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

MathSciNet review:
1370857

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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 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.

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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