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On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions


Authors: A. H. Schatz and L. B. Wahlbin
Journal: Math. Comp. 40 (1983), 47-89
MSC: Primary 65N30
DOI: https://doi.org/10.1090/S0025-5718-1983-0679434-4
MathSciNet review: 679434
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Abstract: Second order elliptic boundary value problems which are allowed to degenerate into zero order equations are considered. The behavior of the ordinary Galerkin finite element method without special arrangements to treat singularities is studied as the problem ranges from true second order to singularly perturbed.


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  • [1] J. Baranger, "On the thickness of the boundary layer in elliptic-elliptic singular perturbation problems," Numerical Analysis of Singular Perturbation Problems (P. W. Hemker and J. J. H. Miller, eds.), Academic Press, London, New York, San Francisco, 1979, pp. 395-400. MR 556528 (80m:35008)
  • [2] J. Bergh & J. Löfström, Interpolation Spaces: An Introduction, Springer-Verlag, Berlin, Heidelberg, New York, 1976. MR 0482275 (58:2349)
  • [3] L. Bers, F. John & M. Schechter, Partial Differential Equations, Interscience, New York, London, Sydney, 1964. MR 0163043 (29:346)
  • [4] J. G. Besjes, "Singular perturbation problems for linear elliptic differential operators of arbitrary order. I. Degeneration to elliptic operators," J. Math. Anal. Appl., v. 49, 1975, pp. 24-46. MR 0509049 (58:22967a)
  • [5] H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam, 1973.
  • [6] Ph. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. MR 0520174 (58:25001)
  • [7] J. Descloux, "On finite element matrices," SIAM J. Numer. Anal., v. 9, 1972, pp. 260-265. MR 0309292 (46:8402)
  • [8] J. Douglas, Jr., T. Dupont & L. B. Wahlbin, "The stability in $ {L^q}$ of the $ {L^2}$-projection into finite element function spaces," Numer. Math., v. 23, 1975, pp. 193-197. MR 0388799 (52:9633)
  • [9] W. Eckhaus, Matched Asymptotic Expansions and Singular Perturbations, North-Holland, Amsterdam, 1973. MR 0670800 (58:32369)
  • [10] W. Eckhaus, Asymptotic Analysis of Singular Perturbations, North-Holland, Amsterdam, 1979. MR 553107 (81a:34048)
  • [11] P. Grisvard, Boundary Value Problems in Nonsmooth Domains, Lecture Notes 19, Department of Mathematics, University of Maryland, College Park, Maryland, 1980.
  • [12] P. W. Hemker, A Numerical Study of Stiff Two-Point Boundary Problems, Math. Centre Tracts 80, Amsterdam, 1977. MR 0488784 (58:8294)
  • [13] A. M. Ilin, "A difference scheme for a differential equation with a small parameter multiplying the highest derivative." Mat. Zametki, v. 6, 1969, pp. 237-248; English transl. in Math. Notes, v. 6, 1969, pp. 596-602. MR 0260195 (41:4823)
  • [14] J. L. Lions, Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal, Lecture Notes in Math., vol. 323, Springer-Verlag, Berlin, Heidelberg, New York, 1973. MR 0600331 (58:29078)
  • [15] J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969.
  • [16] J. L. Lions & E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. I, Springer-Verlag, New York, Heidelberg, Berlin, 1972.
  • [17] J. J. H. Miller, "On the convergence, uniformly in $ \varepsilon $, of difference schemes for a two point boundary singular perturbation problem," Numerical Analysis of Singular Perturbation Problems (P. W. Hemker and J. J. H. Miller, eds.), Academic Press, London, New York, San Francisco, 1979, pp. 467-474. MR 556537 (81f:65061)
  • [18] W. L. Miranker, The Computational Theory of Stiff Differential Equations, Istituto per le Applicazioni del Calcolo "Mauro Picone", Rome, 1975.
  • [19] K. Niijima, "On a three-point difference scheme for a singular perturbation problem without a first derivative term. I," Mem. Num. Math., v. 7, 1980.
  • [20] K. Niijima, "On a three-point difference scheme for a singular perturbation problem without a first derivative term. II". Mem. Num. Math., v. 7, 1980. MR 588462 (82a:65059)
  • [21] J. Nitsche & A. H. Schatz, "On local approximation properties of $ {L_2}$ projection on spline subspaces," Applicable Anal., v. 2, 1972, pp. 161-168. MR 0397268 (53:1127)
  • [22] H.-J. Reinhardt, "A-posteriori error estimates and adaptive finite element computations for singularly perturbed one space dimensional parabolic problems," Analytical and Numerical Approaches to Asymptotic Problems in Analyis (O. Axelsson, L. S. Frank and A. Van der Sluis, eds.), North-Holland, Amsterdam, 1981, pp. 213-233. MR 605509 (82e:65095)
  • [23] A. H. Schatz, V. Thomée & L. B. Wahlbin, "Maximum norm stability and error estimates in parabolic finite element equations," Comm. Pure Appl. Math., v. 33, 1980, pp. 265-304. MR 562737 (81g:65136)
  • [24] A. H. Schatz & L. B. Wahlbin, "Interior maximum norm estimates for finite element methods," Math. Comp., v. 31, 1977, pp. 414-442. MR 0431753 (55:4748)
  • [25] A. H. Schatz & L. B. Wahlbin, "On the quasi-optimality in $ {L_\infty }$ of the $ \dot{H}^1$ projection into finite element spaces," Math. Comp., v. 38, 1982, pp. 1-22. MR 637283 (82m:65106)
  • [26] G. I. Shiskin & V. A. Titov, "A difference scheme for a differential equation with two small parameters affecting the derivatives," Numer. Meth. Mechs. Cont. Media, v. 7, 1976, pp. 145-155. (Russian) MR 0455427 (56:13665)
  • [27] L. B. Wahlbin, "A quasioptimal estimate in piecewise polynomial Galerkin approximation of parabolic problems," Numerical Analysis (G. A. Watson, ed.), Lecture Notes in Math., vol. 912, Springer-Verlag, New York, Heidelberg, Berlin, 1982, pp. 230-245. MR 654353 (83f:65157)
  • [28] W. L. Wendland, Elliptic Systems in the Plane, Pitman, San Francisco, 1979. MR 518816 (80h:35053)
  • [29] M. Zlamal, "Curved elements in the finite element method. II," SIAM J. Numer. Anal., v. 11, 1974, pp. 347-362. MR 0343660 (49:8400)

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DOI: https://doi.org/10.1090/S0025-5718-1983-0679434-4
Article copyright: © Copyright 1983 American Mathematical Society

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