THE INVESTIGATION OF THE STABILITY OF THE MULTISTEP METHOD FOR THE OPERATOR VALUED ABSTRACT PARABOLIC

EQUATION WITH THE USE OF ASSOCIATIVE POLYNOMIALS

J. Rogava, R. Galdava Georgian Technical University

Let us consider the Cauchy problem in the Hilbert space H:

u′(t) + A(t)u(t) = f(t), t > 0, u(0) = φ,  (1)

where A is a self-adjoint positive definite operator with domain of definition D(A) that is independent of t; f(t) is an abstract function with values from the space H; φ is an initial vector; u(t) is the sought for function.

Consider the multistep method for problem (1), which is described by the following scheme:

∑_{i=0}^{q} α_{i}u_{k−i} + τA(t_{k})u_{k} = τf(t_{k}),  (2)

where

k = q, q + 1, ⋯, (q ≥ 2), t_{k} = k · τ (τ > 0), α_{0} > 0,

∑_{l=1}^{q} α_{i} = 0.

Theorem. Assume the following :

a) A(t) is a self-adjoint positive definite operator for any t on [0, +∞[ ;

b) for any t′, t′′ and s from [0, +∞[, the following inequality holds

∥[A(t′) − A(t′′)]A^{−1}(s)∥ ≤ c_{0}|t′ − t′′|, c_{o} = const > 0;

c) for any x on [0, 1] all the roots of the characteristic equation

α_{0}λ^{q} + α_{1}x^{q−1}λ^{q−1} + ⋯ + α_{q−1}xλ + α_{q} = 0  (3)

possibly, except one root, belong to the some circle, which is settled within the unit circle and the exceptional root belongs to the unit circle, or all the roots of the equation (3), possibly, with the exception of two of them, belong to the same circle, which is settled within the unit circle, and the residual roots λ_{1}(x) and λ_{2}(x) belong to the unit circle and satisfy the following conditions: 1)λ_{1}(x) = \overline{λ_{2}(x)} or both are real; 2) there is a number 0 < δ < 1 that is independent of x, the distances from the point (λ_{1} + λ_{2}; λ_{1}λ_{2}) to the points (2; 1) and (−2; 1) is more or equal to δ. Then the true estimate for the scheme (2) is the following one:

∥u_{k+q−1}∥ ≤ c[e^{ct_{k−1}}(∥u_{0}∥ + ∥u_{1}∥ + ⋯ + ∥u_{q−1}∥) + τ ∑_{i=1}^{k} e^{ct_{k−i}}∥f(t_{i+q−1})∥] (4)

where k = q, q + 1, ⋯, (q ≥ 2), c = const > 0.

The scheme of the proof.

From (2) it follows:

u_{k} = β_{1}L_{k}u_{k−1} + β_{2}L_{k}u_{k−2} + ⋯ + β_{q}L_{k}u_{k−q} + τβ_{0}L_{k}f(t_{k}),  (5)

where

β_{0} = \frac{1}{α_{0}} , β_{i} = −\frac{α_{i}}{α_{0}} , i = 1, 2, ⋯, q,

L_{k} = (I + τβ_{0}A(t_{k}))^{−1}, k = q, q + 1, ⋯

By the method of induction the following equation is obtained from (5):

u_{k+q−1} = (β_{1} U^{q}_{k−1} L_{q} + β_{2} U^{q+1}_{k−2} L_{q+1} + ⋯ + β_{q} U^{2q−1}_{k−q} L_{2q−1})u_{q−1}+

+(β_{2} U^{q}_{k−1} L_{q} + β_{3} U^{q+1}_{k−2} L_{q+1} + ⋯ + β_{q} U^{2q−2}_{k−q+1} L_{2q−2})u_{q−2}+

+(β_{3} U^{q}_{k−1} L_{q} + β_{4} U^{q+1}_{k−2} L_{q+1} + ⋯ + β_{q} U^{2q−3}_{k−q+2} L_{2q−3})u_{q−3}+

+ ⋯ + β_{q} U^{q}_{k−1} L_{q}u_{0} + τβ_{0} ∑_{i=1}^{k} U^{i+q−1}_{k−i} L_{i+q−1}f(t_{i+q−1}),  (6)

where U^{i}_{k} operators are determined by the following recurrent dependence:

U^{i}_{k}= β_{1}L_{k+i} U^{i}_{k−1} +β_{2}L_{k+i} U^{i}_{k−2} + ⋯ + β_{q}L_{k+i} U^{i}_{k−q},  (7)

U^{i}_{0}= I, U^{i}_{−1}=U^{i}_{−2}= ⋯ =U^{i}_{1−q}= 0.

