The and versions of the finite element method for problems with boundary layers
Authors:
Christoph Schwab and Manil Suri
Journal:
Math. Comp. 65 (1996), 14031429
MSC (1991):
Primary 65N30, 35B30, 65N15
MathSciNet review:
1370857
Fulltext PDF Free Access
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Abstract: We study the uniform approximation of boundary layer functions for , , by the and versions of the finite element method. For the version (with fixed mesh), we prove superexponential convergence in the range . We also establish, for this version, an overall convergence rate of in the energy norm error which is uniform in , and show that this rate is sharp (up to the term) when robust estimates uniform in are considered. For the version with variable mesh (i.e., the version), we show that exponential convergence, uniform in , is achieved by taking the first element at the boundary layer to be of size . Numerical experiments for a model elliptic singular perturbation problem show good agreement with our convergence estimates, even when few degrees of freedom are used and when is as small as, e.g., . They also illustrate the superiority of the approach over other methods, including a loworder version with optimal ``exponential" mesh refinement. The estimates established in this paper are also applicable in the context of corresponding spectral element methods.
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 2.
 I. Babu\v{s}ka and M. Suri. On locking and robustness in the finite element method. SIAM J. Numer. Anal., 29:12611293, 1992. MR 94c:65128
 3.
 I. Babu\v{s}ka and B. A. Szabo. Lecture notes on finite element analysis, (to appear).
 4.
 I. A. Blatov and V. V. Strygin. On estimates best possible in order in the Galerkin finite element method for singularly perturbed boundary value problems. Russian Acad. Sci. Dokl. Math., 47:9396, 1993.
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 E. C. Gartland. Uniform highorder difference schemes for a singularly perturbed twopoint boundary value problem. Math. Comp., 48:551564, 1987. MR 89a:65116
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 W. B. Liu and J. Shen. A new efficient spectral Galerkin method for singular perturbation problems, Preprint, Department of Mathematics, Penn State University, State College Pa (1994).
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 11.
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 12.
 H. Kraus. Thin elastic shells: an introduction to the theoretical foundations and the analysis of their static and dynamic behavior. New York, Wiley 1967.
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 K. Scherer. On optimal global error bounds obtained by scaled local error estimates. Numer. Math, 36:151176, 1981. MR 82e:65053
 15.
 C. Schwab and M. Suri. Locking and boundary layer effects in the finite element approximation of the ReissnerMindlin plate model. Proc. Symp. Appl. Math., 48:367371, 1994. MR 95m:65195
 16.
 C. Schwab, M. Suri, and C. Xenophontos. The finite element method for problems in mechanics with boundary layers (to appear).
 17.
 C. Schwab and S. Wright. Boundary layers in hierarchical beam and plate models, report. Journal of Elasticity 38 (1995), 140. MR 96d:73044
 18.
 G.I. Shishkin. Grid approximation of singularly perturbed parabolic equations with internal layers. Soviet J. Numer. Anal. Math. Modelling, 3:393407, 1988. MR 89k:65109
 19.
 R. Vulanovi\'{c}, D. Herceg, and N. Petrovi\'{c}. On the extrapolation for a singularly perturbed boundary value problem. Computing, 36:6979, 1986. MR 88a:65008
 20.
 C. A. Xenophontos. The version of the finite element method for singularly perturbed problems in unsmooth domains. Ph.D. Dissertation, UMBC, 1996.
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Additional Information
Christoph Schwab
Affiliation:
Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, CH8092, Zürich, Switzerland
Email:
schwab@sam.math.ethz.ch
Manil Suri
Affiliation:
Department of Mathematics and Statistics, University of Maryland Baltimore County, 5401 Wilkens Avenue, Baltimore, Maryland 21228
Email:
suri@math.umbc.edu
DOI:
http://dx.doi.org/10.1090/S0025571896007818
PII:
S 00255718(96)007818
Keywords:
Boundary layer,
singularly perturbed problem,
$p$ version,
$hp$ version,
spectral element method
Received by editor(s):
March 7, 1995
Additional Notes:
This work was supported in part by the Air Force Office of Scientific Research, Air Force Systems Command, USAF under Grant F4962092J0100.
Dedicated:
Dedicated to Professor Ivo Babuška on the occasion of his seventieth birthday
Article copyright:
© Copyright 1996
American Mathematical Society
