Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

Nonconforming elements in least-squares mixed finite element methods


Authors: Huo-Yuan Duan and Guo-Ping Liang
Journal: Math. Comp. 73 (2004), 1-18
MSC (2000): Primary 65N30
DOI: https://doi.org/10.1090/S0025-5718-03-01520-5
Published electronically: March 27, 2003
MathSciNet review: 2034108
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we analyze the finite element discretization for the first-order system least squares mixed model for the second-order elliptic problem by means of using nonconforming and conforming elements to approximate displacement and stress, respectively. Moreover, on arbitrary regular quadrilaterals, we propose new variants of both the rotated ${\mathcal Q}_1$ nonconforming element and the lowest-order Raviart-Thomas element.


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

  • 1. Z. Cai, R. Lazarov, T.A. Manteuffel and S.F. McCormick, First order system least-squares for second order partial differential equations: Part 1, SIAM J.Numer.Anal., 31(1994), pp.1785-1799. MR 95i:65133
  • 2. Z. Cai, T.A. Manteuffel and S.F. McCormick, First order system least-squares for second order partial differential equations: Part 2, SIAM J.Numer.Anal., 34(1997), pp.425-454. MR 98m:65039
  • 3. A.I. Pehlivanov, G.F. Carey and R.D. Lazarov, Least-squares mixed finite element methods for second order elliptic problems, SIAM J.Numer.Anal., 31(1994),pp.1368-1374. MR 95f:65206
  • 4. P.B. Bochev and M.D. Gunzburger, Finite element methods of least-squares type, SIAM Rev., 40(91998), pp.789-834. MR 99k:65104
  • 5. P.A. Raviart and J.M. Thomas, A mixed finite element method for second order elliptic problems, in: Mathematical Aspects of The Finite Element Method, Lecture Notes in Mathematics 606, Springer-Verlag, Berlin and New York, 1977, pp. 292-315. MR 58:3547
  • 6. F. Brezzi, J. Douglas Jr., M. Fortin and L.D. Marini, Efficient rectangular mixed finite elements in tw and three space variables, M2AN Math.Model. Anal.Numer., 21(1987), pp.581-604. MR 88j:65249
  • 7. F. Brezzi, J. Douglas Jr. and L.D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer.Math., 47(1985), pp.217-235. MR 87g:65133
  • 8. J.C. Nédélec, Mixed finite elements in $\Re^3$, Numer.Math.,22(1985), pp.315-341. MR 81k:65125
  • 9. -, A new family of mixed finite elements in $\Re^3$, Numer.Math., 50(1986), pp.57-81. MR 88e:65145
  • 10. T. Arbogast and Zhang-xin Chen, On the implementation of mixed methods as nonconforming methods for second order elliptic problems, Math.Comput., 64(1995), pp.943-974.
  • 11. M. Crouzeix and P.A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations, RAIRO Anal.Numer., 7(1973), pp.33-74.MR 95k:65102
  • 12. Han Hou-de, Nonconforming elements in the mixed finite element method, J. Comput. Math. 2(1984), pp.223-234. MR 87d:65130
  • 13. J. Douglas, J. J. Santos, D. Sheen and X. Ye, Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems, RAIRO, Math. Model and Numer. Anal, 33(1999), 747-770. MR 2000k:65206
  • 14. L.E. Payne and H.F. Weinberger, An optimal Poincaré inequality for convex domains, Archive for Rational Mechanics and Analysis, 5(1960), pp.286-294. MR 22:8198
  • 15. P. Kloucek and F. R. Toffoletto, The three dimensional nonconforming finite element solution of the Chapman-Ferraro problem, J. Comp. Phy. 150(1999), pp. 549-560. MR 99m:86009
  • 16. P. Kloucek, Bo Li and M. Luskin, Analysis of a class of nonconforming finite elements for crystalline microstructures, Math. Comput., 65(1996), 1111-1135. MR 97a:73076
  • 17. Bo Li and M. Luskin, Nonconforming finite element approximation of crystalline microstructures, Math. Comput. 67(1998), 917-944. MR 98j:73092
