Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. Part II: The piecewise linear case
HTML articles powered by AMS MathViewer
- by Alfred H. Schatz and Lars B. Wahlbin PDF
- Math. Comp. 73 (2004), 517-523 Request permission
Abstract:
We extend results from Part I about estimating gradient errors elementwise a posteriori, given there for quadratic and higher elements, to the piecewise linear case. The key to our new result is to consider certain technical estimates for differences in the error, $e(x_{1})-e(x_{2})$, rather than for $e(x)$ itself. We also give a posteriori estimators for second derivatives on each element.References
- A. Demlow, Piecewise linear finite elements methods are not localized, Math. Comp. (to appear).
- Kenneth Eriksson and Claes Johnson, An adaptive finite element method for linear elliptic problems, Math. Comp. 50 (1988), no. 182, 361–383. MR 929542, DOI 10.1090/S0025-5718-1988-0929542-X
- W. Hoffmann, A. H. Schatz, L. B. Wahlbin, and G. Wittum, Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. I. A smooth problem and globally quasi-uniform meshes, Math. Comp. 70 (2001), no. 235, 897–909. MR 1826572, DOI 10.1090/S0025-5718-01-01286-8
- Alfred H. Schatz, Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids. I. Global estimates, Math. Comp. 67 (1998), no. 223, 877–899. MR 1464148, DOI 10.1090/S0025-5718-98-00959-4
- A. H. Schatz and L. B. Wahlbin, Pointwise error estimates for differences of piecewise linear finite element approximations, SIAM J. Numer. Anal. (to appear).
Additional Information
- Alfred H. Schatz
- Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
- Email: schatz@math.cornell.edu
- Lars B. Wahlbin
- Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
- Email: wahlbin@math.cornell.edu
- Received by editor(s): April 12, 2002
- Received by editor(s) in revised form: September 7, 2002
- Published electronically: June 17, 2003
- Additional Notes: Both authors were supported by the National Science Foundation, USA, Grant DMS-0071412. They thank a referee for suggesting improvements in the presentation
- © Copyright 2003 American Mathematical Society
- Journal: Math. Comp. 73 (2004), 517-523
- MSC (2000): Primary 65N30, 65N15
- DOI: https://doi.org/10.1090/S0025-5718-03-01570-9
- MathSciNet review: 2028417