If the operators U^{i}_{k} are uniformly bounded, then according to equation (6) a priori estimate (4) holds,thereby the scheme (2) is stable, and on the contrary,

if the scheme (2) is stable, then the operators U^{i}_{k} are uniformly bounded (in this case we assume that t is varying in a finite interval).

Let’s see if the U^{i}_{k} operators are uniformly bounded when the conditions of the theorem hold. To prove this fact the operator-valued polynomials are necessary, which are determined by the following recurrent dependence:

U_{k}(β_{1}L, β_{2}L, ⋯, β_{q}L) = β_{1}LU_{k−1}(β_{1}L, β_{2}L, ⋯, β_{q}L)+
+β_{2}LU_{k−2}(β_{1}L, β_{2}L, ⋯, β_{q}L) + ⋯ + β_{q}LU_{k−q}(β_{1}L, β_{2}L, ⋯, β_{q}L), (8)
U_{0} = I, U_{−1} = U_{−2} = ⋯ = U_{1−q} = 0,
L = (I + τβA(t_{j}))^{−1}.

These polynomials are called polynomials associated to the scheme (2).

The scheme (2) for an operator A which does not depend on t with the use of associated polynomials was investigated in [6].

The operators U^{i}_{k} are related to the operator-valued polynomials U_{k}(β_{1}L, β_{2}L, ⋯, β_{q}L) by the following equation:

U^{q+j−1}_{k}= U_{k} − τβ_{0} ∑_{i=1}^{k} U^{i+q+j−1}_{k−i} L_{i+q+j−1}[A(t_{i+j}) − A(t_{j})]U_{i}.  (9)

The following estimates hold:

∥U_{k}(β_{1}L, β_{2}L, ⋯, β_{q}L)∥ ≤ c;  (10)
∥U_{k}(β_{1}S^{q−1}, β_{2}S^{q−2}, ⋯, β_{q}I)∥ ≤ c;  (11)
∥τA(t_{j})S^{k}∥ ≤ \frac{q}{β_{0}k} ,  (12)

where S = L^{\frac{1}{q}} , c = const > 0.

Taking into account

U_{k} (β_{1}L, β_{2}L, ..., β_{q}L) = L^{\frac{k}{q}} U_{k} (β_{1}S^{q−1}, β_{2}S^{q−2}, ..., β_{q}I) ,

according to inequalities (11), (12) and the conditions of the theorem the following is true:

∥(A(t_{k+j}) − A(t_{j}))U_{k}(β_{1}L, β_{2}L, ⋯, β_{q}L)∥ ≤

≤ ∥(A(t_{k+j}) − A(t_{j}))A^{−1}(t_{j})∥∥A(t_{j})L^{\frac{k}{q}} ∥∥U_{k}(β_{1}S^{q−1}, β_{2}S^{q−2}, ⋯, β_{q}I)∥ ≤

≤ c_{0} \frac{t_{k+j} − t_{j}}{τ} · \frac{q}{β_{0}k} · c = c, c = const > 0.  (13)

With the help of estimates (10) and (13) from (9), we obtain

∥ U^{q+j−1}_{k} ∥ ≤ c + cτ ∑_{i=1}^{k} ∥ U^{i+q+j−1}_{k−i} ∥.  (14)

Applying the discrete analog of the Gronwall lemma the following estimate can be obtained:

∥ U^{i+q+j−1}_{k−i} ∥ ≤ ce^{ct_{k−i}}.  (15)

Considering this estimate a priori estimate (4) follows from equation (6). The proof is complete. To the investigation of multistep method for an abstract parabolic equation

are dedicated the papers [1] − [5].

R e f e r e n c e s

14 Bulletin of TICMI

[1] Crouzeix M., Une Methode Multipas Implicite-explisite pour l’approximation des Equations d’evolution Paraboliques. - Numerische Mathematik, 35, 1980, pp.257-276.

[2] Crouzeix M., Raviart P.A., Approximation des Equations d’evolution Lineaires par des Methodes Pasmultiples.- C.R.Acad. Sci. Paris, 283, A, 1976, 367-370.

[3] Le Roux M.N., Semi-Discretisation en Temps pour les Equations d’evolution Paraboliques Lorsque l’operateur Depend du Temps. - R.A.I.R.O. Analyse Numerique/Numerical Analisis, 13, 1979, 119-137.

[4] Zlamal M., Finite Element Multistep Methods for Parabolic Equations. I.S.N.M. 28, Birkha^{..}tuser Verlag. Basel and Stuttgart, 1975.

[5] Zlamal M., Finite Element Multistep Discretisations of Parabolic Boundary Value Problems. - Math. Comp., 29, 350-359.

[6] Rogava J.L., Semi-discrete Schems for Operator Differential Equations. Tbilisi: Technical University press, 1995, (Russian).