PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 99(113) (2016), 67–76 DOI: 10.2298/PIM1613067S

FINITE DIFFERENCE APPROXIMATION FOR PARABOLIC INTERFACE PROBLEM WITH TIME-DEPENDENT COEFFICIENTS

Bratislav V. Sredojević and Dejan R. Bojović

Abstract. The convergence of difference scheme for two-dimensional initialboundary value problem for the heat equation with concentrated capacity and time-dependent coefficients of the space derivatives, is considered. An estimate of the rate of convergence in a special discrete _{W_{2}^{1,1/2}} Sobolev norm, compatible with the smoothness of the coefficients and solution, is proved.

1. Introduction

The finite-difference method is one of the basic tools for the numerical solution of partial differential equations. In the case of problems with discontinuous coefficients and concentrated factors (Dirac delta functions, free boundaries, etc.) the solution has a weak global regularity and it is impossible to establish convergence of finite difference schemes using the classical Taylor series expansion. Often, the Bramble–Hilbert lemma takes the role of the Taylor formula for functions from the Sobolev spaces [6, 8, 12].

Following Lazarov et al. [12], a convergence rate estimate of the form

∥u − v∥_{W_{2,h}^{k}} ⩽ Ch^{s−k} ∥u∥_{W_{2}^{s}} , s > k,

is called compatible with the smoothness (regularity) of the solution u of the boundary-value problem. Here v is the solution of the discrete problem, h is the spatial mesh step, W_{2}^{s} and W_{2,h}^{k} are Sobolev spaces of functions with continuous and discrete argument, respectively, C is a constant which doesn’t depend on u and h. For the parabolic case typical estimates are of the form

∥u − v∥_{W_{2,hτ}^{k,k/2}} ⩽ C(h + \sqrt{τ})^{s−k} ∥u∥_{W_{2}^{s,s/2}} , s > k,

2010 Mathematics Subject Classification: 65M12, 65M15.

Key words and phrases: interface problem, convergence, Sobolev norm.

Partially supported the Serbian Ministry of Education, Science and Technological Development (Project 174002).

Communicated by Boško Jovanović.

where τ is the time step. In the case of equations with variable coefficients the constant C in the error bounds depends on norms of the coefficients (see, for example, [1, 8, 18]).

One interesting class of parabolic problems model processes in heat-conducting media with concentrated capacity in which the heat capacity coefficient contains a Dirac delta function, or equivalently, the jump of the heat flow in the singular point is proportional to the time-derivative of the temperature [14]. Such problems are nonstandard and the classical tools of the theory of finite difference schemes are difficult to apply to their convergence analysis.

In the present paper a finite-difference scheme, approximating the two-dimen-sonal initial-boundary value problem for the heat equation with concentrated capacity and time dependent coefficients is derived. Special Sobolev norm (corresponding to the norm W_{2}^{1,1/2} for a classical heat-conduction problem) is constructed. In this norm, a convergence rate estimate, compatible with the smoothness of the solution of the boundary value problem, is obtained.

Note that the convergence to classical solutions is studied in [5, 19]. Onedimensional parabolic problem with weak solution is studied in [2, 3, 10]; 2D parabolic problem with variable coefficients (but not time-dependent) is considered in [3, 11].

2. Preliminary results

Let H be a real separable Hilbert space endowed with inner product (·, ·) and norm ∥ · ∥ and S-unbounded self-adjoint positive definite linear operator, with domain D(S) dense in H. It is easy to see that the product (u, v)_{S} = (Su, v), u, v ∈ D(S) satisfies the inner product axioms. The closure of D(S) in the norm ∥u∥_{S} = (u, u)_{S}^{1/2} is a Hilbert space H_{S} ⊂ H. The inner product (u, v) continuously extends to H_{S}^{∗} × H_{S}, where H_{S}^{∗} = H_{S}^{−1} is the dual space for H_{S}. Spaces H_{S}, H and H_{S^{−1}} represent a Gelfand triple H_{S} ⊂ H ⊂ H_{S^{−1}} with continuous imbeddings. Operator S extends to the map S : H_{S} ↦ H_{S}^{∗}. There exist unbounded selfadjoint positive definite linear operator S^{1/2} , such that D(S^{1/2} ) = H_{S} and (u, v)_{S} = (Su, v) = (S^{1/2} u, S^{1/2} v). We also define Sobolev spaces W_{2}^{s}(a, b; H), W_{2}^{0}(a, b; H) = L_{2}(a, b; H), of the functions u = u(t) mapping the interval (a, b) ⊂ R into H (see [13, 20]).

