Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • J. Baranger, On the thickness of the boundary layer in elliptic singular perturbation problems, Numerical analysis of singular perturbation problems (Proc. Conf., Math. Inst., Catholic Univ., Nijmegen, 1978) Academic Press, London-New York, 1979, pp. 395–400. MR 556528
  • Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. MR 0482275
  • Lipman Bers, Fritz John, and Martin Schechter, Partial differential equations, Lectures in Applied Mathematics, Vol. III, Interscience Publishers John Wiley & Sons, Inc. New York-London-Sydney, 1964. With special lectures by Lars Garding and A. N. Milgram. MR 0163043
  • J. G. Besjes, Singular perturbation problems for linear elliptic differential operators of arbitrary order. I. Degeneration to elliptic operators, J. Math. Anal. Appl. 49 (1975), 24–46. MR 509049, DOI https://doi.org/10.1016/0022-247X%2875%2990160-2
  • H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam, 1973.
  • Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174
  • Jean Descloux, On finite element matrices, SIAM J. Numer. Anal. 9 (1972), 260–265. MR 309292, DOI https://doi.org/10.1137/0709025
  • Lars Wahlbin, Error estimates for a Galerkin method for a class of model equations for long waves, Numer. Math. 23 (1975), 289–303. With an appendix by Lars Wahlbin, Jim Douglas, Jr. and Todd Dupont. MR 388799, DOI https://doi.org/10.1007/BF01438256
  • Wiktor Eckhaus, Matched asymptotic expansions and singular perturbations, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. North-Holland Mathematics Studies, No. 6. MR 0670800
  • Wiktor Eckhaus, Asymptotic analysis of singular perturbations, Studies in Mathematics and its Applications, vol. 9, North-Holland Publishing Co., Amsterdam-New York, 1979. MR 553107
  • P. Grisvard, Boundary Value Problems in Nonsmooth Domains, Lecture Notes 19, Department of Mathematics, University of Maryland, College Park, Maryland, 1980.
  • P. W. Hemker, A numerical study of stiff two-point boundary problems, Mathematisch Centrum, Amsterdam, 1977. Mathematical Centre Tracts, No. 80. MR 0488784
  • A. M. Il′in, A difference scheme for a differential equation with a small parameter multiplying the highest derivative, Mat. Zametki 6 (1969), 237–248 (Russian). MR 260195
  • J.-L. Lions, Perturbations singulières dans les problèmes aux limites et en contrôle optimal, Lecture Notes in Mathematics, Vol. 323, Springer-Verlag, Berlin-New York, 1973 (French). MR 0600331
  • J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969. J. L. Lions & E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. I, Springer-Verlag, New York, Heidelberg, Berlin, 1972.
  • John 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 (Proc. Conf., Math. Inst., Catholic Univ., Nijmegen, 1978) Academic Press, London-New York, 1979, pp. 467–474. MR 556537
  • W. L. Miranker, The Computational Theory of Stiff Differential Equations, Istituto per le Applicazioni del Calcolo "Mauro Picone", Rome, 1975. 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.
  • Koichi Niijima, On a three-point difference scheme for a singular perturbation problem without a first derivative term. I, II, Mem. Numer. Math. 7 (1980), 1–27. MR 588462
  • J. Nitsche and A. Schatz, On local approximation properties of $L_{2}$-projection on spline-subspaces, Applicable Anal. 2 (1972), 161–168. MR 397268, DOI https://doi.org/10.1080/00036817208839035
  • Hans-Jürgen Reinhardt, A posteriori error estimates and adaptive finite element computations for singularly perturbed one space dimensional parabolic equations, Analytical and numerical approaches to asymptotic problems in analysis (Proc. Conf., Univ. Nijmegen, Nijmegen, 1980) North-Holland Math. Stud., vol. 47, North-Holland, Amsterdam-New York, 1981, pp. 213–233. MR 605509
  • A. H. Schatz, V. C. Thomée, and L. B. Wahlbin, Maximum norm stability and error estimates in parabolic finite element equations, Comm. Pure Appl. Math. 33 (1980), no. 3, 265–304. MR 562737, DOI https://doi.org/10.1002/cpa.3160330305
  • A. H. Schatz and L. B. Wahlbin, Interior maximum norm estimates for finite element methods, Math. Comp. 31 (1977), no. 138, 414–442. MR 431753, DOI https://doi.org/10.1090/S0025-5718-1977-0431753-X
  • A. H. Schatz and L. B. Wahlbin, On the quasi-optimality in $L_{\infty }$ of the $\dot H^{1}$-projection into finite element spaces, Math. Comp. 38 (1982), no. 157, 1–22. MR 637283, DOI https://doi.org/10.1090/S0025-5718-1982-0637283-6
  • G. I. Šiškin and V. A. Titov, A difference scheme for a differential equation with two small parameters at the derivatives, Čisl. Metody Meh. Splošn. Sredy 7 (1976), no. 2, 145–155 (Russian). MR 0455427
  • Lars B. Wahlbin, A quasioptimal estimate in piecewise polynomial Galerkin approximation of parabolic problems, Numerical analysis (Dundee, 1981) Lecture Notes in Math., vol. 912, Springer, Berlin-New York, 1982, pp. 230–245. MR 654353
  • W. L. Wendland, Elliptic systems in the plane, Monographs and Studies in Mathematics, vol. 3, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. MR 518816
  • Miloš Zlámal, Curved elements in the finite element method. II, SIAM J. Numer. Anal. 11 (1974), 347–362. MR 343660, DOI https://doi.org/10.1137/0711031

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N30

Retrieve articles in all journals with MSC: 65N30


Additional Information

Article copyright: © Copyright 1983 American Mathematical Society