ON THE WELL–POSEDNESS OF DIFFERENCE

SCHEMES WITH GENERALIZED DATA

N. Yu. Bakaev

1. Initial-boundary value problems with non-smooth or generalized data are extensively studied in mathematical physics, and in this connection the concept of a generalized solutions is often used [1]. In the approximation of such problems by difference schemes one uses discrete elements, which may be thought of as approximations of some generalized elements in the original formulation of the problem (for example, discrete δ-functions). Singularity properties of such approximations cannot be exhibited in a fixed mesh, but in a sequence of meshes with sufficiently refined step. A characteristic sign of generalized discrete elements is a lack of uniformity with respect to the mesh size in the bounds of these elements in the original mesh norm. It is possible to obtain uniform bounds for generalized elements only in a weaker discrete norm. Because of this, in the presence of generalized elements in the difference scheme, the use of ordinary stability estimates in the original discrete norm may lead to a lowering of the order of convergence of the scheme. For the construction of maximally accurate convergence estimates for difference schemes with generalized data, Tikhonov and Samarskiĭ [2–5] suggested the use of stability estimates, comparing the solution of the discrete problem in a basic norm to the data in a weaker norm. In the following years these ideas have been further developed [6,7].

In the present work we develop a general approach to the study of the well-posedness of difference schemes with generalized data in Banach space norms. As in [2–5] the discrete solution and the data are considered in discrete spaces, with the data in a space with weaker norm. The approach presented below is based on principles, that were used in [8–12], allowing a uniform approach to the study of wide classes of difference schemes with generalized data. We obtain general results that allow us, in particular, to give conditions for the well-posedness of difference schemes for parabolic partial differential equations in a scale of Banach spaces L_{ph}, 1 ≤ p ≤ ∞ (discrete analogues of the spaces L_{p}).

2. Let there be given two families of Banach spaces E_{h} and Ẽ_{h}, parametrized by a scalar or vectorial parameter h. We call the spaces E_{h} of the first family basic, and the spaces Ẽ_{h} of the second family negative, if the spaces E_{h} and Ẽ_{h} are connected by the imbedding

E_{h} ⊂ Ẽ_{h}, ∀h,

Typeset by AMS-TEX

1980 Mathematics Subject Classification (1985 Revision). Primary 65J10, 65M10.

uniformly in h in the sense that

(1)  ∥u∥Ẽ_{h} ≤ C_{0}∥u∥_{E_{h}},  ∀h, ∀u ∈ E_{h}

with a constant C_{0} independent of h and u.

In the family of pairs of basic and negative spaces (E_{h}, Ẽ_{h}) we consider a difference scheme, written in canonical form,

(2)  \begin{matrix}y(t_{k+1}) = [I − τÂ_{τh}(t_{k})]y(t_{k}) + τQ_{τh}F_{τh}(t_{k}),  k = 0, …, T/τ − 1, \\ y(t_{0}) = R_{τh}y_{0h}, \end{matrix}

where t_{k} = kτ ∈ [0, T], k = 0, …, T/τ is a discrete argument (T > 0 is the upper limit of its size, and τ is the discretization step), y(t_{k}) is a discrete function of the argument t_{k} with values in E_{h}, representing the solution of the discrete problem (2), F_{τh}(t_{k}) is a given function of the discerete argument t_{k} with values in Ẽ_{h}, defining the inhomogeneity of the scheme (2) and depending on τ and h, y_{0h} ∈ Ẽ_{h} is the initial datum of the scheme, Â_{τh}(t_{k}), Q_{τh}(t_{k}) and R_{τh} are linear operators acting in Ẽ_{h} (not necessarily defined in all of Ẽ_{h}), depending on t_{k} (except for R_{τh}), τ and h. If y(t_{k}) does not belong to the space E_{h} for all k = 0, …, T/τ, then we shall say that the scheme (2) is not completely solvable, but the totality of elements y(t_{k}) ∈ E_{h} for all corresponding k defines an incomplete solution of the problem (2) in the family of pairs (E_{h}, Ẽ_{h}). The operator Â_{τh}(t_{k}) is called the generator of the scheme [12], and the operators Q_{τh}(t_{k}) and R_{τh} are naturally called smoothing operators for the right-hand side and initial data, respectively. Since the data F_{τh}(t_{k}) and y_{0h} of the scheme (2) may not belong to the space E_{h}, we are lead to consider generalized data.

