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 ${L^\infty }(\Omega )$.
- F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974), no. R-2, 129–151 (English, with French summary). MR 365287
- J. Douglas Jr. and J. E. Roberts, Mixed finite element methods for second order elliptic problems, Mat. Apl. Comput. 1 (1982), no. 1, 91–103 (English, with Portuguese summary). MR 667620
- Todd Dupont and Ridgway Scott, Polynomial approximation of functions in Sobolev spaces, Math. Comp. 34 (1980), no. 150, 441–463. MR 559195, DOI https://doi.org/10.1090/S0025-5718-1980-0559195-7
- R. S. Falk and J. E. Osborn, Error estimates for mixed methods, RAIRO Anal. Numér. 14 (1980), no. 3, 249–277 (English, with French summary). MR 592753
- Claes Johnson and Vidar Thomée, Error estimates for some mixed finite element methods for parabolic type problems, RAIRO Anal. Numér. 15 (1981), no. 1, 41–78 (English, with French summary). MR 610597
- J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 181. MR 0350177
- J.-C. Nédélec, Mixed finite elements in ${\bf R}^{3}$, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, DOI https://doi.org/10.1007/BF01396415
- P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Springer, Berlin, 1977, pp. 292–315. Lecture Notes in Math., Vol. 606. MR 0483555
- Alfred H. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comp. 28 (1974), 959–962. MR 373326, DOI https://doi.org/10.1090/S0025-5718-1974-0373326-0
- Reinhard Scholz, $L_{\infty }$-convergence of saddle-point approximations for second order problems, RAIRO Anal. Numér. 11 (1977), no. 2, 209–216, 221 (English, with French summary). MR 448942, DOI https://doi.org/10.1051/m2an/1977110202091
- R. Scholz, A remark on the rate of convergence for a mixed finite-element method for second order problems, Numer. Funct. Anal. Optim. 4 (1981/82), no. 3, 269–277. MR 665363, DOI https://doi.org/10.1080/01630568208816117
- R. Scholz, Optimal $L_{\infty }$-estimates for a mixed finite element method for second order elliptic and parabolic problems, Calcolo 20 (1983), no. 3, 355–377 (1984). MR 761790, DOI https://doi.org/10.1007/BF02576470 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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© Copyright 1985
American Mathematical Society