Numerical study of natural superconvergence in least-squares finite element methods for elliptic problems

Runchang Lin; Zhimin Zhang

Applications of Mathematics (2009)

  • Volume: 54, Issue: 3, page 251-266
  • ISSN: 0862-7940

Abstract

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Natural superconvergence of the least-squares finite element method is surveyed for the one- and two-dimensional Poisson equation. For two-dimensional problems, both the families of Lagrange elements and Raviart-Thomas elements have been considered on uniform triangular and rectangular meshes. Numerical experiments reveal that many superconvergence properties of the standard Galerkin method are preserved by the least-squares finite element method.

How to cite

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Lin, Runchang, and Zhang, Zhimin. "Numerical study of natural superconvergence in least-squares finite element methods for elliptic problems." Applications of Mathematics 54.3 (2009): 251-266. <http://eudml.org/doc/37819>.

@article{Lin2009,
abstract = {Natural superconvergence of the least-squares finite element method is surveyed for the one- and two-dimensional Poisson equation. For two-dimensional problems, both the families of Lagrange elements and Raviart-Thomas elements have been considered on uniform triangular and rectangular meshes. Numerical experiments reveal that many superconvergence properties of the standard Galerkin method are preserved by the least-squares finite element method.},
author = {Lin, Runchang, Zhang, Zhimin},
journal = {Applications of Mathematics},
keywords = {least-squares finite element method; mixed finite element method; natural superconvergence; Raviart-Thomas element; Poisson equation; Lagrange elements; triangular and rectangular meshes; numerical experiments; Galerkin method; natural superconvergence; Raviart-Thomas element; least-squares finite element method; Poisson equation; Lagrange elements; triangular and rectangular meshes; numerical experiments; Galerkin method},
language = {eng},
number = {3},
pages = {251-266},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Numerical study of natural superconvergence in least-squares finite element methods for elliptic problems},
url = {http://eudml.org/doc/37819},
volume = {54},
year = {2009},
}

TY - JOUR
AU - Lin, Runchang
AU - Zhang, Zhimin
TI - Numerical study of natural superconvergence in least-squares finite element methods for elliptic problems
JO - Applications of Mathematics
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 3
SP - 251
EP - 266
AB - Natural superconvergence of the least-squares finite element method is surveyed for the one- and two-dimensional Poisson equation. For two-dimensional problems, both the families of Lagrange elements and Raviart-Thomas elements have been considered on uniform triangular and rectangular meshes. Numerical experiments reveal that many superconvergence properties of the standard Galerkin method are preserved by the least-squares finite element method.
LA - eng
KW - least-squares finite element method; mixed finite element method; natural superconvergence; Raviart-Thomas element; Poisson equation; Lagrange elements; triangular and rectangular meshes; numerical experiments; Galerkin method; natural superconvergence; Raviart-Thomas element; least-squares finite element method; Poisson equation; Lagrange elements; triangular and rectangular meshes; numerical experiments; Galerkin method
UR - http://eudml.org/doc/37819
ER -