Theorem 1. Let the operator Â_{τh}(t_{k}), the generator of the scheme (2), be defined in the family of pairs of basic and negative spaces (E_{h}, Ẽ_{h}), linear and bounded in the space E_{h}, and satisfying in Ẽ_{h} the (Λ_{2τ})-condition of [12], uniformly with respect to t_{k} (in the sense that all objects occuring in the (Λ_{2τ})-condition do not depend on t_{k}), and let, in addition, the following condition hold

∥[Â_{τh}(t_{k}) − Â_{τh}(t_{l})]u∥_{E_{h}} ≤ C_{1} ∑_{j=1}^{r_{0}} |t_{k} − t_{l}|^{θ_{j}}∥[Â_{τh}(t_{k}) + μ_{0}I]u∥_{E_{h}}^{χ_{j}}∥u∥_{E_{h}}^{1−χ_{j}},

∀h, ∀τ ∈ (0, τ_{0}], ∀t_{k}, t_{l} ∈ [0, T], ∀u ∈ E_{h}

for some natural number r_{0} ≥ 1 and with some constants C_{1}, μ_{0} ≥ 0, τ_{0} > 0, θ_{j} ∈ [0, 1], χ_{j} ∈ [0, 1], j = 1, 2, …, r_{0}, independent of h, τ, u, t_{k}, t_{l}. Let also the operators Q_{τh} and R_{τh} be bounded and linear from Ẽ_{h} into E_{h}, satisfying the conditions

∥R(λ, τÂ_{τh}(t_{k}))Q_{τh}(t_{k})u∥_{E_{h}} ≤ C_{2}|λ|^{γ−1}τ^{−γ}∥u∥Ẽ_{h}, (3)

∀h, ∀τ ∈ (0, τ_{0}], ∀t_{k} ∈ [0, T], ∀λ ∉ Int Λ_{τ} , ∀u ∈ Ẽ_{h};

∥R(λ, τÂ_{τh}(t_{0}))R_{τh}u∥_{E_{h}} ≤ C_{3}|λ|^{γ−1}τ^{−γ}∥u∥Ẽ_{h}, (4)

∀h, ∀τ ∈ (0, τ_{0}], ∀λ ∉ Int Λ_{τ} , ∀u ∈ Ẽ_{h}

with constants C_{2}, C_{3} and γ ∈ [0, 1], γ < θ_{j} + 1 − χ_{j}, j = 1, 2, …, r_{0}, where Λ_{τ} = {λ : |λ| ≤ aτ} ∪ Λ, a = const ≥ 0, and the fixed set Λ has a configuration of type (Λ_{2}) (cf. [12]).

Then there exists a solution y(t_{k}) ∈ E_{h}, k = 0, 1, …, T/τ, of the difference scheme (2) and it satisfies the following stability estimate

\begin{matrix}∥[Â_{τh}(t_{k}) + μ̄I]^{ξ}y(t_{k})∥_{E_{h}} ≤ C_{4}(ξ)\{[(k + 1)τ]^{−(ξ+γ)}∥y_{0h}∥Ẽ_{h} \\ + τ ∑_{l=1}^{k}[(k − l + 1)τ]^{−(ξ+γ)}∥F_{τh}(τ_{l−1})∥Ẽ_{h}\}, k = 0, 1, …, T/τ, \\ ∀h, ∀τ ∈ (0, τ_{0}], ∀ξ ∈ [0, 1 − γ) \end{matrix} (5)

with nonnegative constants C_{4}(ξ), μ̄, independent of h, τ, and k (C_{4}(ξ) may depend on ξ). If in addition the operator Â_{τh}(t_{k}) can be extended to be linear and bounded on Ẽ_{h}, and instead of (4) the following inequality holds