Let A and B be unbounded self-adjoint positive definite linear operators, A = A(t), B ≠ B(t), in Hilbert space H, in general noncommutative, with D(A) dense in H and H_{A} ⊂ H_{B}. We consider the following abstract Cauchy problems:

(2.1)  B \frac{du}{dt} + Au = f(t), 0 < t < T, u(0) = 0,
(2.2)  B \frac{du}{dt} + Au = \frac{dg}{dt}, 0 < t < T, u(0) = 0,

where f(t) and g(t) are given and u(t) is the unknown function with values in H. Let also assume that A_{0} ⩽ A(t) ⩽ kA_{0} where k = const > 1 and A_{0} ≠ A_{0}(t) is a constant self-adjoint positive definite linear operator in H. The following propositions are proved in [2].

FINITE DIFFERENCE APPROXIMATION FOR PARABOLIC INTERFACE PROBLEM… 69

Lemma 2.1. Let f ∈ L_{2}(0, T; H_{A_{0}^{−1}} ). Then the solution to problem (2.1) satisfies a priori estmate

(2.3)  ∫_{0}^{T} ∥u(t)∥_{A_{0}}^{2} dt + ∫_{0}^{T} ∫_{0}^{T} \frac{∥u(t) − u(t′)∥_{B}^{2}}{|t − t′|^{2}} dt dt′ ⩽ C ∫_{0}^{T} ∥f(t)∥_{A_{0}^{−1}}^{2} dt.

Lemma 2.2. Let g ∈ W_{2}^{1/2} (0, T; H_{B^{−1}} ). Then the solution to problem (2.2) satisfies a priori estimate

∫_{0}^{T} ∥u(t)∥_{A_{0}}^{2} dt + ∫_{0}^{T} ∫_{0}^{T} \frac{∥u(t) − u(t′)∥_{B}^{2}}{|t − t′|^{2}} dt dt′

⩽ C[ ∫_{0}^{T} ∫_{0}^{T} \frac{∥g(t) − g(t′)∥_{B^{−1}}^{2}}{|t − t′|^{2}} dt dt′+ ∫_{0}^{T} ( \frac{1}{t} + \frac{1}{T − t} ) ∥g(t)∥_{B^{−1}}^{2} dt].

An analogous result hold for the operator-difference schemes. Let H_{h} be a finite-dimensional real Hilbert space with the inner product (·, ·)_{h} and the norm ∥ · ∥_{h}. For a self-adjoint positive linear operator S_{h} in H_{h}, by H_{S_{h}} we denote the space H_{S_{h}} = H_{h} with the inner product (v, w)_{S_{h}} = (S_{h}v, w)_{h} and the norm ∥v∥_{S_{h}} = (S_{h}v, v)_{h}^{1/2} .

Let ω_{τ} be a uniform mesh on (0, T) with the step size τ = T/m, ω_{τ}^{+} = ω_{τ} ∪{T } and ω̄_{τ} = ω_{τ} ∪ {0, T }. Further, we shall use standard notation from the theory of the difference schemes [17]. In particular, we set

vt̄ = vt̄(t) = \frac{v(t) − v(t − τ)}{τ} .

We consider the implicit operator-difference scheme

(2.4)  B_{h}v\overline{t} + A_{h}v = φ(t), t ∈ ω_{τ}^{+} , v(0) = 0,

where A_{h} = A_{h}(t) and B_{h} ≠ B_{h}(t) are linear positive definite self-adjoint operators in H_{h}, in general case noncommutative, φ(t) is given and v(t) is an unknown function with values in H_{h}. Let us also consider the scheme

(2.5)  B_{h}v\overline{t} + A_{h}v = ψ\overline{t}, t ∈ ω_{τ}^{+} , v(0) = 0,

where ψ(t) is a given mesh function with values in H_{h}. Analogously, as in the previous case, we assume that A_{h0} ⩽ A_{h}(t) ⩽ kA_{h0} where k = const > 1 and A_{h0} ≠ A_{h0}(t) is a self-adjoint positive linear operator in H_{h}. The following analogs of Lemmas 2.1 and 2.2 are proved in [11].

Lemma 2.3. The solution v of operator-difference scheme (2.4) satisfies a priori estimate

τ ∑_{t∈ω_{τ}^{+}} ∥v(t)∥_{A_{h0}}^{2} + τ^{2} ∑_{t∈ω̄_{τ}} ∑_{t′∈ω̄_{τ} ,t′≠t} \frac{∥v(t) − v(t′)∥_{B_{h}}^{2}}{|t − t′|^{2}} ⩽ Cτ ∑_{t∈ω_{τ}^{+}} ∥φ(t)∥_{A_{h0}^{−1}}^{2} .

