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Global estimates for mixed methods for second order elliptic equations


Authors: Jim Douglas and Jean E. Roberts
Journal: Math. Comp. 44 (1985), 39-52
MSC: Primary 65N30
DOI: https://doi.org/10.1090/S0025-5718-1985-0771029-9
MathSciNet review: 771029
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Abstract: Global error estimates in $ {L^2}(\Omega )$, $ {L^\infty }(\Omega )$, and $ {H^{ - s}}(\Omega )$, $ \Omega $ in $ {{\mathbf{R}}^2}$ or $ {{\mathbf{R}}^3}$, are derived for a mixed finite element method for the Dirichlet problem for the elliptic operator $ Lp = - \operatorname{div}(a\;{\mathbf{grad}}\;p + {\mathbf{b}}p) + cp$ based on the Raviart-Thomas-Nedelec space $ {{\mathbf{V}}_h} \times {W_h} \subset {\mathbf{H}}(\operatorname{div};\Omega ) \times {L^2}(\Omega )$ . Optimal order estimates are obtained for the approximation of p and the associated velocity field $ {\mathbf{u}} = - (a\;{\mathbf{grad}}\;p + {\mathbf{b}}p)$ in $ {L^2}(\Omega )$ and $ {H^{ - s}}(\Omega )$, $ 0 \leqslant s \leqslant k + 1$, and, if $ \Omega \subset {{\mathbf{R}}^2}$ for p in $ {L^\infty }(\Omega )$.


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  • [1] F. Brezzi, "On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers," RAIRO Anal. Numér., v. 2, 1974, pp. 129-151. 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. 91-103. MR 667620 (84b:65111)
  • [3] T. Dupont & R. Scott, "Polynomial approximation of functions in Sobolev space," Math. Comp., v. 34, 1980, pp. 441-463. MR 559195 (81h:65014)
  • [4] R. Falk & J. Osborn, "Error estimates for mixed methods," RAIRO Anal. Numér., v. 14, 1980, pp. 249-277. 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. 41-78. MR 610597 (83c:65239)
  • [6] J. L. Lions & E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, Berlin, 1970. MR 0350177 (50:2670)
  • [7] J. C. Nedelec, "Mixed finite elements in $ {{\mathbf{R}}^3}$," Numer. Math., v. 35, 1980, pp. 315-341. 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, Springer-Verlag, Berlin, 1977. MR 0483555 (58:3547)
  • [9] A. H. Schatz, "An observation concerning Ritz-Galerkin methods with indefinite bilinear forms," Math. Comp., v. 28, 1974, pp. 959-962. MR 0373326 (51:9526)
  • [10] R. Scholz, " $ {L_\infty }$-convergence of saddle-point approximation for second order problems," RAIRO Anal. Numér., v. 11, 1977, pp. 209-216. 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. 169-177. MR 665363 (83i:65084)
  • [12] R. Scholz, "Optimal $ {L_\infty }$-estimates for a mixed finite element method for second order elliptic and parabolic problems," Calcolo, v. 20, 1983, pp. 355-377. 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: https://doi.org/10.1090/S0025-5718-1985-0771029-9
Article copyright: © Copyright 1985 American Mathematical Society

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