|
Local energy estimates for the finite element method on sharply varying grids
Author(s):
Alan
Demlow;
Johnny
Guzmán;
Alfred
H.
Schatz.
Journal:
Math. Comp.
80
(2011),
1-9.
MSC (2010):
Primary 65N30, 65N15
Posted:
June 28, 2010
MathSciNet review:
2728969
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
Local energy error estimates for the finite element method for elliptic problems were originally proved in 1974 by Nitsche and Schatz. These estimates show that the local energy error may be bounded by a local approximation term, plus a global ``pollution'' term that measures the influence of solution quality from outside the domain of interest and is heuristically of higher order. However, the original analysis of Nitsche and Schatz is restricted to quasi-uniform grids. We present local a priori energy estimates that are valid on shape regular grids, an assumption which allows for highly graded meshes and which matches much more closely the typical practical situation. Our chief technical innovation is an improved superapproximation result.
References:
-
- [AL95]
- Douglas N. Arnold and Xiao Bo Liu, Local error estimates for finite element discretizations of the Stokes equations, RAIRO Modél. Math. Anal. Numér. 29 (1995), no. 3, 367-389. MR 1342712 (96d:76055)
- [BH00]
- Randolph E. Bank and Michael Holst, A new paradigm for parallel adaptive meshing algorithms, SIAM J. Sci. Comput. 22 (2000), no. 4, 1411-1443 (electronic). MR 1797889 (2002g:65117)
- [BR01]
- Roland Becker and Rolf Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer. 10 (2001), 1-102. MR 2 009 692
- [BS02]
- Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, second ed., Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 2002. MR 1894376 (2003a:65103)
- [Dem04]
- Alan Demlow, Localized pointwise error estimates for mixed finite element methods, Math. Comp. 73 (2004), no. 248, 1623-1653 (electronic). MR 2059729 (2005e:65184)
- [Dem07]
- -, Local a posteriori estimates for pointwise gradient errors in finite element methods for elliptic problems, Math. Comp. 76 (2007), no. 257, 19-42 (electronic). MR 2261010
- [Guz06]
- Johnny Guzmán, Pointwise error estimates for discontinuous Galerkin methods with lifting operators for elliptic problems, Math. Comp. 75 (2006), no. 255, 1067-1085 (electronic). MR 2219019 (2006m:65269)
- [LN03]
- Xiaohai Liao and Ricardo H. Nochetto, Local a posteriori error estimates and adaptive control of pollution effects, Numer. Methods Partial Differential Equations 19 (2003), no. 4, 421-442. MR 1980188 (2004c:65130)
- [NPV91]
- R. H. Nochetto, M. Paolini, and C. Verdi, An adaptive finite element method for two-phase Stefan problems in two space dimensions. I. Stability and error estimates, Math. Comp. 57 (1991), no. 195, 73-108, S1-S11. MR 1079028 (92a:65322)
- [NS74]
- Joachim A. Nitsche and Alfred H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comp. 28 (1974), 937-958. MR 51:9525
- [STW90]
- Alfred H. Schatz, Vidar Thomée, and Wolfgang L. Wendland, Mathematical theory of finite and boundary element methods, DMV Seminar, vol. 15, Birkhäuser Verlag, Basel, 1990. MR 1116555 (92f:65004)
- [SW77]
- Alfred H. Schatz and Lars B. Wahlbin, Interior maximum norm estimates for finite element methods, Math. Comp. 31 (1977), no. 138, 414-442. MR 0431753 (55:4748)
- [SW95]
- -, Interior maximum-norm estimates for finite element methods, Part II, Math. Comp. 64 (1995), no. 211, 907-928. MR 1297478 (95j:65143)
- [XZ00]
- Jinchao Xu and Aihui Zhou, Local and parallel finite element algorithms based on two-grid discretizations, Math. Comp. 69 (2000), no. 231, 881-909. MR 1654026 (2000j:65102)
Similar Articles:
Retrieve articles in Mathematics of Computation
with
MSC (2010):
65N30, 65N15
Retrieve articles in all Journals with
MSC (2010):
65N30, 65N15
Additional Information:
Alan
Demlow
Affiliation:
Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, Kentucky 40506–0027
Email:
demlow@ms.uky.edu
Johnny
Guzmán
Affiliation:
Division of Applied Mathematics, Brown University, 182 George Street, Providence, Rhode Island 02906
Email:
johnny_guzman@brown.edu
Alfred
H.
Schatz
Affiliation:
Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853
DOI:
10.1090/S0025-5718-2010-02353-1
PII:
S 0025-5718(2010)02353-1
Received by editor(s):
August 14, 2008
Received by editor(s) in revised form:
July 16, 2009
Posted:
June 28, 2010
Additional Notes:
The first author was partially supported by NSF grant DMS-0713770.
The second author was partially supported by NSF grant DMS-0503050.
The third author was partially supported by NSF grant DMS-0612599.
Copyright of article:
Copyright
2010,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