Lemma 2.4. The solution v of operator-difference scheme (2.5) satisfies a priori estimate

τ ∑_{t∈ω_{τ}^{+}} ∥v(t)∥_{A_{h0}}^{2} + τ^{2} ∑_{t∈ω̄_{τ}} ∑_{t′∈ω̄_{τ} ,t′≠t} \frac{∥v(t) − v(t′)∥_{B_{h}}^{2}}{|t − t′|^{2}}

⩽ C[τ^{2} ∑_{t∈ω̄_{τ}} ∑_{t′∈ω̄_{τ} ,t′≠t} \frac{∥ψ(t) − ψ(t′)∥_{B_{h}^{−1}}^{2}}{|t − t′|^{2}} + τ ∑_{t∈ω̄_{τ}} ( \frac{1}{t + τ} + \frac{1}{T − t + τ} ) ∥ψ(t)∥_{B_{h}^{−1}}^{2} ]

3. Differential problem and its approximation

Let us consider the 2D initial-boundary-value problem for the heat equation in the presence of a concentrated capacity at the line x_{2} = ξ, 0 < ξ < 1:

(1 + Kδ_{Σ} (x))\frac{∂u}{∂t} − ∑_{i=1}^{2} \frac{∂}{∂x_{i}} ( a_{i}(x, t)\frac{∂u}{∂x_{i}} ) = f, on Q, (3.1)

u = 0, on ∂Ω × (0, T),
u(x, 0) = u_{0}(x), on Ω,

where δ_{Σ} (x) = δ(x_{2}−ξ) is the Dirac delta function, K = const > 0 and Ω = (0, 1)^{2}, Q = Ω × (0, T). We shall assume that

(3.2)  a_{i} ∈ W_{2}^{2+ε,1+ε/2} (Q_{1}) ∩ W_{2}^{2+ε,1+ε/2} (Q_{2}), ε > 0,
f ∈ W_{2}^{1+ε,1/2+ε/2} (Q),
u_{0} ∈ W_{2}^{2}(Ω_{1}) ∩ W_{2}^{2}(Ω_{2}),
(3.3)  u ∈ W_{2}^{3,3/2} (Q_{1}) ∩ W_{2}^{3,3/2} (Q_{2}) ∩ W_{2}^{3,3/2} (Σ × (0, T)),

where Ω_{1} = (0, 1) × (0, ξ), Ω_{2} = (0, 1) × (ξ, 1), Q_{1} = Ω_{1} × (0, T), Q_{2} = Ω_{2} × (0, T), Σ = {(x_{1}, ξ)|x_{1} ∈ (0, 1)}. Note that conditions (3.2) express the minimal smoothness requirements on the data under which the solution u of (3.1) may belong to the function space stated in (3.3). To guarantee that such u really exists, we also need some additional compatibility conditions at the corners of Ω (see [7]). We also assume that and 0 < c_{1} ⩽ a_{i}(x, t) ⩽ c_{2}, on Q.

Let ω̄_{h} –uniform mesh with step size h in Ω̄, ω_{h} = ω̄_{h} ∩ Ω, ω_{1h} = ω̄_{h} ∩ ([0, 1) × (0, 1)), ω_{2h} = ω̄_{h} ∩((0, 1)×[0, 1)), σ_{h} = ω_{h}∩Σ. Suppose that ξ is a rational number. Then one can choose step h so that σ_{h} ≠ ∅. Also we assume that the condition c_{1}h^{2} ⩽ τ ⩽ c_{2}h^{2} is satisfied. Define the finite differences in the usual way:

vx̄_{i} (x, t) = \frac{v − v^{−i}}{h} , v_{x_{i}} (x, t) = \frac{v^{+i} − v}{h} ,

where v^{±i} (x, t) = v(x ± e_{i}h, t), e_{1} = (1, 0), e_{2} = (0, 1). Problem (3.1) can be approximated on the mesh \overline{Q}_{hτ} = ω̄_{h} × ω̄_{τ} by the following difference scheme with averaged right-hand side:

(1 + Kδ_{σ_{h}} )vt̄ + L_{h}v = T_{1}^{2}T_{2}^{2}T_{t}^{−} f, on Q_{hτ} , (3.4)

v = 0, on γ_{h} × ω_{τ}^{+} , v(x, 0) = u_{0}(x), on ω_{h},

where L_{h}v = −\frac{1}{2} ∑ _{i=1}^{2} ((a_{i}v_{x_{i}} )x̄_{i} + (a_{i}vx̄_{i} )_{x_{i}} ),