∥[R(λ, τÂ_{τh}(t_{k})) − (λ − 1)^{−1}I]u∥_{E_{h}} ≤ C_{5}|λ|^{γ−1}τ^{−γ}∥u∥Ẽ_{h}, C_{5} = const,
∀h, ∀τ ∈ (0, τ_{0}], ∀t_{k} ∈ [0, T], ∀λ ∉ Int Λ_{τ} , ∀u ∈ Ẽ_{h}

for the same values of γ as in (3), then for R_{τh} = I there exists an incomplete solution y(t_{k}) ∈ E_{h}, k = 1, 2, …, T/τ, of problem (2), and the estimate (5) holds for these k. In the case where the operator Â_{τh}(t_{k}) does not depend on t_{k} the same statement holds with no upper bound for ξ (in this case we may formally consider r_{0} = 1, θ_{1} = 1, χ_{1} = 0).

The proof is based on techniques devoloped in [8, 9, 12].

3. In the family of pairs (E_{h}, Ẽ_{h}) of basic and negative spaces we consider a difference scheme of the form

(6)  \begin{matrix}y(t_{k+1}) = y(t_{k}) − τ ∑_{j=1}^{ν_{1}} b_{j}A_{h}(t_{k} + c_{j}τ)Y_{j} + τ ∑_{j=1}^{ν_{2}} b̄_{j}f_{h}(t_{k} + c̄_{j}τ), \\ y(t_{0}) = R_{τh}y_{0h},  k = 0, 1, …, T/τ − 1, \end{matrix}

where Y_{j} are defined by the following system of equations

(7) Y_{j} = β_{j}y(t_{k}) − τ ∑_{l=1}^{ν_{1}} a_{jl}A_{h}(t_{k} + c_{l}τ)Y_{l} + τ ∑_{l=1}^{ν_{2}} ā_{jl}f_{h}(t_{k} + c̄_{l}τ), j = 1, 2, …, ν_{1};

here b_{j}, b̄_{l}, β_{j}, a_{jr}, ā_{jl}, c_{j}, c̄_{l}; j, r = 1, 2, …, ν_{1}, l = 1, 2, …, ν_{2}, are coefficients defining the scheme, A_{h}(t) is an operator depending on t ∈ [0, T] and h, f_{h}(t) is a function of t ∈ [0, T] with values in Ẽ_{h}, depending on h and defining the inhomogeneity of the scheme, and the remaining notation has the same meaning as above (see no. 2). A solution (incomplete solution) y(t_{k}) ∈ E_{h} of the difference problem (6), (7) is understood in the same sense as in the case of the difference scheme (2). The data f_{h}(t) ∈ Ẽ_{h} and y_{0h} ∈ Ẽ_{h} of the scheme (6), (7) are generalized.

Difference schemes of the form (6), (7) arise from the discretization in t of a family of Cauchy problems with parameter h of the form

(8)  \frac{dy}{dt} + A_{h}(t)y = f_{h}(t), 0 < t ≤ T;  y(t = 0) = y_{0h}.

The discretization method, consisting in the passage from the family of problems (8) to the difference scheme (6), (7), was studied from the point of view of stability of (6), (7) in the norms of the basic spaces in [11]. This method includes both the well-known Runge-Kutta method [13], widely popular for the solution of ordinary differential equations, and the method of Vinokurov [14] developed recently.

