Stability of Ritz procedure for nonlinear two-point boundary value problem

Schiop, Alexandru I.

doi:10.1007/bf01436564

Cited by 6 publications

(6 citation statements)

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“…Following Ciarlet, Schultz, and Varga [2] and Schiop [7], we consider the numerical approximation of the solution of the real nonlinear two-point boundary value problem…”

Section: Preliminariesmentioning

confidence: 99%

“…Following Schiop [7], we consider the stability of the Rayleigh-Ritz procedure for the nonlinear (or semilinear) two-point boundary value problem analysed by Ciarlet, Schultz, and Varga [2] (see w 2). After establishing some preliminaries in w 2, we show in w 3 that the proof of Schiop's stability result is only valid, if suitable assumptions about the choice of co-ordinate functions are made.…”

Section: Introductionmentioning

confidence: 99%

“…Though the Rayleigh-Ritz procedure for the approximate solution of nonlinear two-point boundary value problems is often discussed in the literature (see Schultz [t0;w 7] for an introduction and bibliographical details), little attention appears to have been paid to the question of numerical stability, except for Schiop [7] and Mikhlin [6; w 76]. In addition, Schiop's basic result [7; Theorem 2] is in apparent conflict with Mikhlin's [6; Theorem 75.1] in that Schiop's stability result is independent of the choice of co-ordinate functions.…”

Section: Introductionmentioning

confidence: 99%

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Stability of the Rayleigh-Ritz procedure for nonlinear two-point boundary value problems

Omodei

Anderssen

1975

Numer. Math.

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“…Following Ciarlet, Schultz, and Varga [2] and Schiop [7], we consider the numerical approximation of the solution of the real nonlinear two-point boundary value problem…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Stability of the Rayleigh-Ritz procedure for nonlinear two-point boundary value problems

Omodei

Anderssen

1975

Numer. Math.

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“…The ]. [r is a norm in H generated by an inner product, and if we complete the space H in this norm we obtain a Hilbert space (H, [ , Ir) (see [t], [15]). …”

Section: F(u) -----~ I"" I G(t T~ T)u(ta) U(ts) Dr mentioning

confidence: 99%

On the stability of the Ritz procedure for nonlinear problems

Schiop

1974

Numer. Math.

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Abstract. In this paper we establish the stability of the Ritz procedure for the nonlinear equation A u +F(u) --] = 0 where A is a linear positive definite operator in a real Hilbert space H and F is a monotone nonlinear operator. 1~The stability of Ritz procedure has been studied by Mihlin in [7]. In [7] Mihlin also formulated the stability of the Ritz procedure for nonlinear problems and has given a sufficient condition for the stability.Stetter [t3] formulated the stability of nonlinear problems discretizations and has proved sufficient conditions (see also [14] For the study of the stability for discretization procedures see also Spijker [t 2J, Chartres and Stepleman [2].In this paper we shall study the stability of the Ritz procedure for the equation of the form: The problem has been studied by Gishkarianii in [5] by using the second derivative of the functional which we minimize and by the Newton-Kantorovich method for solving perturbated Ritz system, generated by computing errors.We shall use Lemma t for the study of the Ritz perturbated system in afinite dimensional space.We give also a sufficient condition for the stability of the Ritz procedure for a class of nonlinear two-point boundary value problems.

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“…Let us remark that we did not use the existence of the second derivative of the functional (2.3) as has been done by Mikhlin [3] and Schiop [5]. On the other hand, we have to assume (2.14).…”

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confidence: 99%