δ_{σ_{h}} (x) = \{ \begin{matrix}0, & x ∉ σ_{h} \\ 1/h, & x ∈ σ_{h} \end{matrix}

is the mesh Dirac function, and T_{1}, T_{2}, T_{t}^{−} are Steklov averaging operators defined by

T_{1}f(x_{1}, x_{2}) = T_{1}^{±} f(x_{1} ∓ h/2, x_{2}) = \frac{1}{h} ∫_{x_{1} −h/2}^{x_{1} +h/2} f(x′_{1}, x_{2}) dx′_{1},

T_{2}f(x_{1}, x_{2}) = T_{2}^{±} f(x_{1}, x_{2} ∓ h/2) = \frac{1}{h} ∫_{x_{2} −h/2}^{x_{2} +h/2} f(x_{1}, x′_{2}) dx′_{2},

T_{t}^{−} f(x, t) = T_{t}^{+} f(x, t − τ) = \frac{1}{τ} ∫_{t−τ}^{t} f(x, t′) dt′.

Note that these operators are commutative and transform the derivatives to divided differences, for example:

T_{i}^{−} \frac{∂u}{∂x_{i}} = ux̄_{i} , T_{i}^{+} \frac{∂u}{∂x_{i}} = u_{x_{i}} , T_{i}^{2} \frac{∂^{2}u}{∂x_{i}^{2}} = u_{x_{i} x̄_{i}} , T_{t}^{−} \frac{∂u}{∂t} = ut̄.

We also define

T_{2}^{2−} f(x_{1}, x_{2}) = \frac{1}{h} ∫_{x_{2} −h}^{x_{2}} ( 1 + \frac{x′_{2} − x_{2}}{h} ) f(x_{1}, x′_{2}) dx′_{2},

T_{2}^{2+} f(x_{1}, x_{2}) = \frac{1}{h} ∫_{x_{2}}^{x_{2} +h} ( 1 − \frac{x′_{2} − x_{2}}{h} ) f(x_{1}, x′_{2}) dx′_{2}.

We define the following discrete norms and seminorms:

∥v∥_{L_{2} (Q_{hτ} )}^{2} = τ ∑_{t∈ω_{τ}^{+}} ∥v(·, t)∥_{L_{2} (ω_{h} )}^{2} , ∥v∥_{L_{2} (σ_{h} ×ω_{τ} )}^{2} = τ ∑_{t∈ω_{τ}^{+}} ∥v(·, t)∥_{L_{2} (σ_{h} )}^{2}

|v|_{L_{2} (ω_{τ} ; W_{2}^{1/2} (σ_{h} ))}^{2} = τ ∑_{t∈ω_{τ}^{+}} |v(·, t)|_{W_{2}^{1/2} (σ_{h} )}^{2} ,

|v|_{W_{2}^{1/2} (ω_{τ} ; L_{2} (ω_{h} ))}^{2} = τ ∑_{t∈ω̄_{τ}} τ ∑_{t′∈ω̄_{τ} , t′≠t} \frac{∥v(·, t) − v(·, t′)∥_{L_{2} (ω_{h} )}^{2}}{|t − t′|^{2}} ,

|v|_{W_{2}^{1/2} (ω_{τ} ; L_{2} (σ_{h} ))}^{2} = τ ∑_{t∈ω̄_{τ}} τ ∑_{t′∈ω̄_{τ} , t′≠t} \frac{∥v(·, t) − v(·, t′)∥_{L_{2} (σ_{h} )}^{2}}{|t − t′|^{2}},

∥v∥_{_{W_{2}^{1/2}} (ω_{τ} ; L_{2} (ω_{h} ))}^{2} = |v|_{W_{2}^{1/2} (ω_{τ} ; L_{2} (ω_{h} ))}^{2} + τ∑_{t∈ω̄_{τ}} ( \frac{1}{t + τ} + \frac{1}{T − t + τ} ) ∥v(·, t)∥_{L_{2} (ω_{h} )}^{2} ,

∥v∥_{_{W_{2}^{1/2}} (ω_{τ} ; L_{2} (σ_{h} ))}^{2} = |v|_{W_{2}^{1/2} (ω_{τ} ; L_{2} (σ_{h} ))}^{2} + τ∑_{t∈ω̄_{τ}} ( \frac{1}{t + τ} + \frac{1}{T − t + τ} )∥v(·, t)∥_{L_{2} (σ_{h} )}^{2} ,

∥v∥_{_{W_{2}^{1, 1/2}} (Q_{hτ} )}^{2} = τ ∑_{t∈ω_{τ}^{+}} ∥v(·, t)∥_{W_{2}^{1} (ω_{h} )}^{2} + |v|_{W_{2}^{1/2} (ω_{τ} ; L_{2} (ω_{h} ))}^{2} + |v|_{W_{2}^{1/2} (ω_{τ} ; L_{2} (σ_{h} ))}^{2} .

