Global estimates for mixed methods for second order elliptic equations
Authors:
Jim Douglas and Jean E. Roberts
Journal:
Math. Comp. 44 (1985), 3952
MSC:
Primary 65N30
MathSciNet review:
771029
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Abstract: Global error estimates in , , and , in or , are derived for a mixed finite element method for the Dirichlet problem for the elliptic operator based on the RaviartThomasNedelec space . Optimal order estimates are obtained for the approximation of p and the associated velocity field in and , , and, if for p in .
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 [1]
 F. Brezzi, "On the existence, uniqueness and approximation of saddlepoint problems arising from Lagrangian multipliers," RAIRO Anal. Numér., v. 2, 1974, pp. 129151. MR 0365287 (51:1540)
 [2]
 J. Douglas, Jr. & J. E. Roberts, "Mixed finite element methods for second order elliptic problems," Mat. Apl. Comput., v. 1, 1982, pp. 91103. MR 667620 (84b:65111)
 [3]
 T. Dupont & R. Scott, "Polynomial approximation of functions in Sobolev space," Math. Comp., v. 34, 1980, pp. 441463. MR 559195 (81h:65014)
 [4]
 R. Falk & J. Osborn, "Error estimates for mixed methods," RAIRO Anal. Numér., v. 14, 1980, pp. 249277. MR 592753 (82j:65076)
 [5]
 C. Johnson & V. Thomée, "Error estimates for some mixed finite element methods for parabolic type problems," RAIRO Anal. Numér., v. 15, 1981, pp. 4178. MR 610597 (83c:65239)
 [6]
 J. L. Lions & E. Magenes, NonHomogeneous Boundary Value Problems and Applications, Vol. I, SpringerVerlag, Berlin, 1970. MR 0350177 (50:2670)
 [7]
 J. C. Nedelec, "Mixed finite elements in ," Numer. Math., v. 35, 1980, pp. 315341. MR 592160 (81k:65125)
 [8]
 P. A. Raviart & J. M. Thomas, "A mixed finite element method for 2nd order elliptic problems," Mathematical Aspects of the Finite Element Method, Lecture Notes in Math., Vol. 606, SpringerVerlag, Berlin, 1977. MR 0483555 (58:3547)
 [9]
 A. H. Schatz, "An observation concerning RitzGalerkin methods with indefinite bilinear forms," Math. Comp., v. 28, 1974, pp. 959962. MR 0373326 (51:9526)
 [10]
 R. Scholz, " convergence of saddlepoint approximation for second order problems," RAIRO Anal. Numér., v. 11, 1977, pp. 209216. MR 0448942 (56:7247)
 [11]
 R. Scholz, "A remark on the rate of convergence for a mixed finite element method for second order problems," Numer. Funct. Anal. Optim., v. 4, 1981/1982, pp. 169177. MR 665363 (83i:65084)
 [12]
 R. Scholz, "Optimal estimates for a mixed finite element method for second order elliptic and parabolic problems," Calcolo, v. 20, 1983, pp. 355377. MR 761790 (86j:65164)
 [13]
 J. M. Thomas, Sur l'Analyse Numérique des Méthodes d'Éléments Finis Hybrides et Mixtes, Thèse, Université P. et M. Curie, Paris, 1977.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198507710299
PII:
S 00255718(1985)07710299
Article copyright:
© Copyright 1985
American Mathematical Society
