Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Piecewise linear finite element methods are not localized

Author: Alan Demlow
Journal: Math. Comp. 73 (2004), 1195-1201
MSC (2000): Primary 65N30, 65N15
Published electronically: July 14, 2003
MathSciNet review: 2047084
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Recent results of Schatz show that standard Galerkin finite element methods employing piecewise polynomial elements of degree two and higher to approximate solutions to elliptic boundary value problems are localized in the sense that the global dependence of pointwise errors is of higher order than the overall order of the error. These results do not indicate that such localization occurs when piecewise linear elements are used. We show via simple one-dimensional examples that Schatz's estimates are sharp in that localization indeed does not occur when piecewise linear elements are used.

References [Enhancements On Off] (What's this?)

  • [BLR86] Heribert Blum, Qun Lin, and Rolf Rannacher, Asymptotic error expansion and Richardson extrapolation for linear finite elements, Numer. Math. 49 (1986), 11-37. MR 87m:65172
  • [HSWW01] Wolfgang Hoffmann, Alfred H. Schatz, Lars B. Wahlbin, and Gabriel 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 2002a:65178
  • [Sch98] 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 98j:65082
  • [SW02] Alfred H. Schatz and Lars B. Wahlbin, Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. Part II: The piecewise linear case, to appear.
  • [Whe73] Mary Fanett Wheeler, An optimal $L_\infty$ error estimate for Galerkin approximations to solutions of two-point boundary value problems, SIAM J. Numer. Anal. 10 (1973), 914-917. MR 49:8399

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N30, 65N15

Retrieve articles in all journals with MSC (2000): 65N30, 65N15

Additional Information

Alan Demlow
Affiliation: Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853

Received by editor(s): July 22, 2002
Received by editor(s) in revised form: December 15, 2002
Published electronically: July 14, 2003
Additional Notes: This material is based upon work supported under a National Science Foundation graduate fellowship and under NSF grant DMS-0071412.
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society