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On the stability of the $L^2$ projection in $H^1(\Omega)$


Authors: James H. Bramble, Joseph E. Pasciak and Olaf Steinbach
Journal: Math. Comp. 71 (2002), 147-156
MSC (2000): Primary 65D05, 65N30, 65N50
DOI: https://doi.org/10.1090/S0025-5718-01-01314-X
Published electronically: May 7, 2001
MathSciNet review: 1862992
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Abstract:

We prove the stability in $H^1(\Omega)$ of the $L^2$ projection onto a family of finite element spaces of conforming piecewise linear functions satisfying certain local mesh conditions. We give explicit formulae to check these conditions for a given finite element mesh in any number of spatial dimensions. In particular, stability of the $L^2$ projection in $H^1(\Omega)$ holds for locally quasiuniform geometrically refined meshes as long as the volume of neighboring elements does not change too drastically.


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Additional Information

James H. Bramble
Affiliation: Department of Mathematics, Texas A & M University, College Station, Texas 77843
Email: bramble@math.tamu.edu

Joseph E. Pasciak
Affiliation: Department of Mathematics, Texas A & M University, College Station, Texas 77843
Email: pasciak@math.tamu.edu

Olaf Steinbach
Affiliation: Mathematisches Institut A, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
Email: steinbach@mathematik.uni-stuttgart.de

DOI: https://doi.org/10.1090/S0025-5718-01-01314-X
Keywords: $L^2$ projection, finite elements, stability, adaptivity
Received by editor(s): February 11, 2000
Received by editor(s) in revised form: May 24, 2000
Published electronically: May 7, 2001
Additional Notes: This work was supported by the National Science Foundation under grants numbered DMS-9626567 and DMS-9973328 and by the State of Texas under ARP/ATP grant #010366-168. This work was done while the third author was a Postdoctoral Research Associate at the Institute for Scientific Computation (ISC), Texas A & M University. The financial support by the ISC is gratefully acknowledged.
Article copyright: © Copyright 2001 American Mathematical Society

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