_{1}, k_{2} = const > 0, then we obtain the estimate

∥u − v∥_{W_{2}^{r,r/2}(Q_{hτ})} ≤ Ch^{s−r}∥u∥_{W_{2}^{s,s/2}(Q)} , s ≥ r.  (2)

For problems with variable coefficients the constant C depends on the norms of coefficients.

A standard technique for the derivation of such estimates is based on the Bramble-Hilbert lemma [2]. In this paper we expose an alternative technique, based on the theory of interpolation of Banach spaces. Estimate (2) was derived in the paper [3], by the same technique, for r = 2 and 2 ≤ s ≤ 4 in the domain Q = (0, 1) × (0, T]. In this paper we derive estimate (2) for r = 2 and 2 ≤ s ≤ 4 in the domain Q = (0, 1)^{2} × (0, T].

AMS Subject Classification: 65 M 15, 46 B 70

Keywords and phrases: Initial-Boundary Value Problems, Finite Differences, Interpolation of Function Spaces, Sobolev 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

2. Interpolation of Banach spaces

In this paper we use the K-method of real interpolation [10,12]. Let {A_{1}, A_{2}} be an interpolation pair. Define the functional

K(t, a) = K(t, a, A_{1}, A_{2}) = inf{ ∥a_{1}∥_{A_{1}}+t∥a_{2}∥_{A_{2}} | a ∈ A_{1}+A_{2}, a = a_{1}+a_{2}, a_{i} ∈ A_{i} }.

It is obvious that, for a fixed t ∈ (0, ∞), K(t, a) is a norm in A_{1} + A_{2}, equivalent to the standard norm ∥a∥_{A_{1}+A_{2}}. For 0 < θ < 1, 1 ≤ q < ∞, let us define the space (A_{1}, A_{2})_{θ,q} as follows:

(A_{1}, A_{2})_{θ,q} = \{a ∈ A_{1} + A_{2} : ∥a∥_{(A_{1},A_{2})_{θ,q}} = (∫_{0}^{∞} (t^{−θ}K(t, a))^{q} \frac{dt}{t})^{\frac{1}{q}} < ∞\} ,

and for q = ∞

(A_{1}, A_{2})_{θ,∞} = {a ∈ A_{1} + A_{2} : ∥a∥_{(A_{1},A_{2})_{θ,∞}} = sup_{0<t<∞} t^{−θ}K(t, a) < ∞}.

The space (A_{1}, A_{2})_{θ,q} is an interpolation space. The corresponding interpolation functor F({A_{1}, A_{2}}) = (A_{1}, A_{2})_{θ,q} is of the type θ, i.e.

∥L∥_{(A_{1},A_{2})_{θ,q}→(B_{1},B_{2})_{θ,q}} ≤ ∥L∥_{A_{1}→B_{1}}^{1−θ}∥L∥_{A_{2}→B_{2}}^{θ},

An analogous assertion holds true for bilinear operators:

Lemma 1. Let A_{1} ⊂ A_{2}, B_{1} ⊂ B_{2} and C_{1} ⊂ C_{2} and let L: A_{2} × B_{2} → C_{2} be a continuous bilinear form whose restriction on A_{1} × B_{1} is a continuous maping with values in C_{1}. Then L is continuous maping from (A_{1}, A_{2})_{θ,p} ×(B_{1}, B_{2})_{θ,q} into (C_{1}, C_{2})_{θ,r}, 0 < θ < 1, 1/r = 1/p + 1/q − 1 ≥ 0, and

∥L∥_{(A_{1},A_{2})_{θ,p}×(B_{1},B_{2})_{θ,q}→(C_{1},C_{2})_{θ,r}} ≤ ∥L∥_{A_{1}×B_{1}→C_{1}}^{1−θ}∥L∥_{A_{2}×B_{2}→C_{2}}^{θ}.

As an example of interpolation spaces, let us consider the Sobolev spaces W_{p}^{s} [1]. For noninteger positive s one sets W_{p}^{s}(ℝ^{n}) = B_{p,p}^{s}(ℝ^{n}), where B_{pp}^{s} is a Besov space [12].

For 0 ≤ s_{1}, s_{2} < ∞, s_{1} ≠ s_{2}, 0 < θ < 1, 1 ≤ q < ∞ we have [12]:

(W_{p}^{s_{1}}(ℝ^{n}), W_{p}^{s_{2}}(ℝ^{n}))_{θ,q} = B_{p,q}^{s}(ℝ^{n}) , s = (1 − θ)s_{1} + θs_{2} .