4. Convergence of the difference scheme

In this section we prove the convergence of difference scheme (3.4) in the W̃_{2}^{1,1/2} (Q_{hτ}) norm. Let u be the solution to boundary-value problem (3.1) and v the solution of difference problem (3.4). The error z = u − v satisfies the finite difference scheme

(1 + Kδ_{σ_{h}} )zt̄ + L_{h}z = φ, on ω_{h} × ω_{τ}^{+} , (4.1)

z = 0, on γ_{h} × ω_{τ}^{+} , z(x, 0) = 0, on ω_{h}.

where

φ = ∑_{i=1}^{2} η_{i,x̄_{i}} + χ_{bart} + δ_{σ_{h}} μt̄,
χ = u − T_{1}^{2}T_{2}^{2}u,
η_{i} = T_{i}^{+} T_{3−i}^{2} T_{t}^{−} ( a_{i} \frac{∂u}{∂x_{i}} ) − \frac{1}{2}(a_{i} + a_{i}^{+i} )u_{x_{i}} ,
μ = Ku − T_{1}^{2}(Ku).

Let us set η_{1} = η̃_{1} + δ_{σ_{h}} η̂_{1}, χ = χ̃ + δ_{σ_{h}} χ̂, where

η̂_{1} = \frac{h^{2}}{6} T_{1}^{+} T_{t}^{−} ( [a_{1} \frac{∂^{2}u}{∂x_{1}∂x_{2}} + \frac{∂a_{1}}{∂x_{2}} \frac{∂u}{∂x_{1}} ]_{Σ} ) , χ̂ = \frac{h^{2}}{6} [T_{1}^{2} \frac{∂u}{∂x_{2}} ]_{Σ} ,

and [u]_{Σ} = u(x_{1}, ξ + 0, t) − u(x_{1}, ξ − 0, t).

Using Lemmas 2.3 and 2.4, we directly obtain the following a priori estimate for the solution of difference scheme (4.1):

(4.2) ∥z∥_{_{W_{2}^{1,1/2}} (Q_{hτ} )} ⩽ C[∥η_{2}∥_{L_{2} (Q_{hτ} )} + ∥η̃_{1}∥_{L_{2} (Q_{hτ} )} + |η̂_{1}|_{L_{2} (ω_{τ} ; W_{2}^{1/2} (σ_{h} ))}

+ ∥χ̃∥_{_{W_{2}^{1/2}} (ω_{τ} ,L_{2} (ω_{h} ))} + ∥χ̂∥_{_{W_{2}^{1/2}} (ω_{τ} ,L_{2} (σ_{h} ))} + ∥μ∥_{_{W_{2}^{1/2}} (ω_{τ} ,L_{2} (σ_{h} ))} ].

Therefore, in order to estimate the rate of convergence of difference scheme (3.4), it is sufficient to estimate the right-hand side of inequality (4.2).

We decompose η_{i} = η_{i1} + η_{i2} + η_{i3}, where

η_{i1} = T_{i}^{+} T_{3−i}^{2} T_{t}^{−} ( a_{i} \frac{∂u}{∂x_{i}} ) − (T_{i}^{+} T_{3−i}^{2} T_{t}^{−} a_{i})( T_{i}^{+} T_{3−i}^{2} T_{t}^{−} \frac{∂u}{∂x_{i}} ) ,
η_{i2} = [T_{i}^{+} T_{3−i}^{2} T_{t}^{−} a_{i} − 0, 5(a_{i} + a_{i}^{+i} )]( T_{i}^{+} T_{3−i}^{2} T_{t}^{−} \frac{∂u}{∂x_{i}} ) ,
η_{i3} = −0, 5(a_{i} + a_{i}^{+i} )\{T_{i}^{+} T_{3−i}^{2} T_{t}^{−} \frac{∂u}{∂x_{i}} − u_{x_{i}} \}.

