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

Full-text PDF Free Access

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.

**1.**Abdellatif Agouzal and Jean-Marie Thomas,*Une méthode d’éléments finis hybrides en décomposition de domaines*, RAIRO Modél. Math. Anal. Numér.**29**(1995), no. 6, 749–764 (French, with English and French summaries). MR**1360674****2.**James H. Bramble, Joseph E. Pasciak, and Panayot S. Vassilevski,*Computational scales of Sobolev norms with application to preconditioning*, Math. Comp.**69**(2000), no. 230, 463–480. MR**1651742**, 10.1090/S0025-5718-99-01106-0**3.**Guo Ying Wang and Ming Lun Chen,*Second-order accurate difference method for the singularly perturbed problem of fourth-order ordinary differential equations*, Appl. Math. Mech.**11**(1990), no. 5, 431–437 (Chinese, with English summary); English transl., Appl. Math. Mech. (English Ed.)**11**(1990), no. 5, 463–468. MR**1069806**, 10.1007/BF02016376**4.**Philippe G. Ciarlet,*The finite element method for elliptic problems*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR**0520174****5.**Ph. Clément,*Approximation by finite element functions using local regularization*, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. \jname RAIRO Analyse Numérique**9**(1975), no. R-2, 77–84 (English, with Loose French summary). MR**0400739****6.**M. Crouzeix and V. Thomée,*The stability in 𝐿_{𝑝} and 𝑊¹_{𝑝} of the 𝐿₂-projection onto finite element function spaces*, Math. Comp.**48**(1987), no. 178, 521–532. MR**878688**, 10.1090/S0025-5718-1987-0878688-2**7.**W. McLean and O. Steinbach,*Boundary element preconditioners for a hypersingular integral equation on an interval*, Adv. Comput. Math.**11**(1999), no. 4, 271–286. MR**1732138**, 10.1023/A:1018944530343**8.**L. Ridgway Scott and Shangyou Zhang,*Finite element interpolation of nonsmooth functions satisfying boundary conditions*, Math. Comp.**54**(1990), no. 190, 483–493. MR**1011446**, 10.1090/S0025-5718-1990-1011446-7**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), no. 2, 307–316. MR**1696413**, 10.1016/S0377-0427(99)00073-4**11.**Olaf Steinbach,*On a hybrid boundary element method*, Numer. Math.**84**(2000), no. 4, 679–695. MR**1738053**, 10.1007/s002110050014**12.**O. Steinbach and W. L. Wendland,*The construction of some efficient preconditioners in the boundary element method*, Adv. Comput. Math.**9**(1998), no. 1-2, 191–216. Numerical treatment of boundary integral equations. MR**1662766**, 10.1023/A:1018937506719**13.**Lars B. Wahlbin,*Superconvergence in Galerkin finite element methods*, Lecture Notes in Mathematics, vol. 1605, Springer-Verlag, Berlin, 1995. MR**1439050**

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

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:
http://dx.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