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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 Raviart-Thomas-Nedelec space . Optimal order estimates are obtained for the approximation of *p* and the associated velocity field in and , , and, if for *p* in .

**[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 ,"*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, " -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 -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.

Retrieve articles in *Mathematics of Computation*
with MSC:
65N30

Retrieve articles in all journals with MSC: 65N30

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1985-0771029-9

Article copyright:
© Copyright 1985
American Mathematical Society