The term η_{i1} is a bounded bilinear functional of the argument (a_{i}, u) ∈ W_{4}^{1,1/2} (e)× W_{4}^{2,1}(e),

e = e(x, t) = {(x′_{1}, x′_{2}, t′) : x_{i} < x′_{i} < x_{i} + h, |x′_{3−i} − x_{3−i} | < h, t′ ∈ (t − τ, t)},

and for i = 1, x_{2} ≠ ξ. Further, η_{i1} = 0 whenever a_{i} is a constant or u is a polynomial of degree one in x_{1} or x_{2} or a constant. Applying the Bramble–Hilbert lemma [6] we get:

(4.3)  |η_{i1}(x, t)| ⩽ C|a_{i}|_{W_{4}^{1,1/2} (e)} |u|_{W_{4}^{2,1} (e)} .

The term η_{i2} is a bounded bilinear functional of the argument (a_{i}, u)∈W_{q}^{2,1}(e)× W_{2q/(q−2)}^{1,1/2} (e), q = 2 + ε. Further, η_{i2} = 0 whenever a_{i} is a polynomial of degree one in x_{1} or x_{2} or constant or u is constant. Applying the Bramble–Hilbert lemma we get the following estimate:

(4.4)  |η_{i2}(x, t)| ⩽ C|a_{i}|_{W_{q}^{2, 1} (e)} |u|_{W_{2q/(q−2)}^{1, 1/2} (e)} .

The term η_{i3} is a bounded bilinear functional of the argument (a_{i}, u) ∈ ℂ(\overline{Q_{k}})× W_{2}^{3,3/2} (e), k = 1, 2, Further, η_{i3} = 0 whenever u is a polynomial of degree two in x_{1} or x_{2} and a polynomial of arbitrary degree in t. Applying the Bramble–Hilbert lemma, we get the estimate

(4.5)  |η_{i3}(x, t)| ⩽ C∥a_{i}∥_{ℂ(\overline{Q_{k}} )}|u|_{W_{2}^{3, 3/2} (e)} .

From estimates (4.3)–(4.5), choosing i = 2, after summation and using the imbeddings

W_{2}^{2+ε,1+ε/2} ⊂ W_{4}^{1,1/2} , W_{2}^{3,3/2} ⊂ W_{4}^{2,1},
(∗)  W_{2}^{2+ε,1+ε/2} ⊂ W_{q}^{2,1}, W_{2}^{3,3/2} ⊂ W_{2q/(q−2)}^{1,1/2} ,
W_{2}^{2+ε,1+ε/2} ⊂ ℂ,  for q = 2 + ε,

we get

(4.6)  ∥η_{2}∥_{L_{2} (Q_{hτ} )} ⩽ Ch^{2}(∥a_{2}∥_{W_{2}^{2+ε,1+ε/2} (Q_{1} )} ∥u∥_{W_{2}^{3,3/2} (Q_{1} )}
+ ∥a_{2}∥_{W_{2}^{2+ε,1+ε/2} (Q_{2} )} ∥u∥_{W_{2}^{3,3/2} (Q_{2} )} ).

Let us estimate the term η̃_{1}. At the point x ∉ σ_{h}, we have η̃_{1} = η_{1} and estimates (4.3)–(4.5) are valid. At the point x ∈ σ_{h}, we decompose

η̃_{1} = ∑_{k=1}^{3} (η_{1,k}^{−} + η_{1,k}^{+} ),

where η_{1,k}^{±} are defined at the point x_{2} = ξ ± 0

η_{1,1}^{±} = T_{1}^{+} T_{2}^{2±} T_{t}^{−} ( a_{1} \frac{∂u}{∂x_{1}} ) − 2(T_{1}^{+} T_{2}^{2±} T_{t}^{−} a_{1})( T_{1}^{+} T_{2}^{2±} T_{t}^{−} \frac{∂u}{∂x_{1}} )

± \frac{h}{6}( T_{1}^{+} T_{t}^{−} \frac{∂a_{1}}{∂x_{2}} ) [2( T_{1}^{+} T_{2}^{2±} T_{t}^{−} \frac{∂u}{∂x_{1}} ) − ( T_{1}^{+} T_{t}^{−} \frac{∂u}{∂x_{1}} ) ]

± \frac{h}{6}[ \frac{a_{1} + a_{1}^{+1}}{2} ( T_{1}^{+} T_{t}^{−} \frac{∂^{2}u}{∂x_{1}∂x_{2}} ) − ( T_{1}^{+} T_{t}^{−} a_{1} \frac{∂^{2}u}{∂x_{1}∂x_{2}} ) ]