  • 18. R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element, Numerical Methods for Partial Differential Equations, 8(1992),pp.97-111. MR 92i:65170
  • 19. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag.(1991) MR 92d:65187
  • 20. P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam. (1978) MR 58:25001
  • 21. P. Clément, Approximation by finite elements using local regularization, RAIRO Anal.Numer. 8(1975),pp.77-84. MR 53:4569
  • 22. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Texts Appl.Math.15, Springer-Verlag, New York, 1994. MR 95f:65001
  • 23. B. Lee, First order system least-squares for elliptic problems with Robin boundary conditions, SIAM J.Numer.Anal., 37(1999),pp.70-104. MR 2001b:65126
  • 24. G. Starke, Multilevel boundary functionals for least-squares mixed finite element methods, SIAM J.Numer.Anal., 36(1999),pp.1065-1074. MR 2000d:65220
  • 25. J. Wang and T. Mathew, Mixed finite element methods over quadrilaterals, in: Proceedings of 3rd Internat.Conference on Advances in Numerical Methods and Applications, J.T.Dimov, B.Sendov, and P.Vassilevski, eds, World Scientific, River Edge, NJ, 1994,pp.203-214. MR 2001f:65001
  • 26. J.H. Bramble, R.D. Lazarov and J.E. Pasciak, Least-squares for second order elliptic problems, Comput.Methods Appl.Mech.Engrg., 152(1998),pp.195-210. MR 99a:65145
  • 27. D.N. Arnold and R.S. Falk, A uniformly accurate finite element method for the Reissner-Mindlin plate, SIAM J. Numer. Anal., 26(1989), pp.1276-1290. MR 91c:65068
  • 28. V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlarg, Berlin, New York, 1986. MR 88b:65129
  • 29. G. Strang and G.J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973. MR 56:1747
  • 30. F. Stummel, The generalized patch test, SIAM J.Numer.Anal., 16(1979), pp.449-471. MR 80e:65106
  • 31. A.I. Pehlivanov, G.F. Carey and P.S. Vassilevski, Least-squares mixed finite element methods for nonselfadjoint elliptic problems: I. Error estimates, Numer. Math., 72(1996), pp.501-522. MR 97f:65068
  • 32. D.N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: Implemenetation, postprocessing and error estimates, M2AN Math.Modelling Anal.Numer., 19(1985), pp.7-32. MR 87g:65126
  • 33. R. Stenberg, Postprocessing schemes for some mixed finite elements, M2AN Math.Modelling Anal.Numer., 25(1991), pp. 151-167. MR 92a:65303
  • 34. R.A. Adams, Soblev Spaces, Academic Press, New York, 1975. MR 56:9247
  • 35. J. Douglas Jr. and J.E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comput., 44(1985), pp. 39-52. MR 86b:65122
  • 36. M. Crouzeix and R.S. Falk, Nonconforming finite elements for the Stokes problem, Math. Comput., 52(1989), pp. 437-456. MR 89i:65113
  • 37. D. Boffi, Fortin operator and discrete compactness for edge elements, Numer.Math., 87(2000), pp. 229-246. MR 2001k:65168
  • 38. Huoyuan Duan, Studies On Mixed Finite Element Methods, Ph.D. thesis, Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, P.R. China, March, 2002.

Similar Articles

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

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


Additional Information

Huo-Yuan Duan
Affiliation: Institute of Mathematics, Chinese Academy of Sciences, Beijing 100080, Peoples Republic of China
Email: dhymath@yahoo.com.cn

Guo-Ping Liang
Affiliation: Institute of Mathematics, Chinese Academy of Sciences, Beijing 100080, Peoples Republic of China
Email: lin@fegen.com

DOI: https://doi.org/10.1090/S0025-5718-03-01520-5
Keywords: Second-order elliptic problem, least-squares mixed finite element method, nonconforming element, normal continuous element
Received by editor(s): May 29, 2001
Received by editor(s) in revised form: May 7, 2002
Published electronically: March 27, 2003
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society