The and versions of the finite element method for problems with boundary layers

Authors:
Christoph Schwab and Manil Suri

Journal:
Math. Comp. **65** (1996), 1403-1429

MSC (1991):
Primary 65N30, 35B30, 65N15

DOI:
https://doi.org/10.1090/S0025-5718-96-00781-8

MathSciNet review:
1370857

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 super-exponential 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 low-order version with optimal ``exponential" mesh refinement. The estimates established in this paper are also applicable in the context of corresponding spectral element methods.

**1.**D. N. Arnold and R. S. Falk. Asymptotic analysis of the boundary layer for the Reissner-Mindlin plate model.*SIAM J. Math. Anal.*27: 486--514, 1996.**2.**I. Babu\v{s}ka and M. Suri. On locking and robustness in the finite element method.*SIAM J. Numer. Anal.*, 29:1261--1293, 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.*, 47:93--96, 1993.**5.**C. Canuto. Spectral methods and a maximum principle.*Math. Comp.*, 51:615--629, 1988. MR**89d:65099****6.**E. C. Gartland. Uniform high-order difference schemes for a singularly perturbed two-point boundary value problem.*Math. Comp.*, 48:551--564, 1987. MR**89a:65116****7.**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).**8.**W. B. Liu and T. Tang. Boundary layer resolving methods for singularly perturbed problems, submitted to*I.M.A. J. Numer. Anal.***9.**F. W. J. Olver. Error bounds for the Liouville-Green (or WKB) approximation.*Proc. Cambridge Philos. Soc.*, 57:790--810, 1961. MR**24:A313****10.**I. S. Gradshteyn and I. M. Ryzhik.*Table of Series, Integrals and Products, enlarged edition*. Wiley, New York, 1980. MR**81g:33001****11.**H. Hakula, Y. Leino, and J. Pitkäranta. Scale resolution, layers and high-order numerical modeling of shells, to appear in*Comp. Meth. Appl. Mech. Eng.*, 1996.**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.**13.**A. H. Schatz and L. B. Wahlbin. On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions.*Math. Comp.*, 40:47--89, 1983. MR**84c:65137****14.**K. Scherer. On optimal global error bounds obtained by scaled local error estimates.*Numer. Math*, 36:151--176, 1981. MR**82e:65053****15.**C. Schwab and M. Suri. Locking and boundary layer effects in the finite element approximation of the Reissner-Mindlin plate model.*Proc. Symp. Appl. Math.*, 48:367--371, 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), 1--40. MR**96d:73044****18.**G.I. Shishkin. Grid approximation of singularly perturbed parabolic equations with internal layers.*Soviet J. Numer. Anal. Math. Modelling*, 3:393--407, 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*, 36:69--79, 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.

Retrieve articles in *Mathematics of Computation of the American Mathematical Society*
with MSC (1991):
65N30,
35B30,
65N15

Retrieve articles in all journals with MSC (1991): 65N30, 35B30, 65N15

Additional Information

**Christoph Schwab**

Affiliation:
Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, CH-8092, 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:
https://doi.org/10.1090/S0025-5718-96-00781-8

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 F49620-92-J-0100.

Dedicated:
Dedicated to Professor Ivo Babuška on the occasion of his seventieth birthday

Article copyright:
© Copyright 1996
American Mathematical Society