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Theory and Applications of Singular Perturbations pp 207–227Cite as

Analysis of adaptive finite element methods for −εU″+U′=F based on a-posteriori error estimates

  • Part I: Theory of Singular Perturbations
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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 942))

Abstract

A-posteriori error estimates containing realistic bounds provide a basis for adaptive numerical methods solving differential equations. In this paper, for a singularly perturbed convection-diffusion model problem, a finite element method is analysed which is based on a technique of approximate symmetrization of the given unsymmetric problem. Realistic a-posteriori error estimates with respect to an appropriate energy-norm are presented. A series of numerical examples demonstrate that our adaptive methods detect and resolve the boundary layer.

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References

  1. Axelsson, O., Frank, L., van der Sluis, A. (eds.): Analytical and numerical approaches to asymptotic problems in analysis (Proc. Conf., Nijmegen, 1980). North-Holland, Amsterdam-New York-Oxford, 1981.

    Google Scholar 

  2. Babuška, I., Aziz, A.K.: Survey lectures on the mathematical foundations of the Finite Element Method. In: The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Conf. Baltimore, 1972), pp. 5–359. Academic Press, New York-London, 1972.

    Google Scholar 

  3. Babuška, I., Rheinboldt, W.C.: Error estimates for adaptive finite element comptuations. SIAM J. Numer. Anal. 15 (1978), 736–754.

    Article  MathSciNet  MATH  Google Scholar 

  4. Babuška, I., Rheinboldt, W.C.: Analysis of optimal finite element meshes in R1. Math. Comp. 33 (1979), 435–463.

    MathSciNet  MATH  Google Scholar 

  5. Babuška, I., Rheinboldt, W.C.: A posteriori error analysis of finite element solutions for one-dimensional problems. SIAM J. Numer. Anal 18 (1981), 565–589.

    Article  MathSciNet  MATH  Google Scholar 

  6. Babuška, I., Szymczak, W.G.: An error analysis for the finite element method applied to convection diffusion problems. Techn. Note BN-962, Inst. Physical Sc. and Techn., Univ. Maryland, College Park, March 1981.

    MATH  Google Scholar 

  7. Barrett, J.W., Morton, K.W.: Optimal finite element solutions to diffusion-convection problems in one dimension. Internat. J. Numer. Methods. Engrg. 15 (1980), 1457–1474.

    Article  MathSciNet  MATH  Google Scholar 

  8. Barrett, J.W., Morton, K.W.: Optimal Petrov-Galerkin methods through approximate symmetrization. Numerical Analysis Report 4/80, Dept. of Math., Univ. of Reading, 1980.

    Google Scholar 

  9. Berger, A.E., Solomon, J.M., Ciment, M., Leventhal, S.H., Weinberg, B.C.: Generalized OCI schemes for boundary layer problems. Math. Comp. 35 (1980), 695–731.

    Article  MathSciNet  Google Scholar 

  10. Brandt, A.: Multi-level adaptive techniques for singular-perturbation problems. In-[15],, pp. 53–142.

    Google Scholar 

  11. Christie, I., Griffiths, D.F., Mitchell, A.R., Zienkiewicz, O.C.: Finite element methods for second order differential equations with significant first derivatives. Internat. J. Numer. Methods Engrg. 10 (1976), 1389–1396.

    Article  MathSciNet  MATH  Google Scholar 

  12. Ciarlet, P.G.: The finite element method for elliptic problems. Studies in Math. and its Applications, Vol. 4. North-Holland, Amsterdam-New York-Oxford, 1978.

    Book  MATH  Google Scholar 

  13. Doolan, E.P., Miller, J.J.H., Schilders, W.H.A.: Uniform numerical methods for problems with initial and boundary layers. Boole, Dublin, 1980.

    MATH  Google Scholar 

  14. Griffiths, D.F., Lorenz, J.: An analysis of the Petrov-Galerkin finite element method. Comput. Methods Appl. Mech. Engrg. 14 (1978), 39–64.

    Article  MathSciNet  MATH  Google Scholar 

  15. Hemker, P.W., Miller, J.J.H. (eds.): Numerical analysis of singular perturbation problems. (Proc. Conf., Nijmegen, 1978). Academic Press, London-New York-San Francisco, 1979.

    Google Scholar 

  16. Ilin, A.M.: Differencing scheme for a differential equation with a small parameter affecting the highest derivatives. Math. Notes Acad. Sci. USSR 6 (1969), 596–602.

    Article  Google Scholar 

  17. Kellogg, R.B., Tsan, A.: Analysis of some difference approximations for a singular perturbation problem without turning points. Math. Comp. 32 (1978), 1025–1039.

    Article  MathSciNet  MATH  Google Scholar 

  18. Kreiss, B., Kreiss, H.-O.: Numerical methods for singular perturbation problems. SIAM J. Numer. Anal. 18 (1981), 262–276.

    Article  MathSciNet  MATH  Google Scholar 

  19. Lentini, M., Pereyra, V.: An adaptive finite difference solver for nonlinear two-point boundary problems with mild boundary layers. SIAM J. Numer. Anal. 14 (1977), 91–111.

    Article  MathSciNet  MATH  Google Scholar 

  20. Miller, J.J.H. (ed.): Boundary and interior layers—Computational and asymptotic methods. (Proc. BAIL I Conf., Trinity College, Dublin, June 1980), Boole, Dublin, 1980.

    Google Scholar 

  21. Pearson, C.E.: On a differential equation of the boundary layer type. J. Math. and Phys. 47 (1968), 134–154.

    Article  MathSciNet  MATH  Google Scholar 

  22. Reinhardt, H.-J.: A-posteriori error estimates for the finite element solution of a singularly perturbed linear ordinary differential equation. SIAM J. Numer. Anal. 18 (1981), 406–430.

    Article  MathSciNet  MATH  Google Scholar 

  23. Reinhardt, H.-J.: A-posteriori error estimates and adaptive finite element computations for singularly perturbed one space dimensional parabolic equations-In [1], pp. 213–233.

    Google Scholar 

  24. Reinhardt, H.-J.: A-posteriori error analysis and adaptive finite element methods for singularly perturbed convection-diffusion equations. Submitted to Math. Methods Appl. Sci.

    Google Scholar 

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Authors
  1. Hans-Jürgen Reinhardt
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W. Eckhaus E. M. de Jager

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© 1982 Springer-Verlag

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Reinhardt, HJ. (1982). Analysis of adaptive finite element methods for −εU″+U′=F based on a-posteriori error estimates. In: Eckhaus, W., de Jager, E.M. (eds) Theory and Applications of Singular Perturbations. Lecture Notes in Mathematics, vol 942. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0094749

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  • DOI: https://doi.org/10.1007/BFb0094749

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-11584-7

  • Online ISBN: 978-3-540-39332-0

  • eBook Packages: Springer Book Archive

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