Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. Part II: The piecewise linear case

Authors:
Alfred H. Schatz and Lars B. Wahlbin

Journal:
Math. Comp. **73** (2004), 517-523

MSC (2000):
Primary 65N30, 65N15

DOI:
https://doi.org/10.1090/S0025-5718-03-01570-9

Published electronically:
June 17, 2003

MathSciNet review:
2028417

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Abstract | References | Similar Articles | Additional Information

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, , rather than for itself. We also give a posteriori estimators for second derivatives on each element.

**[1]**A. Demlow,*Piecewise linear finite elements methods are not localized*, Math. Comp. (to appear).**[2]**K. Eriksson and C. Johnson,*An adaptive finite element method for linear elliptic problems*, Math. Comp.**50**(1988), 361-384. MR**89c:65119****[3]**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. Part I: A smooth problem and globally quasi-uniform meshes*, Math. Comp.**70**(2001), 897-909. MR**2002a:65178****[4]**A. H. Schatz,*Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids: Part I. Global estimates*, Math. Comp.**67**(1998), 877-899. MR**98j:65082****[5]**A. H. Schatz and L. B. Wahlbin,*Pointwise error estimates for differences of piecewise linear finite element approximations*, SIAM J. Numer. Anal. (to appear).

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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

DOI:
https://doi.org/10.1090/S0025-5718-03-01570-9

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

Article copyright:
© Copyright 2003
American Mathematical Society