MATEMATIQKI VESNIK 49 (1997), 99–107

UDK 519.633.6

originalni nauqni rad research paper

APPLICATION OF INTERPOLATION THEORY TO THE ANALYSIS

OF THE CONVERGENCE RATE FOR FINITE DIFFERENCE SCHEMES OF PARABOLIC TYPE

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. We assume that the solution of the problem and the coefficients of the equation belong to the 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 problems, the estimate of the convergence rate consistent with the smoothness of the 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 the 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 the discrete Sobolev space, and C is a positive generic constant, independent of h, τ and u. For problems with variable coefficients the constant C depends on the norms of the 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.

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.

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

2. Interpolation of Banach spaces

Let A_{1} and A_{2} be two Banach spaces, linearly and continuosly imbedded in a topological linear space A. Two such spaces are called interpolation pair {A_{1}, A_{2}} (see [11]). Consider also the space A_{1} ∩ A_{2}, with the norm

∥a∥_{A_{1}∩A_{2}} = max{∥a∥_{A_{1}}, ∥a∥_{A_{2}}} ,

and the space A_{1} + A_{2} = {a ∈ A : a = a_{1} + a_{2}, a_{i} ∈ A_{i}, i = 1, 2}, with the norm

∥a∥_{A_{1}+A_{2}} = inf_{\begin{matrix}a=a_{1}+a_{2} \\ a_{i}∈A_{i} \end{matrix}}\{∥a_{1}∥_{A_{1}} + ∥a_{2}∥_{A_{2}}\}.

Obviously, A_{1} ∩ A_{2} ⊂ A_{i} ⊂ A_{1} + A_{2}, i = 1, 2.

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

A functor F: C_{2} → C_{1} is called an interpolation functor if

A_{1} ∩ A_{2} ⊂ F({A_{1}, A_{2}}) ⊂ A_{1} + A_{2}

for every interpolation pair {A_{1}, A_{2}}, while for every morphism L ∈ L({A_{1}, A_{2}}, {B_{1}, B_{2}}), F(L) is the restriction of the operator L on F({A_{1}, A_{2}}).

The corresponding Banach space A = F({A_{1}, A_{2}}) is called interpolation space.

Note that A_{1} ∩ A_{2} and A_{1} + A_{2} are interpolation spaces.

If the inequality

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

where 0 < θ < 1 and C = const ≥ 1, is satisfied for every morphism L of category C_{2} the interpolation functor is said to be of the type θ. (Here ∥L∥_{A_{i}→B_{i}} denotes the standard operator norm of L: A_{i} → B_{i}, i = 1, 2).

One of the most often used interpolation methods is so called K-method [9,11]. Let {A_{1}, A_{2}} be an interpolation pair. Define the functional

K(t, a) = K(t, a, A_{1}, A_{2}) = inf_{\begin{matrix}a∈A_{1}+A_{2} \\ a=a_{1}+a_{2} \\ a_{i}∈A_{i} \end{matrix}}\{∥a_{1}∥_{A_{1}} + t∥a_{2}∥_{A_{2}}\}

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} defined in such a way is an interpolation space. The following relations hold:

(A_{1}, A_{2})_{θ,q} = (A_{2}, A_{1})_{1−θ,q} ,
(A, A)_{θ,q} = A ,
∥a∥_{(A_{1},A_{2})_{θ,q}} ≤ C_{θ,q}∥a∥_{A_{1}}^{1−θ}∥a∥_{A_{2}}^{θ} , ∀a ∈ A_{1} ∩ A_{2}.

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 mapping with values in C_{1}. Than L is a continuous mapping from (A_{1}, A_{2})_{θ,p} × (B_{1}, B_{2})_{θ,q} into (C_{1}, C_{2})_{θ,r}, 0 < θ < 1, \frac{1}{r} = \frac{1}{p} + \frac{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 [11].

For 0 ≤ s_{1}, s_{2} < ∞, s_{1} ≠ s_{2}, 0 < θ < 1, 1 ≤ q < ∞ we have [11]:

(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}) for all s ∈ (s_{1}, s_{2}).

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

Let us define anisotropic Sobolev spaces W_{2}^{s,s/2}(Q), Q = Ω × I, I = (0, T), as follows [4]:

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})^{\frac{1}{2}} .

These spaces are interpolation spaces, too. For s_{1}, s_{2}, r_{1}, r_{2} ≥ 0, 0 < θ < 1, we have [7,11]

(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 aproximation

Let us consider, as a model problem, the first initial-boundary value problem for a parabolic equation with variable coefficient in the rectangular domain Q = Ω × (0, T] = (0, 1) × (0, T]:

\frac{∂u}{∂t} − \frac{∂}{∂x}(a\frac{∂u}{∂x}) = f ,  (x, t) ∈ Q ,
u = 0 ,  (x, t) ∈ ∂Ω × [0, T] ,  (2)
u(x, 0) = u_{0}(x) , x ∈ Ω ,

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

W_{2}^{s−1}(Ω) [8].

