Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids:

Part I. Global estimates

Author:
Alfred H. Schatz

Journal:
Math. Comp. **67** (1998), 877-899

MSC (1991):
Primary 65N30

DOI:
https://doi.org/10.1090/S0025-5718-98-00959-4

MathSciNet review:
1464148

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Abstract | References | Similar Articles | Additional Information

Abstract: This part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in . In a sense to be discussed below these sharpen known quasi-optimal and estimates for the error on irregular quasi-uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution . We note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces. As a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities. These expansion inequalities are valid for large classes of finite elements defined on irregular grids in and have applications to superconvergence and extrapolation and a posteriori estimates. Part II of this series will contain local estimates applicable to non-smooth problems.

**[1]**H. Blum, Q. Lin and R. Rannacher,*Asymptotic error expansions and Richardson extrapolation for linear finite elements*, Numer. Math.**49**(1986), 11-37. MR**87m:65172****[2]**Ju. P. Krasovskii,*Properties of Green's function and generalized solutions of elliptic boundary value problems*, Soviet Mathematics (translation of Doklady Academy of Sciences of the USSR) (1969), 54-120. MR**38:6233****[3]**F. Natterer,*Uber die punktweise konvergenz finiter elemente*, Numer. Math.**25**(1975), 67-77. MR**57:14514****[4]**J. A. Nitsche,*convergence of finite element approximations*, Proceedings Second Conference on Finite Elements, Rennes, France (1975). MR**81e:65058****[5]**J. A. Nitsche,*convergence of finite element approximations*, Mathematical Aspects of Finite Element Methods, Lecture Notes in Math., vol. 606, Springer-Verlag, 1977, pp. 261-274. MR**58:8351****[6]**J. A. Nitsche and A. H. Schatz,*Interior estimates for Ritz-Galerkin methods*, Math. Comp.**28**(1974), 937-958. MR**51:9525****[7]**R. Rannacher,*Zur konvergenz linearer finiter elemente*, Math. Z.**149**, 69-77. MR**58:8361****[8]**R. Rannacher and R. Scott,*Some optimal error estimates for piecewise linear finite element approximations*, Math. Comp.**38**(1982), 437-445. MR**83e:65180****[9]**A. H. Schatz and L. B. Wahlbin,*Interior maximum norm estimates for finite element methods*, Math. Comp.**31**(1977), 414-442. MR**55:4748****[10]**A. H. Schatz and L. B. Wahlbin,*Interior maximum norm estimates for finite element methods Part II*, Math. Comp.**64**(1995), 907-928. MR**95j:65143****[11]**A. H. Schatz and L. B. Wahlbin,*On the quasi-optimality in of the projection into finite element spaces*, Math. Comp.**38**(1982), 1-21. MR**82m:65106****[12]**A. H. Schatz, I. Sloan and L. B. Wahlbin,*Superconvergence in the finite element method and meshes which are locally symmetric with respect to a point*, SIAM J. Numer. Anal.**33**(1996), 505-521. CMP**96:12****[13]**R. Scott,*Optimal estimates for the finite element method on irregular grids*, Math. Comp.**30**(1976), 681-697. MR**55:9560****[14]**W. Hoffman, A. H. Schatz, L. B. Wahlbin, and G. Wittum,*The analysis of some local pointwise a posteriore error estimators for elliptic problems*, in preparation.**[15]**D. Gilbarg and N. S. Trudinger,*Elliptic partial differential equations of second order*, Grundlehren Math. Wiss., 2nd ed., vol. 224, Springer, 1983. MR**86c:35035**

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Additional Information

**Alfred H. Schatz**

Affiliation:
Department of Mathematics, White Hall, Cornell University, Ithaca, New York 14853

Email:
schatz@math.cornell.edu

DOI:
https://doi.org/10.1090/S0025-5718-98-00959-4

Received by editor(s):
February 7, 1997

Additional Notes:
Supported in part by the National Science Foundation Grant DMS 9403512.

Article copyright:
© Copyright 1998
American Mathematical Society