Improved accuracy by adapted meshrefinements in the finite element method
Author:
Kenneth Eriksson
Journal:
Math. Comp. 44 (1985), 321343
MSC:
Primary 65N30; Secondary 65N50
MathSciNet review:
777267
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Abstract: For appropriately adapted meshrefinements, optimal order error estimates are proved for the finite element approximate solution of the Neumann problem for the secondorder elliptic equation , where is the Dirac distribution.
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 R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. MR 0450957 (56:9247)
 [2]
 I. Babuška, "Error bounds for the finite element method," Numer. Math., v. 16, 1971, pp. 322333. MR 0288971 (44:6166)
 [3]
 I. Babuška & A. K. Aziz, "Survey lectures on the mathematical foundations of the finite element method," Section 6.3.6 of The Mathematical Foundations of the Finite Element Method with Applications to Parital Differential Equations, (A. K. Aziz, ed.), Academic Press, New York, 1973.
 [4]
 J. H. Bramble & S. R. Hilbert, "Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation," SIAM J. Numer. Anal., v. 7, 1970, pp. 112124. MR 0263214 (41:7819)
 [5]
 J. H. Bramble & A. H. Schatz, "Estimates for spline projections," RAIRO Anal. Numér., v. 10, 1976, pp. 537. MR 0436620 (55:9563)
 [6]
 P. G. Ciarlet, The Finite Element Method for Elliptic Problems, NorthHolland, Amsterdam, 1978. MR 0520174 (58:25001)
 [7]
 P. G. Ciarlet, "Discrete variational Green's function. I," Aequationes Math., v. 4, 1970, pp. 7482. MR 0273838 (42:8714)
 [8]
 K. Eriksson, Improved Convergence by MeshRefinement in the Finite Element Method, Thesis, Chalmers University of Technology and the University of Göteborg, 1981.
 [9]
 S. R. Hilbert, "A mollifier useful for approximations in Sobolev spaces and some applications to approximating solutions of differential equations," Math. Comp., v. 27, 1973, pp. 8189. MR 0331715 (48:10047)
 [10]
 Ju. P. Krasovskii, "Isolation of singularities of the Green's function," Math. USSRIzv., v. 1, 1967, pp. 935966.
 [11]
 J. L. Lions & E. Magenes, Problèmes aux Limites Non Homogènes et Applications, Vol. 1, Dunod, Paris, 1968. MR 0247243 (40:512)
 [12]
 J. A. Nitsche & A. H. Schatz, "Interior estimates for RitzGalerkin methods," Math. Comp., v. 28, 1974, pp. 937958. MR 0373325 (51:9525)
 [13]
 A. H. Schatz & L. B. Wahlbin, "Interior maximum norm estimates for finite element methods", Math. Comp., v. 31, 1977, pp. 414442. MR 0431753 (55:4748)
 [14]
 R. Scott, "Finite element convergence for singular data," Numer. Math., v. 21, 1973, pp. 317327. MR 0337032 (49:1805)
 [15]
 R. Scott, "Optimal estimates for the finite element method on irregular meshes," Math. Comp., v. 30, 1976, pp. 681697. MR 0436617 (55:9560)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198507772673
PII:
S 00255718(1985)07772673
Keywords:
Neumann problem,
Green's function,
finite element method,
meshrefinement,
error estimate
Article copyright:
© Copyright 1985
American Mathematical Society
