Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



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
Published electronically: May 7, 2001
MathSciNet review: 1862992
Full-text PDF

Abstract | References | Similar Articles | Additional Information


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

  • 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

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65D05, 65N30, 65N50

Retrieve articles in all journals with MSC (2000): 65D05, 65N30, 65N50

Additional Information

James H. Bramble
Affiliation: Department of Mathematics, Texas A & M University, College Station, Texas 77843

Joseph E. Pasciak
Affiliation: Department of Mathematics, Texas A & M University, College Station, Texas 77843

Olaf Steinbach
Affiliation: Mathematisches Institut A, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

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

American Mathematical Society