DOI: 10.1002/nme.1620121010. <i>A‐posteriori</i> error estimates for the finite element method

A‐posteriori error estimates for the finite element method

10.1002/nme.1620121010

Crossref journal-article

Wiley

International Journal for Numerical Methods in Engineering (311)

Abstract

AbstractComputable a‐posteriori error estimates for finite element solutions are derived in an asymptotic form for h → 0 where h measures the size of the elements. The approach has similarity to the residual method but differs from it in the use of norms of negative Sobolev spaces corresponding to the given bilinear (energy) form. For clarity the presentation is restricted to one‐dimensional model problems. More specifically, the source, eigenvalue, and parabolic problems are considered involving a linear, self‐adjoint operator of the second order. Generalizations to more general one‐dimensional problems are straightforward, and the results also extend to higher space dimensions; but this involves some additional considerations. The estimates can be used for a practical a‐posteriori assessment of the accuracy of a computed finite element solution, and they provide a basis for the design of adaptive finite element solvers.

Bibliography

BabuÅ¡ka, I., & Rheinboldt, W. C. (1978). Aâposteriori error estimates for the finite element method. International Journal for Numerical Methods in Engineering, 12(10), 1597â1615. Portico.

Authors 2

I. Babuška (first)
W. C. Rheinboldt (additional)

References 11 Referenced 930

10.1137/0709052
10.1137/1018075
10.1145/321784.321786
{'key': 'e_1_2_1_5_2', 'first-page': '223', 'volume-title': 'Mathematical Software—III', 'author': 'Babuška I.', 'year': '1973'} / Mathematical Software—III by Babuška I. (1973)
I.BabuškaandW.Rheinboldt ‘Error estimates for adaptive finite element computations’ University of Maryland Institute for Physical Science and Technology Technical Note BN‐854(1977);
SIAM J. Num. Anal. 1978 15
I.BabuškaandW.Rheinboldt Theoretical and Computational Analysis of the Finite Element Method in preparation.
{'key': 'e_1_2_1_8_2', 'series-title': 'Technical Note BN‐869', 'volume-title': 'Analysis of optimal finite element meshes in R1', 'author': 'Babuška I.', 'year': '1977'} / Analysis of optimal finite element meshes in R1 / Technical Note BN‐869 by Babuška I. (1977)
10.1016/B978-0-12-068650-6.50006-X
{'key': 'e_1_2_1_10_2', 'volume-title': 'Sobolev Spaces', 'author': 'Adams R. A.', 'year': '1975'} / Sobolev Spaces by Adams R. A. (1975)
{'key': 'e_1_2_1_11_2', 'volume-title': 'Modern Numerical Methods for Ordinary Differential Equations', 'author': 'Hall C. A.', 'year': '1976'} / Modern Numerical Methods for Ordinary Differential Equations by Hall C. A. (1976)

Dates

Type	When
Created	20 years ago (Aug. 8, 2005, 2:17 p.m.)
Deposited	1 year, 10 months ago (Oct. 8, 2023, 10:59 a.m.)
Indexed	3 minutes ago (Sept. 6, 2025, 8:14 a.m.)
Issued	47 years, 8 months ago (Jan. 1, 1978)
Published	47 years, 8 months ago (Jan. 1, 1978)
Published Online	20 years, 2 months ago (June 30, 2005)
Published Print	47 years, 8 months ago (Jan. 1, 1978)

Funders 0

None

BibTeX

@article{Babu_ka_1978, title={A‐posteriori error estimates for the finite element method}, volume={12}, ISSN={1097-0207}, url={http://dx.doi.org/10.1002/nme.1620121010}, DOI={10.1002/nme.1620121010}, number={10}, journal={International Journal for Numerical Methods in Engineering}, publisher={Wiley}, author={Babuška, I. and Rheinboldt, W. C.}, year={1978}, month=jan, pages={1597–1615} }

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