In such a way, for q = p and noninteger s = (1 − θ)s_{1} + θs_{2}, we obtain

(W_{p}^{s_{1}}(ℝ^{n}), W_{p}^{s_{2}}(ℝ^{n}))_{θ,p} = W_{p}^{s}(ℝ^{n}) , s = (1 − θ)s_{1} + θs_{2} .

For p = 2 this relation holds without restrictions, i.e.:

(W_{2}^{s_{1}}(ℝ^{n}), W_{2}^{s_{2}}(ℝ^{n}))_{θ,2} = W_{2}^{s}(ℝ^{n}) .

Hence, W_{2}^{s}(ℝ^{n}) are interpolation spaces. The same result holds for Sobolev spaces in a domain Ω with sufficiently smooth boundary.

Let us define the anisotropic Sobolev space W_{2}^{s,s/2}(Q), Q = Ω × I, I = (0, T), as follows [5]: W_{2}^{s,s/2}(Q) = L_{2}(I, W_{2}^{s}(Ω)) ∩ W_{2}^{s/2}(I, L_{2}(Ω)), with the norm

∥f∥_{W_{2}^{s,s/2}(Q)} = (∫_{0}^{T} ∥f(t)∥_{W_{2}^{s}(Ω)}^{2}dt + ∥f∥_{W_{2}^{s/2}(I,L_{2}(Ω))}^{2})^{1/2} .

These spaces are interpolation spaces, too. For s_{1}, s_{2}, r_{1}, r_{2} ≥ 0, 0 < θ < 1, we have [8,12] (W_{2}^{s_{1},r_{1}}(Q), W_{2}^{s_{2},r_{2}}(Q))_{θ,2} = W_{2}^{s,r}(Q) , s = (1 − θ)s_{1} + θs_{2}, r = (1 − θ)r_{1} + θr_{2}.

3. Initial-boundary value problem and its approximation

Let us consider the first initial-boundary value problem for parabolic equation with variable coefficients in the cylinder Q = Ω × (0, T] = (0, 1)^{2} × (0, T]:

\frac{∂u}{∂t} − ∑_{i,j=1}^{2} D_{i}(a_{ij}D_{j}u) = f , (x, t) ∈ Q,
u = 0 , (x, t) ∈ ∂Ω × [0, T],  (3)
u(x, 0) = u_{0}(x) , x ∈ Ω,

We assume that the generalized solution of the problem (3) belongs to the Sobolev space W_{2}^{s,s/2}(Q), 2 ≤ s ≤ 4, with the right-hand side f(x, t) which belongs to W_{2}^{s−2,s/2−1}(Q). Consequently, the coefficients a_{ij} = a_{ij}(x) belong to the space of multipliers M (W_{2}^{s−1,(s−1)/2}(Q)), i.e. it is sufficient that [9]:

a_{ij} ∈ W_{2}^{s−1}(Ω) ,  for 2 < s ≤ 4 ,
a_{ij} ∈ W_{2}^{1+δ}(Ω) , δ > 0 , for s = 2 .

Let ω̄ be the uniform mesh in \overline{Ω} = [0, 1]^{2} with the step size h, ω = ω̄ ∩ Ω, γ = ω̄ ∩ ∂Ω. Let θ_{τ} be the uniform mesh in (0, T) with the step size τ, θ_{τ}^{+} = θ_{τ} ∪ {T }, θ̄_{τ} = θ_{τ} ∪ {0, T }. We define the uniform mesh in Q: Q_{hτ} = ω × θ_{τ}, Q_{hτ}^{+} = ω × θ_{τ}^{+} and \overline{Q}_{hτ} = ω̄ × θ̄_{τ}. We assume that the condition:

k_{1}h^{2} ≤ τ ≤ k_{2}h^{2} , k_{1}, k_{2} = const > 0

is satisfied. We define finite differences in the usual manner:

v_{x_{i}} = \frac{v^{+i} − v}{h} = v+i ; x̄_{i} , v_{t}(x, t) = \frac{v(x, t + τ) − v(x, t)}{τ} = vt̄(x, t + τ) ,

where v^{±i}(x, t) = v(x ± hr_{i}, t), and r_{i} is the unit vector along the x_{i} axis. We also define the Steklov smoothing operators:

T_{i}^{+}f(x, t) = ∫_{0}^{1} f(x + hx′r_{i}, t) dx′ = T_{i}^{−}f(x + hr_{i}, t) ,
T_{i}^{2}f(x, t) = T_{i}^{+}T_{i}^{−}f(x, t) = ∫_{−1}^{1}(1 − |x′|)f(x + hx′r_{i}, t) dx′ ,
T_{t}^{+}f(x, t) = ∫_{0}^{1} f(x, t + τt′) dt′ = T_{t}^{−}f(x, t + τ) .