± \frac{h}{6}[( T_{1}^{+} T_{t}^{−} \frac{∂a_{1}}{∂x_{2}} ) ( T_{1}^{+} T_{t}^{−} \frac{∂u}{∂x_{1}} ) − ( T_{1}^{+} T_{t}^{−} \frac{∂a_{1}}{∂x_{2}} \frac{∂u}{∂x_{1}} ) ],

η_{1,2}^{±} = [2(T_{1}^{+} T_{2}^{2±} T_{t}^{−} a_{1}) − \frac{a_{1} + a_{1}^{+1}}{2} ∓ \frac{h}{3}( T_{1}^{+} T_{t}^{−} \frac{∂a_{1}}{∂x_{2}} ) ] × ( T_{1}^{+} T_{2}^{2±} T_{t}^{−} \frac{∂u}{∂x_{1}} ) ,
η_{1,3}^{±} = \frac{a_{1} + a_{1}^{+1}}{4} [2( T_{1}^{+} T_{2}^{2±} T_{t}^{−} \frac{∂u}{∂x_{1}} ) − u_{x_{1}} ∓ \frac{h}{3}( T_{1}^{+} T_{t}^{−} \frac{∂^{2}u}{∂x_{1}∂x_{2}} ) ].

The term η_{1,1}^{±} is a bounded bilinear functional of the argument (a_{1}, u) ∈ W_{4}^{1,1/2} (e_{1}^{±} ) × W_{4}^{2,1}(e_{1}^{±} ) where

e_{1}^{+} = (x_{1}, x_{1} + h) × (ξ, ξ + h) × (t − τ, t),
e_{1}^{−} = (x_{1}, x_{1} + h) × (ξ − h, ξ) × (t − τ, t).

Further, η_{1,1}^{±} = 0 whenever a_{1} is a constant or u is a polynomial of degree one in x_{1} or x_{2} or a constant. Applying the Bramble–Hilbert lemma we get

(4.7)  |η_{1,1}^{±}(x, t)| ⩽ C|a_{1}|_{W_{4}^{1,1/2} (e_{1}^{±} )} |u|_{W_{4}^{2,1} (e_{1}^{±} )} .

The term η_{1,2}^{±} is a bounded bilinear functional of the argument (a_{1}, u) ∈ W_{q}^{2,1}(e_{1}^{±} ) × W_{2q/(q−2)}^{1,1/2} (e_{1}^{±} ), q = 2 + ε. Further, η_{1,2}^{±} = 0 whenever a_{1} is a polynomial of degree one in x_{1} or x_{2} or constant or u is a constant. Applying the Bramble–Hilbert lemma we get the estimate

(4.8)  |η_{1,2}^{±}(x, t)| ⩽ C|a_{1}|_{W_{q}^{2,1} (e_{1}^{±} )} |u|_{W_{2q/(q−2)}^{1,1/2} (e_{1}^{±} )} .

The term η_{1,3}^{±} is a bounded bilinear functional of the argument (a_{1}, u) ∈ ℂ(\overline{Q_{k}}) × W_{2}^{3,3/2} (e_{1}^{±} ), k = 1, 2. Further, η_{1,3}^{±} = 0 whenever u is a polynomial of degree two in x_{1} or x_{2} and polynomial of arbitraty degree in t. Applying the Bramble–Hilbert lemma we get

(4.9)  |η_{1,3}^{±}(x, t)| ⩽ C∥a_{1}∥_{ℂ(\overline{Q_{k}} )}|u|_{W_{2}^{3, 3/2} (e_{1}^{±} )} .

From estimates (4.3)–(4.5) and (4.7)–(4.9), after summation and using imbeddings (∗), we get

(4.10)  ∥η̃_{1}∥_{L_{2} (Q_{hτ} )} ⩽ Ch^{2}(∥a_{1}∥_{W_{2}^{2+ε,1+ε/2} (Q_{1} )} ∥u∥_{W_{2}^{3, 3/2} (Q_{1} )}
+ ∥a_{1}∥_{W_{2}^{2+ε,1+ε/2} (Q_{2} )} ∥u∥_{W_{2}^{3,3/2} (Q_{2} )} ).

Let us estimate the term η̂_{1}. For ϕ ∈ W_{2}^{1/2} (Σ), the following estimate is valid

|T_{1}^{+} ϕ|_{W_{2}^{1/2} (σ_{h} )} ⩽ C|ϕ|_{W_{2}^{1/2} (Σ)} ⩽ C∥ϕ∥_{W_{2}^{1} (Ω_{k})} , k = 1, 2,

wherefrom

|η̂_{1}(·, t)|_{W_{2}^{1/2} (σ_{h} )} ⩽ Ch^{2}(∥T_{t}^{−} ν(·, t)∥_{W_{2}^{1} (Ω_{1} )} + ∥T_{t}^{−} ν(·, t)∥_{W_{2}^{1} (Ω_{2} )} ),

where ν = ν_{1} + ν_{2} , and

ν_{1} = a_{1} \frac{∂^{2}u}{∂x_{1}∂x_{2}} , ν_{2} = \frac{∂a_{1}}{∂x_{2}} \frac{∂u}{∂x_{1}} .

