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

Author(s): James H. Bramble; Joseph E. Pasciak; Olaf Steinbach.
Journal: Math. Comp. 71 (2002), 147-156.
MSC (2000): Primary 65D05, 65N30, 65N50
Posted: May 7, 2001
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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.

References:

1.
A. Agouzal, J.-M. Thomas, Une methode d'elements finis hybrides en decomposition de domains. Math. Modell. Numer. Anal. 29 (1995) 749-764. MR 96g:65115
2.
J. H. Bramble, J. E. Pasciak, P. S. Vassilevski, Computational scales of Sobolev norms with applications to preconditioning. Math. Comp. 69 (2000), 463-480. MR 2000k:65088
3.
J. H. Bramble, J. Xu, Some estimates for a weighted $L^2$ projection. Math. Comp. 56 (1991) 463-476. MR 91i:65140
4.
P. G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, 1978. MR 58:25001
5.
P. Clement, Approximation by finite element functions using local regularization. RAIRO Anal. Numer. 9 R-2 (1975) 77-84. MR 53:4569
6.
M. Crouzeix, V. Thomeé, The stability in $L_p$ and $W_p^1$ of the $L^2$-projection onto finite element function spaces. Math. Comp. 48 (1987) 521-532. MR 88f:41016
7.
W. McLean, O. Steinbach, Boundary element preconditioners for a hypersingular integral equation on a curve. Adv. Comput. Math. 11 (1999) 271-286. MR 2000k:65236

8.
L. R. Scott, S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483-493. MR 90j:65021
9.
H. Schulz, O. Steinbach, A new a posteriori error estimator in direct boundary element methods. The Neumann problem. Multifield Problems. State of the Art. (A.-M. Sändig, W. Schiehlen, and W. L. Wendland, eds.) Springer-Verlag, Berlin, 201-208, 2000.
10.
O. Steinbach, Adaptive finite element-boundary element solution of boundary value problems. J. Comput. Appl. Math. 106 (1999) 307-316. MR 2000b:65225
11.
O. Steinbach, On a hybrid boundary element method. Numer. Math. 84 (2000), 679-695. MR 2001a:65154
12.
O. Steinbach, W. L. Wendland, The construction of some efficient preconditioners in the boundary element method. Adv. Comput. Math. 9 (1998) 191-216. MR 99j:65219
13.
L. B. Wahlbin, Superconvergence in Galerkin Finite Element Methods. Lecture Notes in Mathematics 1605, Springer, Berlin, 1995. MR 98j:65083

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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: 10.1090/S0025-5718-01-01314-X
PII: S 0025-5718(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
Posted: 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.
Copyright of article: Copyright 2001, American Mathematical Society

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