We approximate problem (3) with the following finite–difference scheme:

vt̄ + L_{h}v = T_{1}^{2}T_{2}^{2}T_{t}^{−}f , in Q_{hτ}^{+} ,
v = 0 ,  on γ × θ̄_{τ} ,  (4)
v = u_{0} ,  on ω × {0} ,

where

L_{h}v = −0.5 ∑_{i,j=1}^{2}((a_{ij}vx̄_{j})_{x_{i}} + (a_{ij}v_{x_{j}})x̄_{i}) .

The finite-difference scheme (4) is the standard symmetric scheme with the averaged right-hand side. Note that for s ≤ 4 the right-hand side may be a discontinuous function, so without averaging the scheme is not well defined.

4. Convergence of the finite-difference scheme

Let u be the solution of the initial-boundary value problem (3) and v the solution of the finite difference scheme (4). The error z = u − v satisfies the conditions

zt̄ + L_{h}z = ∑_{i,j=1}^{2} η_{ij} + φ , in Q_{hτ}^{+} ,
z = 0 ,  on ω × {0} ,  (5)
z = 0 ,  on γ × θ̄_{τ} ,

where

η_{ij} = T_{1}^{2}T_{2}^{2}T_{t}^{−} (D_{i}(a_{ij}D_{j}u)) − 0.5((a_{ij}ux̄_{j})_{x_{i}} + (a_{ij}u_{x_{j}})x̄_{i}) , φ = ut̄ − T_{1}^{2}T_{2}^{2}ut̄.

We define the discrete inner products:

(v, w)_{ω} = (v, w)_{L_{2}(ω)} = h^{2} ∑_{x∈ω} v(·, t)w(·, t) ,

(v, w)_{Q_{hτ}} = (v, w)_{L_{2}(Q_{hτ})} = h^{2}τ ∑_{x∈ω} ∑_{t∈θ_{τ}^{+}} v(x, t)w(x, t) = τ ∑_{t∈θ_{τ}^{+}}(v, w)_{ω} ,

and the discrete Sobolev norms:

∥v∥_{ω}^{2} = (v, v)_{ω} , ∥v∥_{Q_{hτ}}^{2} = (v, v)_{Q_{hτ}} ,

∥v∥_{W_{2}^{2,1}(Q_{hτ})}^{2} = ∥v∥_{Q_{hτ}}^{2} + ∑_{i=1}^{2} ∥v_{x_{i}}∥_{Q_{hτ}}^{2} + ∑_{i,j=1}^{2} ∥v_{x_{i}x_{j}}∥_{Q_{hτ}}^{2} + ∥vt̄∥_{Q_{hτ}}^{2} .

The following assertion holds true :

Lemma 2. Finite-difference scheme (5) satisfies a priori estimate

∥z∥_{W_{2}^{2,1}(Q_{hτ})} ≤ ∑_{i,j=1}^{2} ∥η_{ij}∥_{Q_{hτ}} + ∥φ∥_{Q_{hτ}} .  (6)

In such a way, the problem of deriving the convergence rate estimate for finitedifference scheme (4) is now reduced to estimating the right-hand side terms in (6).

We decompose term η_{ij} in the following manner: η_{ij} = ∑_{k=1}^{7} η_{ijk}, where