Let ω be a uniform mesh in Ω = (0, 1) with the step size h, ω̄ = ω ∪ {0, 1} = ω ∪ γ. Let θ_{τ} be a uniform mesh in (0, T) with step size τ, θ_{τ}^{+} = θ_{τ} ∪ {T }, θ̄_{τ} = θ_{τ} ∪ {0, T }. We define the following 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} = \frac{v^{+} − v}{h} = v+ ; x̄ , v_{xx̄} = \frac{v^{+} − 2v + v^{−}}{h^{2}} , where v^{±}(x, t) = v(x ± h, t) ,

v_{t}(x, t) = \frac{v(x, t + τ) − v(x, t)}{τ} = vt̄(x, t + τ).

We also define the Steklov smoothing operators:

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

These operators commute with derivatives and transform derivatives into differences:

T_{x}^{2} (D_{x}^{2}f(x, t)) = D_{x}^{2} (T_{x}^{2}f(x, t)) = f_{xx̄}(x, t) ,
T_{t}^{−} (D_{t}f(x, t)) = D_{t} (T_{t}^{−}f(x, t)) = ft̄(x, t) , etc.

We approximate problem (2) with the following finite–difference scheme:

vt̄ + L_{h}v = T_{x}^{2}T_{t}^{−}f , in Q_{hτ}^{+} ,
v = 0 ,  on γ × θ̄_{τ} ,  (3)
v = u_{0} ,  on ω × {0} ,

where

L_{h}v = −\frac{1}{2}((av_{x})x̄ + (avx̄)_{x}) .

The finite-difference scheme (3) is the the standard symmetric scheme with the averaged right-hand side. Note that for s ≤ 3.5 the right-hand side may be 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 (2) and v—the solution of the finite difference scheme (3). The error z = u − v satisfies the conditions

zt̄ + L_{h}z = η + φ , in Q_{hτ}^{+} ,
z = 0 ,  on ω × {0} ,  (4)
z = 0 ,  on γ × θ̄_{τ} ,

where

η = T_{x}^{2}T_{t}^{−} (D_{x}(aD_{x}u)) − \frac{1}{2} ((au_{x})x̄ + (aux̄)_{x}) and φ = ut̄ − T_{x}^{2}ut̄.

We define the discrete inner products

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

where v(·, t) = v(x, t), (x, t) ∈ ω × {t}, t ∈ θ_{τ}^{+}–fixed,

(v, w)_{Q_{hτ}} = (v, w)_{L_{2}(Q_{hτ})} = hτ ∑_{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} + ∥v_{x}∥_{Q_{hτ}}^{2} + ∥v_{xx̄}∥_{Q_{hτ}}^{2} + ∥vt̄∥_{Q_{hτ}}^{2} .

The following assertion holds true:

Lemma 2. Finite-difference scheme (4) satisfies a priori estimate

∥z∥_{W_{2}^{2,1}(Q_{hτ})} ≤ ∥η∥_{Q_{hτ}} + ∥φ∥_{Q_{hτ}} .  (5)

In such a way, the problem of deriving a convergence rate estimate for the finite-difference scheme (3) is now reduced to estimating the right-hand side terms in (5).

Let us derive an estimate (1) for s = r = 2. We decompose η in the following manner:

η = T_{x}^{2}T_{t}^{−}(D_{x}(aD_{x})) − \frac{1}{2}((aux̄)_{x} + (au_{x})x̄)

= T_{x}^{2}(aT_{t}^{−}D_{x}^{2}u) + T_{x}^{2}(D_{x}aT_{t}^{−}D_{x}u) − au_{xx̄} − \frac{1}{2}(ax̄u_{x}^{−} + a_{x}u+ ; x̄) = ∑_{k=1}^{4} η_{k}

where:

η_{1} = T_{x}^{2}(aT_{t}^{−}D_{x}^{2}u) , η_{2} = T_{x}^{2}(D_{x}aT_{t}^{−}D_{x}u) ,
η_{3} = −au_{xx̄} , η_{4} = −\frac{1}{2}(ax̄u_{x}^{−} + a_{x}u+ ; x̄) .

The value η_{1} in the node (·, t) ∈ ω × {t} can be represented in the form

η_{1}(·, t) = \frac{1}{h} ∫_{x−h}^{x+h} (1 − \frac{|ξ − x|}{h}) a(ξ)T_{t}^{−}\frac{∂^{2}u(ξ, t)}{∂x^{2}} dξ .

Applying the Cauchy-Schwartz inequality we obtain

|η_{1}(·, t)| ≤ \frac{C}{h^{1/2}} (∫_{x−h}^{x+h} |a(ξ)T_{t}^{−}\frac{∂^{2}u(ξ, t)}{∂x^{2}} |^{2} dξ)^{\frac{1}{2}} .