We extend somewhat the concept of a uniformly strongly semi-positive operator defined in [11]. We say that a linear operator A_{h}(t) acting in the Banach space E_{h} is uniformly (with respect to t ∈ [0, T] and h) strongly semi-positive with an angle φ ∈ (0, π/2), if it is closed, has a domain of definition D_{h}, which is independent of t and dense in E_{h}, and in addition

∥R(λ, A_{h}(t))∥_{E_{h}} ≤ C_{6}|λ|^{−1}, ∀h, ∀t ∈ [0, T], ∀λ : | arg λ| ≥ φ, |λ| ≥ b_{0}

with some constants C_{6} and b_{0} independent of h, t and λ. In addition we will use below the following concepts introduced in [11]: the symbol α(z) of the generator of the scheme, the correcting symbols ω_{j}(z), j = 1, 2, …, ν_{2}, and difference scheme of type R(φ). We also define the concept of underlying symbols, by which we understand functions η_{j}(z), j = 1, 2, …, ν_{1}, coinciding with the corresponding correcting symbols ω_{j}(z) when the parameters ν_{2}, ā_{jr}, b̄_{r} of the scheme (6), (7) are prescribed in a special way, namely, when ν_{2} = ν_{1}, ā_{jr} = a_{jr}, b̄_{r} = b_{r}, j, r = 1, 2, …, ν_{1}. Finally, we let the range of the scheme be the complex number λ_{0} defined by

λ_{0} = lim_{|z|→∞} α(z),

where α(z) is the symbol of the generator of the scheme.

Theorem 2. Let the difference scheme (6), (7) be given in a family of pairs (E_{h}, Ẽ_{h}) of basic and negative spaces, and belong to the type R(φ_{0}) for some angle φ_{0} ∈ (0, π/2), whereby deg[α(z)] = 0, where α(z) is the symbol of the generator of the scheme (the parameter deg[ρ(z)] is defined in [11] for an arbitrary rational function ρ(z)). Let, in addition, the following conditions hold for the scheme (6), (7)

deg[g_{jl}(z)] ≤ 0, deg [∑_{k=1}^{ν_{1}} g_{jk}(z)ā_{kr}] ≤ −1, j, l = 1, 2, …, ν_{1}, r = 1, 2, …, ν_{2}; (9)

(10) deg [∑_{r=1}^{ν_{1}} g_{jr}(z)β_{r}] ≤ −1, j = 1, 2, …, ν_{1},

where g_{jl}(z) are the elements of the inverse matrix of the matrix (δ_{jl} + za_{jl}) (δ_{jl} is the Kronecker symbol), and we have the following representations of the underlying symbols η_{j}(z) and the correcting symbols ω_{l}(z)

η_{j}(z) = [1 − λ_{0}^{−1}α(z)]η̄_{j}(z), j = 1, 2, …, ν_{1}, (11)

ω_{l}(z) = [1 − λ_{0}^{−1}α(z)]ω̄_{l}(z), j = 1, 2, …, ν_{2}, (12)

where λ_{0} is the range of the scheme, and η̄_{j}(z), j = 1, 2, …, ν_{1}, ω̄_{l}(z), j = 1, 2, …, ν_{2}, are rational functions such that deg[η̄_{j}(z)] ≤ 0, j = 1, 2, …, ν_{1}, deg[ω̄_{l}(z)] ≤ 0, j = 1, 2, …, ν_{2}, and the poles of these functions are outside the closed sector {z : | arg z| ≤ φ_{0}}. We assume also that the operator A_{h}(t) is uniformly strongly semi-positive with angle φ_{0} in the space E_{h}, admits an extension to an operator in the space Ẽ_{h} which is uniformly strongly semi-positive with angle φ_{0} (the value of φ_{0} is the same as in the previous conditions of the theorem), and satisfies the conditions

∥[A_{h}(t) − A_{h}(s)]u∥_{E_{h}} ≤ C_{7} ∑_{j=1}^{q_{0}} |t − s|^{σ_{j}}∥[A_{h}(t) + μ_{1}I]u∥_{E_{h}}^{κ_{j}}∥u∥_{E_{h}}^{1−κ_{j}},

∀h, ∀t, s ∈ [0, T], ∀u ∈ D_{h},

∥[A_{h}(t) − A_{h}(s)]u∥Ẽ_{h} ≤ C_{8}{|t − s|^{δ_{1}}∥[A_{h}(t) + μ_{1}I]u∥Ẽ_{h} + ∥[A_{h}(t) + μ_{1}I]u∥δ_{2} ; Ẽ_{h}∥u∥_{E_{h}}^{1−δ_{2}}},