η_{ij1} = T_{1}^{2}T_{2}^{2}(a_{ij}T_{t}^{−}D_{i}D_{j}u) − (T_{1}^{2}T_{2}^{2}a_{ij})(T_{1}^{2}T_{2}^{2}T_{t}^{−}D_{i}D_{j}u) ,
η_{ij2} = (T_{1}^{2}T_{2}^{2}a_{ij} − a_{ij})(T_{1}^{2}T_{2}^{2}T_{t}^{−}D_{i}D_{j}u) ,
η_{ij3} = a_{ij}(T_{1}^{2}T_{2}^{2}T_{t}^{−}D_{i}D_{j}u − 0.5(u_{x̄_{i}x_{j}} + u_{x_{i}x̄_{j}})) ,
η_{ij4} = T_{1}^{2}T_{2}^{2}(D_{i}a_{ij}T_{t}^{−}D_{j}u) − (T_{1}^{2}T_{2}^{2}D_{i}a_{ij})(T_{1}^{2}T_{2}^{2}T_{t}^{−}D_{j}u) ,
η_{ij5} = (T_{1}^{2}T_{2}^{2}D_{i}a_{ij} − 0.5(a_{ij,x_{i}} + a_{ij,x̄_{i}}))(T_{1}^{2}T_{2}^{2}T_{t}^{−}D_{j}u) ,
η_{ij6} = 0.5(a_{ij,x_{i}} + a_{ij,x̄_{i}})(T_{1}^{2}T_{2}^{2}T_{t}^{−}D_{j}u − 0.5(u_{x_{j}}^{−i} + u+i ; x̄_{j})) ,
η_{ij7} = 0.25(a_{ij,x_{i}} − a_{ij,x̄_{i}})(u_{x_{j}}^{−i} − u+i ; x̄_{j}) .

Let us derive the estimate (2) for s = 2 , r = 2.

The value η_{ij1} in the node (·, t) ∈ ω × {t} can be represented in the form

η_{ij1}(·, t) = \frac{1}{h^{2}} ∫∫_{e} k(ξ_{1}, ξ_{2})a_{ij}(ξ_{1}, ξ_{2})T_{t}^{−}D_{i}D_{j}u(ξ_{1}, ξ_{2}, t)dξ_{1}dξ_{2}−
−\frac{1}{h^{2}} ∫∫_{e} k(σ_{1}, σ_{2})a_{ij}(σ_{1}, σ_{2})dσ_{1}dσ_{2} × \frac{1}{h^{2}} ∫∫_{e} k(ξ_{1}, ξ_{2})T_{t}^{−}D_{i}D_{j}u(ξ_{1}, ξ_{2}, t)dξ_{1}dξ_{2} (7)

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

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

From (7) immediately follows:

|η_{ij1}(·, t)| ≤ \frac{C}{h} ∥a_{ij}∥_{C(ē)}∥T_{t}^{−}u(·, t)∥_{W_{2}^{2}(e)}

Summation over the mesh ω yields:

∥η_{ij1}(·, t)∥_{ω} ≤ C∥a_{ij}∥_{C(\overline{Ω})}∥T_{t}^{−}u(·, t)∥_{W_{2}^{2}(Ω)} ≤ C∥a_{ij}∥_{W_{2}^{1+δ}(Ω)}∥T_{t}^{−}u(·, t)∥_{W_{2}^{2}(Ω)}

From here, summing over the mesh θ_{τ}^{+} we obtain

∥η_{ij1}∥_{Q_{hτ}} ≤ C∥a_{ij}∥_{W_{2}^{1+δ}(Ω)}∥u∥_{W_{2}^{2,1}(Q)} .

Analogous estimates hold true also for the other terms η_{ijk} and for term φ. In these estimates we assume that u ∈ W_{2}^{2+ε,1+ε/2}(Q), ε > 0. In such a way we obtain the estimates:

∥η_{ij}∥_{Q_{hτ}} ≤ C∥a_{ij}∥_{W_{2}^{1+δ}(Ω)}∥u∥_{W_{2}^{2+ε,1+ε/2}(Q)} , and  (8)
∥φ∥_{Q_{hτ}} ≤ C∥u∥_{W_{2}^{2+ε,1+ε/2}(Q)} .  (9)

From (6), (8) and (9) we obtain estimate (2) for s = 2 , r = 2.

Let us derive estimate (2) for s = 3 , r = 2.

The value η_{ij1} in the node (·, t) ∈ ω × {t} can be represented in the form

η_{ij1}(·, t) = \frac{1}{2h^{4}} ∫∫∫∫_{e×e}k(ξ_{1}, ξ_{2})k(σ_{1}, σ_{2})( ∫_{σ_{1}}^{ξ_{1}}D_{1}a_{ij}(τ_{1}, σ_{2})dτ_{1}+∫_{σ_{2}}^{ξ_{2}}D_{2}a_{ij}(ξ_{1}, τ_{2})dτ_{2})
× (T_{t}^{−}D_{i}D_{j}u(ξ_{1}, ξ_{2}, t) − T_{t}^{−}D_{i}D_{j}u(σ_{1}, σ_{2}, t))dξ_{1}dξ_{2}dσ_{1}dσ_{2} ,  (10)

From here, using Cauchy–Schwarz’s and Hölder’s inequality we obtain

|η_{ij1}(·, t)| ≤ C∥a_{ij}∥_{W_{p}^{1}(e)}∥T_{t}^{−}u(·, t)∥_{W 2 ; \frac{2p}{p−2}(e)} , p > 2 .