From here, summing over the mesh ω, we obtain:

∥η_{1}(·, t)∥_{ω} ≤ C∥a∥_{C(\overline{Ω})}∥T_{t}^{−}u(·, t)∥_{W_{2}^{2}(Ω)} .

Using the imbedding W_{2}^{1}(Ω) ⊆ C(\overline{Ω}) we have

∥η_{1}(·, t)∥_{ω} ≤ C∥a∥_{W_{2}^{1}(Ω)}∥T_{t}^{−}u(·, t)∥_{W_{2}^{2}(Ω)} .

Summation over the mesh θ_{τ}^{+} yields:

∥η_{1}∥_{Q_{hτ}} ≤ C∥a∥_{W_{2}^{1}(Ω)}∥u∥_{W_{2}^{2,1}(Q)} .

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

∥η∥_{Q_{hτ}} ≤ C∥a∥_{W_{2}^{1}(Ω)}∥u∥_{W_{2}^{2,1}(Q)} , and  (6)
∥φ∥_{Q_{hτ}} ≤ C∥u∥_{W_{2}^{2,1}(Q)} .  (7)

From (5), (6) and (7) we obtain estimate (1) for s = r = 2.

Application of interpolation theory … 105

Let us derive estimate (1) for s = 4, r = 2. We decompose term η in the following manner: η = ∑_{k=5}^{11} η_{k}, where

η_{5} = T_{x}^{2}(aT_{t}^{−}D_{x}^{2}u) − (T_{x}^{2}a)(T_{x}^{2}T_{t}^{−}D_{x}^{2}u) ,
η_{6} = (T_{x}^{2}a − a)(T_{x}^{2}T_{t}^{−}D_{x}^{2}u) ,
η_{7} = a(T_{x}^{2}T_{t}^{−}D_{x}^{2}u − u_{xx̄}) ,
η_{8} = T_{x}^{2}(D_{x}aT_{t}^{−}D_{x}u) − (T_{x}^{2}D_{x}a)(T_{x}^{2}T_{t}^{−}D_{x}u) ,
η_{9} = (T_{x}^{2}D_{x}a − 0.5(a_{x} + ax̄))(T_{x}^{2}T_{t}^{−}D_{x}u) ,
η_{10} = 0.5(a_{x} + ax̄)(T_{x}^{2}T_{t}^{−}D_{x}u − 0.5(u_{x}^{−} + u+ ; x̄)) ,
η_{11} = 0.25(a_{x} − ax̄)(u_{x}^{−} − u+ ; x̄) .

The value of η_{5} in the node (·, t) ∈ ω × {t} can be represented in the form

η_{5}(·, t) = \frac{1}{h^{2}} ∫_{x−h}^{x+h} ∫_{x−h}^{x+h} ∫_{σ}^{ξ} ∫_{σ}^{ξ} (1 − \frac{|ξ − x|}{h}) (1 − \frac{|σ − x|}{h}) ×

× a′(ρ)T_{t}^{−}\frac{∂^{3}u(ρ_{1}, t)}{∂x^{3}} dρ_{1}dρdσdξ .

From here, using the Cauchy-Schwartz inequality we obtain

|η_{5}(·, t)| ≤ Ch^{3/2}∥a∥_{W_{∞}^{1}(Ω)}∥T_{t}^{−}u(·, t)∥_{W_{2}^{3}(x−h,x+h)} .

Summation over the mesh ω yields:

∥η_{5}(·, t)∥_{ω} ≤ Ch^{2}∥a∥_{W_{∞}^{1}(Ω)}∥T_{t}^{−}u(·, t)∥_{W_{2}^{3}(Ω)} .

Using the imbedding W_{2}^{3}(Ω) ⊂ W_{∞}^{1}(Ω) we obtain

∥η_{5}(·, t)∥_{ω} ≤ Ch^{2}∥a∥_{W_{2}^{3}(Ω)}∥T_{t}^{−}u(·, t)∥_{W_{2}^{3}(Ω)} .

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

∥η_{5}∥_{Q_{hτ}} ≤ Ch^{2}∥a∥_{W_{2}^{3}(Ω)}∥u∥_{W_{2}^{4,2}(Q)} . The value of η_{7} in the node (x, t) ∈ Q_{hτ}^{+} can be represented in the form:

η_{7}(x, t) = \frac{1}{hτ} a(x) ∫_{x−h}^{x+h} ∫_{t−τ}^{t} ∫_{t}^{ν} (1 − \frac{|ξ − x|}{h}) \frac{∂^{3}u(ξ, ν_{1})}{∂x^{2}∂t} dν_{1}dνdξ .