∀h, ∀t, s ∈ [0, T], ∀u ∈ D̃_{h},

for some fixed natural number q_{0} ≥ 1 and with some constants C_{7}, C_{8}, μ_{1} ≥ 0, σ_{j} ∈ [0, 1], κ_{j} ∈ [0, 1], j = 1, 2, …, q_{0}, δ_{1} ∈ (0, 1], δ_{2} ∈ [0, 1), independent of h, t, s, u, and the bound

∥R(λ, A_{h}(t))u∥_{E_{h}} ≤ C_{9}|λ|^{γ_{1}−1}∥u∥Ẽ_{h}, ∀h, ∀t ∈ [0, T], ∀u ∈ Ẽ_{h},
∀λ : | arg λ| ≥ φ_{0}, |λ| ≥ b_{1}

holds with constants C_{9}, b_{1} and γ_{1} ∈ [0, 1), γ_{1} < 1 + σ_{j} − κ_{j}, j = 1, 2, …, q_{0}, independent of h, t, λ, u.

Then the difference scheme (6), (7) admits an equivalent representation in the canonical form (2) with some linear operators Â_{τh}(t_{k}), Q_{τh}(t_{k}) and inhomogeneity F_{τh}(t_{k}), and

Q_{τh}(t_{k}) = I − λ_{0}^{−1}α(τA_{h}(t_{k})), F_{τh}(t_{k}) = ∑_{r=1}^{ν_{2}} Ω_{τhr}(t_{k})f_{h}(t_{k} + c̄τ)

(functions of operators are understood in the sense of the functional calculus of closed operators [15]) and the operator Â_{h}(t_{k}) is bounded both in E_{h} and Ẽ_{h}; Q_{τh}(t_{k}) is bounded as an operator from Ẽ_{h} to E_{h}, and the linear operators Ω_{τhr}(t_{k}), r = 1, 2, …, ν_{2}, are bounded in Ẽ_{h}. If we take R_{τh} = Q_{τh}(t_{0}), then there exists a solution y(t_{k}) ∈ E_{h}, k = 0, 1, …, T/τ, of the discrete problem (6), (7), satisfying the stability estimates

\begin{matrix}∥[Â_{τh}(t_{k}) + μ̄I]^{ξ}y(t_{k})∥_{E_{h}} + ∥[τ^{−1}α(τA_{h}(t_{k})) + μ̄I]^{ξ}y(t_{k})∥_{E_{h}} \\ ≤ C_{10}(ξ)\{[(k + 1)τ]^{−(ξ+γ_{1})}∥y_{0h}∥Ẽ_{h} \\ + τ ∑_{l=1}^{k}[(k − l + 1)τ]^{−(ξ+γ_{1})} max_{r=1,2,...,ν_{2}} ∥f_{h}(t_{k} + c̄τ)∥Ẽ_{h}\}, \\ k = 0, 1, …, T/τ, ∀h, ∀τ ∈ (0, τ_{1}], ∀ξ ∈ [0, 1 − γ_{1}) \end{matrix} (13)

with some constants C_{10}(ξ), μ̄ ≥ 0, τ_{1} > 0, independent of k, h, τ (C_{10}(ξ) may depend on ξ). If furthermore λ_{0} = 1 (λ_{0} is the range of the scheme), then for R_{τh} = I there exists an incomplete solution y(t_{k}), k = 1, 2, …, T/τ, of the problem (6), (7) and for this the bound (13) is true. In the case where the operator A_{h}(t) does not depend on t, then the statement holds with no upper bound on ξ (in this case we may formally assume q_{0} = 1, σ_{1} = 1, κ_{1} = 0).

The proof is based on transforming the scheme (6), (7) into the canonical form (2) and using the result of Theorem 1.

Applications of the theory developed here to the study of the well-posedness of difference schemes for parabolic partial differential equations will appear in a separate work.

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Translated by S. LARSSON