Summing over the meshes ω and θ_{τ}^{+}, using the imbeddings W_{2}^{2}(Ω) ⊂ W_{p}^{1}(Ω), we simply obtain

∥η_{ij1}∥_{Q_{hτ}} ≤ Ch∥a_{ij}∥_{W_{2}^{2}(Ω)}∥u∥_{W_{2}^{3,3/2}(Q)} .

Analogous estimates hold true also for the other terms η_{ijk} and for term φ. In such a way we obtain the estimates:

∥η_{ij}∥_{Q_{hτ}} ≤ Ch∥a_{ij}∥_{W_{2}^{2}(Ω)}∥u∥_{W_{2}^{3,3/2}(Q)} , and  (11)
∥φ∥_{Q_{hτ}} ≤ Ch∥u∥_{W_{2}^{3,3/2}(Q)} .  (12)

From (6), (11) and (12) we obtain estimate (2) for s = 3 , r = 2.

Let us derive estimate (2) for s = 4, r = 2.

From (10), using the representation T_{t}^{−}D_{i}D_{j}u(ξ_{1}, ξ_{2}, t) − T_{t}^{−}D_{i}D_{j}u(σ_{1}, σ_{2}, t) =

= ∫_{σ_{1}}^{ξ_{1}} T_{t}^{−}D_{1}D_{i}D_{j}u(ρ_{1}, σ_{2}, t)dρ_{1} + ∫_{σ_{2}}^{ξ_{2}} T_{t}^{−}D_{2}D_{i}D_{j}u(ξ_{1}, ρ_{2}, t)dρ_{2} ,

and Cauchy-Schwarz’s and Hölder’s inequality we obtain

|η_{ij1}(·, t)| ≤ Ch∥a_{ij}∥_{W_{p}^{1}(e)}∥T_{t}^{−}u(·, t)∥_{W 3 ; \frac{2p}{p−2}(e)} , p > 2 .

Summing over the meshes ω and θ_{τ}^{+}, using the imbeddings W_{2}^{3}(Ω) ⊂ W_{p}^{1}(Ω) and W_{2}^{4}(Ω) ⊂ W_{2p/(p−2)}^{3}(Ω), we simply obtain

∥η_{ij1}∥_{Q_{hτ}} ≤ Ch^{2}∥a_{ij}∥_{W_{2}^{3}(Ω)}∥u∥_{W_{2}^{4,2}(Q)} .

Analogous estimates hold true also for the other terms η_{ijk} and for term φ. In such a way we obtain the estimates:

∥η_{ij}∥_{Q_{hτ}} ≤ Ch^{2}∥a_{ij}∥_{W_{2}^{3}(Ω)}∥u∥_{W_{2}^{4,2}(Q)} , and  (13)
∥φ∥_{Q_{hτ}} ≤ Ch^{2}∥u∥_{W_{2}^{4,2}(Q)} .  (14)

From (6), (13) and (14) we obtain estimate (2) for s = 4 , r = 2.

Let us define the operators A_{ij} and B as follows:

η_{ij} = A_{ij}(a_{ij}, u) and φ = B(u) .

The operator A_{ij} is, obviously, bilinear. From (8), (11) and (13) it follows that it is a bounded bilinear operator from W_{2}^{1+δ}(Ω) × W_{2}^{2+ε,1+ε/2}(Q) to L_{2}(Q_{hτ}), from W_{2}^{2}(Ω) × W_{2}^{3,3/2}(Q) to L_{2}(Q_{hτ}) and from W_{2}^{3}(Ω) × W_{2}^{4,2}(Q) to L_{2}(Q_{hτ}) with the norm:

∥A_{ij}∥_{W_{2}^{1+δ}(Ω)×W_{2}^{2+ε,1+ε/2}(Q)→L_{2}(Q_{hτ})} ≤ C ,  (15)
∥A_{ij}∥_{W_{2}^{2}(Ω)×W_{2}^{3,3/2}(Q)→L_{2}(Q_{hτ})} ≤ Ch .  (16)
∥A_{ij}∥_{W_{2}^{3}(Ω)×W_{2}^{4,2}(Q)→L_{2}(Q_{hτ})} ≤ Ch^{2} .  (17)

Applying Lemma 1, from (16) and (17) it follows that A_{ij} is a bounded bilinear operator from

(W_{2}^{3}(Ω), W_{2}^{2}(Ω))_{θ,2} × (W_{2}^{4,2}(Q), W_{2}^{3,3/2}(Q))_{θ,2} = W_{2}^{3−θ}(Ω) × W_{2}^{4−θ,2−θ/2}(Q)

to

(L_{2}(Q_{hτ}), L_{2}(Q_{hτ}))_{θ,∞} = L_{2}(Q_{hτ}) ,

and

∥A_{ij}∥_{W_{2}^{3−θ}(Ω)×W_{2}^{4−θ,2−θ/2}(Q)→L_{2}(Q_{hτ})} ≤ Ch^{2−θ} , 0 < θ < 1 .

Finally, we obtain the estimate:

∥η_{ij}∥_{Q_{hτ}} ≤ Ch^{2−θ}∥a_{ij}∥_{W_{2}^{3−θ}(Ω)}∥u∥_{W_{2}^{4−θ,2−θ/2}(Q)} , 0 < θ < 1 .

Setting 4 − θ = s, we obtain the estimate:

∥η_{ij}∥_{Q_{hτ}} ≤ Ch^{s−2}∥a_{ij}∥_{W_{2}^{s−1}(Ω)}∥u∥_{W_{2}^{s,s/2}(Q)} , 3 < s < 4 .  (18)

Similarly, from (15) and (16), by interpolation, we obtain the estimate:

∥η_{ij}∥_{Q_{hτ}} ≤ Ch^{s−2}∥a_{ij}∥_{W_{2}^{s−1+δ(3−s)}(Ω)}∥u∥_{W_{2}^{s+ε(3−s),(s+ε(3−s))/2}(Q)} , 2 < s < 3 . (19)

Analogously, we obtain the estimate of term φ:

∥φ∥_{Q_{hτ}} ≤ Ch^{s−2}∥u∥_{W_{2}^{s,s/2}(Q)} , 3 < s < 4 ,  (20)
∥φ∥_{Q_{hτ}} ≤ Ch^{s−2}∥u∥_{W_{2}^{s+ε(3−s),(s+ε(3−s))/2}(Q)} , 2 < s < 3.  (21)

Finally, from (8)–(14), (18)–(21) and (6) we obtain the main result of this paper:

Theorem. Finite-difference scheme (4) converges in the norm of the space W_{2}^{2,1}(Q_{hτ}) and, with condition k_{1}h^{2} ≤ τ ≤ k_{2}h^{2}, the following estimate holds true:

∥u − v∥_{W_{2}^{2,1}(Q_{hτ})} ≤ Ch^{s−2}(max_{i,j} ∥a_{ij}∥_{W_{2}^{s−1+δ(3−s)}(Ω)} + 1)∥u∥_{W_{2}^{s+ε(3−s),(s+ε(3−s))/2}(Q)} ,

2 ≤ s ≤ 3 ,

∥u − v∥_{W_{2}^{2,1}(Q_{hτ})} ≤ Ch^{s−2}(max_{i,j} ∥a_{ij}∥_{W_{2}^{s−1}(Ω)} + 1)∥u∥_{W_{2}^{s,s/2}(Q)} , 3 ≤ s ≤ 4 .

The second estimate is consistent with the smoothness of data, while the first estimate is “almost” consistent with the smoothness of data.

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[11] A.A. Samarski, Theory of Difference Schemes. Nauka, Moscow 1983. (in Russian)

[12] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. Deutscher Verlag der Wissenschaften, Berlin 1978.

[13] A.A. Zlotnik, Convergence rate estimates of projection difference methods for second–order hyperbolic equations. Vychisl. protsessy sist. 8 (1991), 116–167. (in Russian)

(received 12.06.1997.)

University of Kragujevac, Faculty of Science, Radoja Domanovića 12, 34000 Kragujevac, Yugoslavia University of Belgrade, Faculty of Mathematics, Studentski trg 16, 11000 Belgrade, Yugoslavia