From here, using the Cauchy-Schwartz inequality we have

|η_{7}(x, t)| ≤ C (\frac{τ}{h})^{1/2} ∥a∥_{C(\overline{Ω})} (∫_{x−h}^{x+h} ∫_{t−τ}^{t} |\frac{∂^{3}u(ξ, ν)}{∂x^{2}∂t}|^{2} dνdξ)^{\frac{1}{2}} .

Summation over the mesh Q_{hτ}^{+} yields:

∥η_{7}∥_{Q_{hτ}} ≤ Cτ∥a∥_{C(\overline{Ω})} ∥\frac{∂^{3}u}{∂x^{2}∂t}∥_{L_{2}(Q)} .

Using the imbedding W_{2}^{3}(Ω) ⊆ C(\overline{Ω}) and the imbedding theorems for anisotropic spaces W_{2}^{s,s/2}(Q) [4] we have

∥η_{7}∥_{Q_{hτ}} ≤ Ch^{2}∥a∥_{W_{2}^{3}(Ω)}∥u∥_{W_{2}^{4,2}(Q)} .

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

∥η∥_{Q_{hτ}} ≤ Ch^{2}∥a∥_{W_{2}^{3}(Ω)}∥u∥_{W_{2}^{4,2}(Q)} , and  (8)
∥φ∥_{Q_{hτ}} ≤ Ch^{2}∥u∥_{W_{2}^{4,2}(Q)} .  (9)

From (5), (8) and (9) we obtain estimate (1) for s = 4, r = 2.

Let us define the operators A_{1} and A_{2} as follows:

η = A_{1}(a, u) and φ = A_{2}(u) .

The operator A_{1} is, obviously, bilinear. From (6) it follows that A_{1} is a bounded bilinear operator from W_{2}^{1}(Ω) × W_{2}^{2,1}(Q) to L_{2}(Q_{hτ}), and

∥A_{1}∥_{W_{2}^{1}(Ω)×W_{2}^{2,1}(Q)→L_{2}(Q_{hτ})} ≤ C .  (10)

From (8) it follows that A_{1} is a bounded bilinear operator from W_{2}^{3}(Ω) × W_{2}^{4,2}(Q) to L_{2}(Q_{hτ}), and

∥A_{1}∥_{W_{2}^{3}(Ω)×W_{2}^{4,2}(Q)→L_{2}(Q_{hτ})} ≤ Ch^{2} .  (11)

Applying lemma 1, from (10) and (11) it follows that A_{1} is a bounded bilinear operator from

(W_{2}^{3}(Ω), W_{2}^{1}(Ω))_{θ,2} × (W_{2}^{4,2}(Q), W_{2}^{2,1}(Q))_{θ,2} = W_{2}^{3−2θ}(Ω) × W_{2}^{4−2θ,2−θ}(Q)

to

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

and

∥A_{1}∥_{W_{2}^{3−2θ}(Ω)×W_{2}^{4−2θ,2−θ}(Q)→L_{2}(Q_{hτ})} ≤ Ch^{2−2θ} , 0 < θ < 1.  (12)

Finally, from (12) and the inequality

∥η∥_{Q_{hτ}} ≤ ∥A_{1}∥_{W_{2}^{3−2θ}(Ω)×W_{2}^{4−2θ,2−θ}(Q)→L_{2}(Q_{hτ})}∥a∥_{W_{2}^{3−2θ}(Ω)}∥u∥_{W_{2}^{4−2θ,2−θ}(Q)} ,

we obtain the estimate

∥η∥_{Q_{hτ}} ≤ Ch^{2−2θ}∥a∥_{W_{2}^{3−2θ}(Ω)}∥u∥_{W_{2}^{4−2θ,2−θ}(Q)} , 0 < θ < 1 .  (13)

Analogously, we obtain the following estimate of term φ:

∥φ∥_{Q_{hτ}} ≤ Ch^{2−2θ}∥u∥_{W_{2}^{4−2θ,2−θ}(Q)} , 0 < θ < 1 .  (14)

Setting 4 − 2θ = s, we obtain the estimates:

∥η∥_{Q_{hτ}} ≤ Ch^{s−2}∥a∥_{W_{2}^{s−1}(Ω)}∥u∥_{W_{2}^{s,s/2}(Q)} ,  (15)
∥φ∥_{Q_{hτ}} ≤ Ch^{s−2}∥u∥_{W_{2}^{s,s/2}(Q)} , 2 < s < 4 .  (16)

Finally, from (6)–(9), (15), (16) and (5) we obtain the main result of this paper:

Theorem. Finite-difference scheme (3) 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}(∥a∥_{W_{2}^{s−1}(Ω)} + 1)∥u∥_{W_{2}^{s,s/2}(Q)} , 2 ≤ s ≤ 4 .

This estimate is consistent with the smoothness of data.

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(received 08.10.1996.)

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