After summation, we have

(4.11)  |η̂_{1}|_{L_{2} (ω_{τ} ,W_{2}^{1/2} (σ_{h} ))} ⩽ Ch^{2}(∥ν∥_{W_{2}^{1,0} (Q_{1} )} + ∥ν∥_{W_{2}^{1,0} (Q_{2} )} ).

Using the Hölder inequality and the imbeddings (∗), we get

(4.12)  \begin{matrix}∥ν_{1}∥_{W_{2}^{1,0} (Q_{k})} & ⩽ ∥a_{1}∥_{W_{2}^{2+ε,1+ε/2} (Q_{k})} ∥u∥_{W_{2}^{3,3/2} (Q_{k})} , \\ ∥ν_{2}∥_{W_{2}^{1,0} (Q_{k})} & ⩽ ∥a_{1}∥_{W_{2}^{2+ε,1+ε/2} (Q_{k})} ∥u∥_{W_{2}^{3,3/2} (Q_{k})} , \end{matrix} k = 1, 2.

From (4.11)–(4.12) we obtain

(4.13)  |η̂_{1}|_{L_{2} (ω_{τ} ,W_{2}^{1/2} (σ_{h} ))} ⩽ Ch^{2}(∥a_{1}∥_{W_{2}^{2+ε,1+ε/2} (Q_{1} )} ∥u∥_{W_{2}^{3,3/2} (Q_{1} )}
+ ∥a_{1}∥_{W_{2}^{2+ε,1+ε/2} (Q_{2} )} ∥u∥_{W_{2}^{3,3/2} (Q_{2} )} ).

The estimates of terms χ̃, μ and χ̂ are obtained in [11]:

(4.14)  ∥χ̃∥_{_{W_{2}^{1/2}} (ω_{τ} ,L_{2} (ω_{h} ))} ⩽ Ch^{2} \sqrt{1/h}(∥u∥_{W_{2}^{3,3/2} (Q_{1} )} + ∥u∥_{W_{2}^{3,3/2} (Q_{2} )} ),
(4.15)  ∥μ∥_{_{W_{2}^{1/2}} (ω_{τ} ,L_{2} (σ_{h} ))} ⩽ Ch^{2} \sqrt{1/h}∥u∥_{W_{2}^{3,3/2} (Σ×(0,T))} ,
(4.16)  ∥χ̂∥_{_{W_{2}^{1/2}} (ω_{τ} ,L_{2} (σ_{h} ))} ⩽ Ch^{2} \sqrt{1/h}∥u∥_{W_{2}^{2,1} (Σ×(0,T))} .

Finally, from (4.2)–(4.16) we obtain the following result.

Theorem 4.1. The solution of problem (3.4) converges in W̃_{2}^{1,1/2} (Q_{hτ}) to the solution of differential problem (3.1), provided c_{1}h^{2} ⩽ τ ⩽ c_{2}h^{2}. Furthermore,

∥u−v∥_{_{W_{2}^{1,1/2}} (Q_{hτ} )} ⩽ Ch^{2}( max_{i}∥a_{i}∥_{W_{2}^{2+ε,1+ε/2} (Q_{1} )} +max_{i}∥a_{i}∥_{W_{2}^{2+ε,1+ε/2} (Q_{2} )} +l(h))

× (∥u∥_{W_{2}^{3,3/2} (Q_{1} )} + ∥u∥_{W_{2}^{3,3/2} (Q_{2} )} + ∥u∥_{W_{2}^{3,3/2} (Σ×(0,T))} ),

where l(h) = \sqrt{log1/h}.

Remark 4.1. The previous estimate is “almost" compatible with the smoothness of the coefficients and solution of differential problem (3.1). The compatibility is spoiled only by the term l(h), which slowly increases when h → 0.

Remark 4.2. Convergence in W̃_{2}^{2,1}(Q_{hτ}) is proved in [4].

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Faculty of Science (Received 06 10 2015) University of Kragujevac (Revised 04 01 2016) Kragujevac Serbia

    bratislavsredojevic9@gmail.com
    bojovicd@ptt.rs