References

top
  1. Ainsworth, M., Oden, J. T., A Posteriori Error Estimation in Finite Element Analysis. Pure and Applied Mathematics, Wiley Interscience, John Wiley & Sons New York (2000). (2000) MR1885308
  2. Babuška, I., 10.1007/BF02165003, Numer. Math. 16 (1971), 322-333. (1971) MR0288971DOI10.1007/BF02165003
  3. Babuška, I., Strouboulis, T., The Finite Element Method and its Reliability, Clarendon Press Oxford (2001). (2001) MR1857191
  4. Babuška, I., Strouboulis, T., Upadhyay, C. S., Gangaraj, S. K., 10.1002/num.1690120303, Numer. Methods Partial Differ. Equations 12 (1996), 347-392. (1996) MR1388445DOI10.1002/num.1690120303
  5. Bedivan, D. M., 10.1016/S0898-1221(02)80009-8, Comput. Math. Appl. 43 (2002), 1003-1020. (2002) Zbl1050.65098MR1892481DOI10.1016/S0898-1221(02)80009-8
  6. Bochev, P. B., Gunzburger, M. D., 10.1137/S0036144597321156, SIAM Rev. 40 (1998), 789-837. (1998) Zbl0914.65108MR1659689DOI10.1137/S0036144597321156
  7. Brandts, J. H., 10.1007/s002110050064, Numer. Math. 68 (1994), 311-324. (1994) Zbl0823.65103MR1313147DOI10.1007/s002110050064
  8. Brandts, J. H., 10.1016/S0168-9274(99)00034-3, Appl. Numer. Math. 34 (2000), 39-58. (2000) Zbl0948.65120MR1755693DOI10.1016/S0168-9274(99)00034-3
  9. Brandts, J. H., Chen, Y. P., Superconvergence of least-squares mixed finite element methods, Int. J. Numer. Anal. Model. 3 (2006), 303-311. (2006) MR2237884
  10. Brandts, J. H., Chen, Y. P., Yang, J., 10.1093/imanum/dri048, IMA J. Numer. Anal. 26 (2006), 779-789. (2006) Zbl1106.65102MR2269196DOI10.1093/imanum/dri048
  11. Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Franc. Automat. Inform. Rech. Operat. 8 (1974), 129-151. (1974) Zbl0338.90047MR0365287
  12. Brezzi, F., J. Douglas, Jr., Fortin, M., Marini, L. D., 10.1051/m2an/1987210405811, Mathematical Modelling and Numerical Analysis 21 (1987), 581-604. (1987) MR0921828DOI10.1051/m2an/1987210405811
  13. Brezzi, F., J. Douglas, Jr., Marini, L. D., 10.1007/BF01389710, Numer. Math. 47 (1985), 217-235. (1985) Zbl0599.65072MR0799685DOI10.1007/BF01389710
  14. Cai, Z., Ku, J., 10.1137/050636504, SIAM J. Numer. Anal. 44 (2006), 1721-1734. (2006) Zbl1138.76053MR2257124DOI10.1137/050636504
  15. Cai, Z., Lazarov, R. D., Manteuffel, T. A., McCormick, S. F., 10.1137/0731091, SIAM J. Numer. Anal. 31 (1994), 1785-1799. (1994) Zbl0813.65119MR1302685DOI10.1137/0731091
  16. Carey, G. F., Shen, Y., 10.1002/cnm.1630050702, Commun. Appl. Numer. Methods 5 (1989), 427-434. (1989) Zbl0684.65083DOI10.1002/cnm.1630050702
  17. Chen, C. M., Structure Theory of Superconvergence of Finite Elements, Hunan Science Press Hunan (2001), Chinese. (2001) MR0840307
  18. Chen, C. M., Huang, Y. Q., High Accuracy Theory of Finite Element Methods, Hunan Science and Technology Press Hunan (1995), Chinese. (1995) 
  19. Chen, Y., 10.1090/S0025-5718-08-02104-2, Math. Comput. 77 (2008), 1269-1291. (2008) Zbl1193.49029MR2398768DOI10.1090/S0025-5718-08-02104-2
  20. Chen, Z., Finite Element Methods and Their Applications. Scientific Computation, Springer Berlin (2005). (2005) MR2158541
  21. Douglas, J., Dupont, T., 10.1007/BF01436724, Numer. Math. 22 (1974), 99-109. (1974) Zbl0331.65051MR0362922DOI10.1007/BF01436724
  22. Douglas, J., Dupont, T., Wahlbin, L., Optimal L ∞ error estimates for Galerkin approximations to solutions of two-point boundary value problems, Math. Comput. 29 (1975), 475-483. (1975) MR0371077
  23. Douglas, J., Wang, J., 10.1007/BF02575724, Calcolo 26 (1989), 121-133. (1989) Zbl0714.65084MR1083049DOI10.1007/BF02575724
  24. Durán, R., 10.1007/BF01385626, Numer. Math. 58 (1990), 287-298. (1990) MR1075159DOI10.1007/BF01385626
  25. Ewing, R. E., Lazarov, R. D., Wang, J., 10.1137/0728054, SIAM J. Numer. Anal. 28 (1991), 1015-1029. (1991) Zbl0733.65065MR1111451DOI10.1137/0728054
  26. Ewing, R. E., Liu, M. M., Wang, J., 10.1137/S0036142997322801, SIAM J. Numer. Anal. 36 (1998), 772-787. (1998) Zbl0926.65107MR1681041DOI10.1137/S0036142997322801
