×

Superconvergence of external approximation for two-point boundary problems. (English) Zbl 0641.65064

The author considers the problem \(-u''+b(x)u=f(x),\) \(x\in \ell =[0,1]\), \(u(0)=u(1)=0\), where \(b(x)\geq \beta >0\), b,f\(\in L\) 2(\(\ell)\). The superconvergence properties of a certain external method (a generalization of the Galerkin method) for solving this problem is established. In the case when piecewise polynomial spaces are applied, it is proved that the rate of convergence of the approximate solution at the knot points can be exceed the global one.
Reviewer: D.Herceg

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
PDF BibTeX XML Cite
Full Text: EuDML

References:

[1] J. P. Aubin: Approximation of elliptic boundary-value problems. Wiley-Interscience, New York, 1972. · Zbl 0248.65063
[2] C. De Boor B. Swartz: Collocation at Gausian points. SIAM J. Numer. Anal. 10 (1973), 582-606. · Zbl 0232.65065
[3] P. Ciarlet: The finite element method for elliptic problems. North-Holland, Publishing Company (1978). · Zbl 0383.65058
[4] J. Douglas, Jr. T. Dupont: Collocation method for parabolic equations in a single space variable. Lecture Notes in Math., 385 (1974).
[5] J. Douglas, Jr. T. Dupont: Some superconvergence results for Galerkin methods for the approximate solution of two-point boundary problems. - Topics in numerical analysis J.J.M.F. Miller, pp. 89-92 (1973).
[6] P. J. Davis: Interpolation and approximation. Blaisdell Publishing Company (1963). · Zbl 0111.06003
[7] M. Křížek P. Neittaanmäki: Superconvergence phenomenon in the finite element method arising from averaging gradients. Numer. Math. 45 (1984), pp. 105-116. · Zbl 0575.65104
[8] T. Regińska: Superconvergence of eigenvalue external approximation for ordinary differential operators. IMA Jour. Numer. Anal. 6 (1986), pp. 309-323. · Zbl 0619.65076
[9] M. Zlámal: Some superconvergence results in the finite element method. - Mathematical Aspects of f.e.m., Lecture Notes 606 (1977), pp. 353 - 362.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.