- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Annual Issues
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 523909, 10 pages
Local Projection-Based Stabilized Mixed Finite Element Methods for Kirchhoff Plate Bending Problems
College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China
Received 5 December 2012; Accepted 18 February 2013
Academic Editor: Carlos Vazquez
Copyright © 2013 Xuehai Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, The Netherlands, 1978.
- P. G. Ciarlet and P.-A. Raviart, “A mixed finite element method for the biharmonic equation,” in Mathematical Aspects of Finite Elements in Partial Differential Equations: Proceedings of a Symposium Conducted by the Mathematics Research Center, the University of Wisconsin, Madison, Wis, USA, 1974, pp. 125–145, Academic Press, New York, NY, USA, 1974.
- R. Glowinski and O. Pironneau, “Numerical methods for the first biharmonic equation and the two-dimensional Stokes problem,” SIAM Review, vol. 21, no. 2, pp. 167–212, 1979.
- I. Babuška, J. Osborn, and J. Pitkäranta, “Analysis of mixed methods using mesh dependent norms,” Mathematics of Computation, vol. 35, no. 152, pp. 1039–1062, 1980.
- R. S. Falk and J. E. Osborn, “Error estimates for mixed methods,” RAIRO Analyse Numérique, vol. 14, no. 3, pp. 249–277, 1980.
- T. Gudi, “Residual-based a posteriori error estimator for the mixed finite element approximation of the biharmonic equation,” Numerical Methods for Partial Differential Equations, vol. 27, no. 2, pp. 315–328, 2011.
- T. Gudi, N. Nataraj, and A. K. Pani, “Mixed discontinuous Galerkin finite element method for the biharmonic equation,” Journal of Scientific Computing, vol. 37, no. 2, pp. 139–161, 2008.
- B. Cockburn, J. Gopalakrishnan, and R. Lazarov, “Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems,” SIAM Journal on Numerical Analysis, vol. 47, no. 2, pp. 1319–1365, 2009.
- B. Cockburn, B. Dong, and J. Guzmán, “A hybridizable and superconvergent discontinuous Galerkin method for biharmonic problems,” Journal of Scientific Computing, vol. 40, no. 1–3, pp. 141–187, 2009.
- F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer, New York, NY, USA, 1991.
- D. N. Arnold and R. Winther, “Mixed finite elements for elasticity,” Numerische Mathematik, vol. 92, no. 3, pp. 401–419, 2002.
- S. Adams and B. Cockburn, “A mixed finite element method for elasticity in three dimensions,” Journal of Scientific Computing, vol. 25, no. 3, pp. 515–521, 2005.
- D. N. Arnold, G. Awanou, and R. Winther, “Finite elements for symmetric tensors in three dimensions,” Mathematics of Computation, vol. 77, no. 263, pp. 1229–1251, 2008.
- J. Guzmán and M. Neilan, “Symmetric and conforming mixed finite elements for plane elasticity using rational bubble functions,” Numerische Mathematik. In press.
- C. Johnson, “On the convergence of a mixed finite-element method for plate bending problems,” Numerische Mathematik, vol. 21, pp. 43–62, 1973.
- K. Hellan, Analysis of Elastic Plates in Flexure by a Simplified Finite Element Method, Acta Polytechnica Scandinavica. Civil Engineering and Building Construction Series, Norges Tekniske Vitenskapsakademi, 1967.
- K. Herrmann, “Finite element bending analysis for plates,” Journal of the Engineering Mechanics Division, vol. 93, pp. 49–83, 1967.
- E. M. Behrens and J. Guzmán, “A mixed method for the biharmonic problem based on a system of first-order equations,” SIAM Journal on Numerical Analysis, vol. 49, no. 2, pp. 789–817, 2011.
- J. Huang, X. Huang, and W. Han, “A new discontinuous Galerkin method for Kirchhoff plates,” Computer Methods in Applied Mechanics and Engineering, vol. 199, no. 23-24, pp. 1446–1454, 2010.
- S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer, New York, NY, USA, 3rd edition, 2008.
- P. B. Bochev, C. R. Dohrmann, and M. D. Gunzburger, “Stabilization of low-order mixed finite elements for the Stokes equations,” SIAM Journal on Numerical Analysis, vol. 44, no. 1, pp. 82–101, 2006.
- J. Li and Y. He, “A stabilized finite element method based on two local Gauss integrations for the Stokes equations,” Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 58–65, 2008.
- P. B. Bochev and C. R. Dohrmann, “A computational study of stabilized, low-order C0 finite element approximations of Darcy equations,” Computational Mechanics, vol. 38, no. 4-5, pp. 323–333, 2006.
- K. Nafa, “Local projection finite element stabilization for Darcy flow,” International Journal of Numerical Analysis and Modeling, vol. 7, no. 4, pp. 656–666, 2010.
- F. Shi, J. Yu, and K. Li, “A new stabilized mixed finite-element method for Poisson equation based on two local Gauss integrations for linear element pair,” International Journal of Computer Mathematics, vol. 88, no. 11, pp. 2293–2305, 2011.
- R. An and K. T. Li, “Stabilized mixed finite element approximation for a fourth-order obstacle problem,” Acta Mathematicae Applicatae Sinica, vol. 32, no. 6, pp. 1068–1078, 2009.
- J. Guzmán and M. Neilan, “Conforming and divergence free stokes elements on general triangular meshes,” Mathematics of Computation. In press.
- K. Feng and Z.-C. Shi, Mathematical Theory of Elastic Structures, Springer, Berlin, Germany, 1996.
- J. N. Reddy, Theory and Analysis of Elastic Plates and Shells, CRC Press, New York, NY, USA, 2nd edition, 2006.
- P. Clément, “Approximation by finite element functions using local regularization,” Revue Française d'Automatique, Informatique, Recherche Opérationnelle. Analyse Numérique, vol. 9, no. 2, pp. 77–84, 1975.
- M. I. Comodi, “The Hellan-Herrmann-Johnson method: some new error estimates and postprocessing,” Mathematics of Computation, vol. 52, no. 185, pp. 17–29, 1989.
- M. Dauge, Elliptic Boundary Value Problems on Corner Domains, Springer, Berlin, Germany, 1988.
- P. Grisvard, Singularities in Boundary Value Problems, Masson, Paris, France, 1992.
- S. C. Brenner, T. Gudi, and L. Sung, “An a posteriori error estimator for a quadratic -interior penalty method for the biharmonic problem,” IMA Journal of Numerical Analysis, vol. 30, no. 3, pp. 777–798, 2010.
- M. Wang, “On the necessity and sufficiency of the patch test for convergence of nonconforming finite elements,” SIAM Journal on Numerical Analysis, vol. 39, no. 2, pp. 363–384, 2001.
- R. Verfürth, A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, Chichester, UK, 1996.