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

Schatz, A. H.; Wahlbin, L. B.

doi:10.1090/S0025-5718-1983-0679434-4

On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions
HTML articles powered by AMS MathViewer

by A. H. Schatz and L. B. Wahlbin PDF

Math. Comp. 40 (1983), 47-89 Request permission

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

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, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275
Lipman Bers, Fritz John, and Martin Schechter, Partial differential equations, Lectures in Applied Mathematics, Vol. III, Interscience Publishers, a division of 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 10.1016/0022-247X(75)90160-2

Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert

Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
Jean Descloux, On finite element matrices, SIAM J. Numer. Anal. 9 (1972), 260–265. MR 309292, DOI 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 10.1007/BF01438256
Wiktor Eckhaus, Matched asymptotic expansions and singular perturbations, North-Holland Mathematics Studies, No. 6, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. 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

Boundary Value Problems in Nonsmooth Domains

P. W. Hemker, A numerical study of stiff two-point boundary problems, Mathematical Centre Tracts, No. 80, Mathematisch Centrum, Amsterdam, 1977. 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

Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires

Non-Homogeneous Boundary Value Problems and Applications

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

The Computational Theory of Stiff Differential Equations

Mem. Num. Math.

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 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 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 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 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 10.1137/0711031