On the stability of the projection in

Authors:
James H. Bramble, Joseph E. Pasciak and Olaf Steinbach

Journal:
Math. Comp. **71** (2002), 147-156

MSC (2000):
Primary 65D05, 65N30, 65N50

Published electronically:
May 7, 2001

MathSciNet review:
1862992

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Abstract | References | Similar Articles | Additional Information

We prove the stability in of the 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 projection in 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