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The $p$ and $hp$ 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
MathSciNet review: 1370857
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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.

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Additional Information

Christoph Schwab
Affiliation: Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, CH-8092, Zürich, Switzerland

Manil Suri
Affiliation: Department of Mathematics and Statistics, University of Maryland Baltimore County, 5401 Wilkens Avenue, Baltimore, Maryland 21228

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