Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Error analysis of variable degree mixed methods for elliptic problems via hybridization


Authors: Bernardo Cockburn and Jayadeep Gopalakrishnan
Journal: Math. Comp. 74 (2005), 1653-1677
MSC (2000): Primary 65N30
DOI: https://doi.org/10.1090/S0025-5718-05-01741-2
Published electronically: March 1, 2005
MathSciNet review: 2164091
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A new approach to error analysis of hybridized mixed methods is proposed and applied to study a new hybridized variable degree Raviart-Thomas method for second order elliptic problems. The approach gives error estimates for the Lagrange multipliers without using error estimates for the other variables. Error estimates for the primal and flux variables then follow from those for the Lagrange multipliers. In contrast, traditional error analyses obtain error estimates for the flux and primal variables first and then use it to get error estimates for the Lagrange multipliers. The new approach not only gives new error estimates for the new variable degree Raviart-Thomas method, but also new error estimates for the classical uniform degree method with less stringent regularity requirements than previously known estimates. The error analysis is achieved by using a variational characterization of the Lagrange multipliers wherein the other unknowns do not appear. This approach can be applied to other hybridized mixed methods as well.


References [Enhancements On Off] (What's this?)

  • 1. T. ARBOGAST AND Z. CHEN, On the implementation of mixed methods as nonconforming methods for second-order elliptic problems, Math. Comp., 64 (1995), pp. 943-972. MR 1303084 (95k:65102)
  • 2. D. N. ARNOLD AND F. BREZZI, Mixed and non-conforming finite element methods: implementation, post-processing and error estimates, Modél. Math. Anal.Numér., 19 (1985), pp. 7-35. MR 0813687 (87g:65126)
  • 3. P. BRENNER AND L. R. SCOTT, The mathematical theory of finite element methods, vol. 15 of Texts in Applied Mathematics, Springer Verlag, 1994. MR 1278258 (95f:65001)
  • 4. F. BREZZI, J. DOUGLAS, JR., AND D. MARINI, Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47 (1985), pp. 217-235. MR 0799685 (87g:65133)
  • 5. -, Variable degree mixed methods for second order elliptic problems, Mat. Aplic. e Comp., 4 (1985), pp. 19-34. MR 0808322 (87d:65129)
  • 6. F. BREZZI AND M. FORTIN, Mixed and Hybrid finite element methods, Springer Verlag, 1991. MR 1115205 (92d:65187)
  • 7. B. COCKBURN AND J. GOPALAKRISHNAN, A characterization of hybridized mixed methods for the Dirichlet problem, SIAM J. Numer. Anal., 42 (2004), pp. 283-301. MR 2051067
  • 8. L. DEMKOWICZ, P. MONK, L. VARDAPETYAN, AND W. RACHOWICZ, De Rham diagram for $hp$ finite element spaces, Comput. Math. Appl., 39 (2000), pp. 29-38. MR 1746160 (2000m:78052)
  • 9. B. M. FRAEJIS DE VEUBEKE, Displacement and equilibrium models in the finite element method, in Stress Analysis, O. Zienkiewicz and G. Holister, eds., Wiley, New York, 1977, pp. 145-197.
  • 10. G. N. GATICA, Solvability and Galerkin approximations of a class of nonlinear operator equations, Z. Anal. Anwendungen, 21 (2002), pp. 761-781. MR 1929431 (2003h:65074)
  • 11. G. N. GATICA, N. HEUER, AND S. MEDDAHI, On the numerical analysis of nonlinear twofold saddle point problems, IMA J. Numer. Anal., 23 (2003), pp. 301-330. MR 1975268 (2004b:65183)
  • 12. J. GOPALAKRISHNAN, A Schwarz preconditioner for a hybridized mixed method, Computational Methods in Applied Mathematics, 3 (2003), pp. 116-134. MR 2002260 (2004g:65033)
  • 13. R. KIRBY AND C. DAWSON, Private Communication (2003).
  • 14. I. PERUGIA AND D. SCHÖTZAU, An $hp$-analysis of the local discontinuous Galerkin method for diffusion problems, J. Sci. Comput. (Special Issue: Proceedings of the Fifth International Conference on Spectral and High Order Methods (ICOSAHOM-01), Uppsala, Sweden), 17 (2002), pp. 561-571. MR 1910752
  • 15. P. RAVIART AND J. THOMAS, A mixed method for second order elliptic problems, in Mathematical Aspects of the Finite Element Method, I. Galligani and E. Magenes, eds., Lecture Notes in Math. 606, Springer-Verlag, New York, 1977. MR 0483555 (58:3547)
  • 16. M. SURI, On the stability and convergence of higher-order mixed finite element methods for second-order elliptic problems, Math. Comp., 54 (1990), pp. 1-19. MR 0990603 (90e:65164)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N30

Retrieve articles in all journals with MSC (2000): 65N30


Additional Information

Bernardo Cockburn
Affiliation: School of Mathematics, University of Minnesota, Vincent Hall, Minneapolis, Minnesota 55455
Email: cockburn@math.umn.edu

Jayadeep Gopalakrishnan
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611–8105
Email: jayg@math.ufl.edu

DOI: https://doi.org/10.1090/S0025-5718-05-01741-2
Keywords: Mixed finite elements, hybrid methods, elliptic problems
Received by editor(s): November 26, 2003
Received by editor(s) in revised form: August 1, 2004
Published electronically: March 1, 2005
Additional Notes: The first author is supported in part by the National Science Foundation (Grant DMS-0411254) and by the University of Minnesota Supercomputing Institute
The second author is supported by the National Science Foundation (Grant DMS-0410030).
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society