The $p$ and $hp$ versions of the finite element method for problems with boundary layers
HTML articles powered by AMS MathViewer
- by Christoph Schwab and Manil Suri PDF
- Math. Comp. 65 (1996), 1403-1429 Request permission
Abstract:
We study the uniform approximation of boundary layer functions $\exp (-x/d)$ for $x\in (0,1)$, $d\in (0,1]$, by the $p$ and $hp$ versions of the finite element method. For the $p$ version (with fixed mesh), we prove super-exponential convergence in the range $p + 1/2 > e/(2d)$. We also establish, for this version, an overall convergence rate of ${\mathcal O}(p^{-1}\sqrt {\ln p})$ in the energy norm error which is uniform in $d$, and show that this rate is sharp (up to the $\sqrt {\ln p}$ term) when robust estimates uniform in $d\in (0,1]$ are considered. For the $p$ version with variable mesh (i.e., the $hp$ version), we show that exponential convergence, uniform in $d\in (0,1]$, is achieved by taking the first element at the boundary layer to be of size ${\mathcal O}(pd)$. 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 $d$ is as small as, e.g., $10^{-8}$. They also illustrate the superiority of the $hp$ approach over other methods, including a low-order $h$ version with optimal “exponential" mesh refinement. The estimates established in this paper are also applicable in the context of corresponding spectral element methods.References
- 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.
- Ivo Babuška and Manil Suri, On locking and robustness in the finite element method, SIAM J. Numer. Anal. 29 (1992), no. 5, 1261–1293. MR 1182731, DOI 10.1137/0729075
- I. Babuška and B. A. Szabo. Lecture notes on finite element analysis, (to appear).
- 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.
- Claudio Canuto, Spectral methods and a maximum principle, Math. Comp. 51 (1988), no. 184, 615–629. MR 930226, DOI 10.1090/S0025-5718-1988-0930226-2
- Eugene C. Gartland Jr., Uniform high-order difference schemes for a singularly perturbed two-point boundary value problem, Math. Comp. 48 (1987), no. 178, 551–564, S5–S9. MR 878690, DOI 10.1090/S0025-5718-1987-0878690-0
- 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).
- W. B. Liu and T. Tang. Boundary layer resolving methods for singularly perturbed problems, submitted to I.M.A. J. Numer. Anal.
- F. W. J. Olver, Error bounds for the Liouville-Green (or $WK\,B$) approximation, Proc. Cambridge Philos. Soc. 57 (1961), 790–810. MR 130452
- I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980. Corrected and enlarged edition edited by Alan Jeffrey; Incorporating the fourth edition edited by Yu. V. Geronimus [Yu. V. Geronimus] and M. Yu. Tseytlin [M. Yu. Tseĭtlin]; Translated from the Russian. MR 582453
- 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.
- H. Kraus. Thin elastic shells: an introduction to the theoretical foundations and the analysis of their static and dynamic behavior. New York, Wiley 1967.
- 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 (1983), no. 161, 47–89. MR 679434, DOI 10.1090/S0025-5718-1983-0679434-4
- Karl Scherer, On optimal global error bounds obtained by scaled local error estimates, Numer. Math. 36 (1980/81), no. 2, 151–176. MR 611490, DOI 10.1007/BF01396756
- Christoph Schwab and Manil Suri, Locking and boundary layer effects in the finite element approximation of the Reissner-Mindlin plate model, Mathematics of Computation 1943–1993: a half-century of computational mathematics (Vancouver, BC, 1993) Proc. Sympos. Appl. Math., vol. 48, Amer. Math. Soc., Providence, RI, 1994, pp. 367–371. MR 1314872, DOI 10.1090/psapm/048/1314872
- C. Schwab, M. Suri, and C. Xenophontos. The $hp$ finite element method for problems in mechanics with boundary layers (to appear).
- C. Schwab and S. Wright, Boundary layers of hierarchical beam and plate models, J. Elasticity 38 (1995), no. 1, 1–40. MR 1323554, DOI 10.1007/BF00121462
- G. I. Shishkin, Grid approximation of singularly perturbed parabolic equations with internal layers, Soviet J. Numer. Anal. Math. Modelling 3 (1988), no. 5, 393–407. Translated from the Russian. MR 974091
- R. Vulanović, D. Herceg, and N. Petrović, On the extrapolation for a singularly perturbed boundary value problem, Computing 36 (1986), no. 1-2, 69–79 (English, with German summary). MR 832931, DOI 10.1007/BF02238193
- C. A. Xenophontos. The $hp$ version of the finite element method for singularly perturbed problems in unsmooth domains. Ph.D. Dissertation, UMBC, 1996.
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
- 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.
- © Copyright 1996 American Mathematical Society
- 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
Dedicated: Dedicated to Professor Ivo Babuška on the occasion of his seventieth birthday