## Global estimates for mixed methods for second order elliptic equations

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- by Jim Douglas and Jean E. Roberts PDF
- Math. Comp.
**44**(1985), 39-52 Request permission

## 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 ${L^\infty }(\Omega )$.

## References

- 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 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*, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth. MR**0350177** - J.-C. Nédélec,
*Mixed finite elements in $\textbf {R}^{3}$*, Numer. Math.**35**(1980), no. 3, 315–341. MR**592160**, DOI 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) Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977, pp. 292–315. MR**0483555** - Alfred H. Schatz,
*An observation concerning Ritz-Galerkin methods with indefinite bilinear forms*, Math. Comp.**28**(1974), 959–962. MR**373326**, DOI 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 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 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 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.

## Additional Information

- © Copyright 1985 American Mathematical Society
- Journal: Math. Comp.
**44**(1985), 39-52 - MSC: Primary 65N30
- DOI: https://doi.org/10.1090/S0025-5718-1985-0771029-9
- MathSciNet review: 771029