  27. Ewing, R. E., Wang, J., 10.1007/BF01385855, Numer. Math. 63 (1992), 183-194. (1992) Zbl0772.65071MR1182973DOI10.1007/BF01385855
  28. Gastaldi, L., Nochetto, R. H., 10.1007/BF01408578, Numer. Math. 50 (1987), 587-611. (1987) MR0880337DOI10.1007/BF01408578
  29. Gastaldi, L., Nochetto, R. H., 10.1051/m2an/1989230101031, RAIRO, Modélisation Math. Anal. Numér. 23 (1989), 103-128. (1989) Zbl0673.65060MR1015921DOI10.1051/m2an/1989230101031
  30. Jiang, B.-N., The Least-Squares Finite Element Method. Theory and Applications in Computational Fluid Dynamics and Electromagnetics, Springer Berlin (1998). (1998) MR1639101
  31. Křížek, M., Neittaanmäki, P., Bibliography on superconvergence, Finite element methods. Superconvergence, post-processing, and a posteriori estimates M. Kř'ižek, P. Neittaanmäki, R. Stenberg Marcel Dekker New York (1998), 315-348. (1998) MR1602730
  32. Křížek, M., Neittaanmäki, P., Finite Element Approximation of Variational Problems and Applications. Pitman Monographs and Surveys in Pure and Applied Mathematics, 50, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York (1990). (1990) MR1066462
  33. Li, J., Wheeler, M. F., 10.1137/S0036142999351212, SIAM J. Numer. Anal. 38 (2000), 770-798. (2000) Zbl0974.65106MR1781203DOI10.1137/S0036142999351212
  34. Lin, Q., Lin, J., Finite Element Methods: Accuracy and Improvement, Science Press Beijing (2006). (2006) 
  35. Lin, Q., Pan, J. H., High accuracy for mixed finite element methods in Raviart-Thomas element, J. Comput. Math. 14 (1996), 175-182. (1996) Zbl0846.65062MR1399911
  36. Lin, Q., Yan, N., Construction and Analysis of High Efficient Finite Elements, Hebei University Press Hebei (1996), Chinese. (1996) 
  37. Lin, R., Zhang, Z., 10.1002/num.20013, Numer. Methods Partial Differ. Equations 20 (2004), 864-906. (2004) Zbl1068.65123MR2092411DOI10.1002/num.20013
  38. Lin, R., Zhang, Z., Convergence analysis for least-squares approximations to solutions of second-order two-point boundary value problems, Submitted. 
  39. Pehlivanov, A. I., Carey, G. F., 10.1051/m2an/1994280504991, RAIRO, Modélisation Math. Anal. Numér. 28 (1994), 499-516. (1994) Zbl0820.65065MR1295584DOI10.1051/m2an/1994280504991
  40. Pehlivanov, A. I., Carey, G. F., Lazarov, R. D., 10.1137/0731071, SIAM J. Numer. Anal. 31 (1994), 1368-1377. (1994) Zbl0806.65108MR1293520DOI10.1137/0731071
  41. Pehlivanov, A. I., Carey, G. F., Lazarov, R. D., Shen, Y., 10.1007/BF02243846, Computing 51 (1993), 111-123. (1993) Zbl0790.65079MR1248894DOI10.1007/BF02243846
  42. Raviart, P. A., Thomas, J. M., A mixed finite element method for second order elliptic problems, In: Mathematical Aspects of the Finite Element Method. Lecture Notes Math. 606 I. Galligani, E. Magenes Springer Berlin (1977), 292-315. (1977) MR0483555
  43. Verfürth, R., A Review of Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner Chichester-Stuttgart (1996). (1996) 
  44. Wahlbin, L. B., Superconvergence in Galerkin Finite Flement Methods. Lecture Notes Math. 1605, Springer Berlin (1995). (1995) MR1439050
  45. Wheeler, M. F., 10.1137/0710077, SIAM J. Numer. Anal. 10 (1973), 914-917. (1973) Zbl0266.65061MR0343659DOI10.1137/0710077
  46. Yan, N., Superconvergence Analysis and a Posteriori Error Estimation in Finite Element Methods, Science Press Beijing (2008). (2008) 
  47. Zhang, Z., 10.1090/S0025-5718-98-00942-9, Math. Comput. 67 (1998), 541-552. (1998) MR1459393DOI10.1090/S0025-5718-98-00942-9
  48. Zhang, Z., Recovery techniques in finite element methods, In: Adaptive Computations: Theory and Algorithms T. Tang, J. Xu Science Publisher (2007), 297-365. (2007) 
  49. Zhu, Q., High Accuracy and Post-Processing Theory of the Finite Element Method, Science Press Beijing (2008), Chinese. (2008) 
  50. Zienkiewicz, O. C., Taylor, R. L., Zhu, J. Z., The Finite Element Method, 6th ed, McGraw-Hill London (2005). (2005) MR3292660

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