×

zbMATH — the first resource for mathematics

Singularly perturbed finite element methods. (English) Zbl 0569.65065
The paper deals with the approximation of singularly perturbed ordinary differential equations by new piecewise linear finite elements (hinged elements). The purpose of introducing one or more hinges into a standard element is to allow the element to modify itself as the perturbation parameter \(\epsilon\) changes. As a result, pointwise and global error estimates (independent of \(\epsilon)\) are presented.
Reviewer: J.Haslinger

MSC:
65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
34E15 Singular perturbations, general theory for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Berger, A.E., Solomon, J.M., Ciment, M.: An analysis of a uniformly accurate difference method for a singular perturbation problem. Math. Comput37, 79-94 (1981) · Zbl 0471.65062 · doi:10.1090/S0025-5718-1981-0616361-0
[2] El-Mistikawy, T.M., Werle, M.J.: Numerical Method for Boundary Layers with blowing ? the exponential box scheme. AIAA J.16, 749-751 (1978) · Zbl 0383.76018 · doi:10.2514/3.7573
[3] de Groen, P.P.N.: A finite element method with a large mesh-width for a stiff two-point boundary value problem. J. Comput. Appl. Math.7, 3-15 (1981) · Zbl 0452.65056 · doi:10.1016/0771-050X(81)90001-2
[4] Hegarty, A.F., Miller, J.J.H., O’Riordan, E.: Uniform second order difference schemes for singular perturbation problems. Boundary and interior layers ? computational and asymptotic methods. Miller, J.J.H. (ed.). Dublin: Boole Press, pp. 301-305, 1980
[5] Hemker, P.W.: A numerical study of stiff two-point boundary value problems. Math. Centre Tracts No. 80, Amsterdam, 1977 · Zbl 0426.65043
[6] Kellogg, R.B., Tsan, A.: Analysis of some difference approximations for a singular perturbation problem without turning points. Math. Comput.32, 1025-1039 (1978) · Zbl 0418.65040 · doi:10.1090/S0025-5718-1978-0483484-9
[7] Miller, J.J.H.: A Finite Element Method for a Two Point Boundary Value Problem with a small Parameter affecting the Highest Derivative. Trinity College, Dublin, School of Mathematics Report, TCD-1975-11
[8] O’Riordan, E.: Finite element methods for singularity perturbed problems. PhD Thesis, Trinity College, Dublin, 1982a
[9] O’Riordan, E.: Finite element methods for singularly perturbed problems. BAIL II conference. Miller, J.J.H. (ed.). Dublin: Boole Press, pp. 336-341, 1982b
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.