Analysis of a non-standard mixed finite element method with applications to superconvergence

Jan Brandts

Applications of Mathematics (2009)

  • Volume: 54, Issue: 3, page 225-235
  • ISSN: 0862-7940

Abstract

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We show that a non-standard mixed finite element method proposed by Barrios and Gatica in 2007, is a higher order perturbation of the least-squares mixed finite element method. Therefore, it is also superconvergent whenever the least-squares mixed finite element method is superconvergent. Superconvergence of the latter was earlier investigated by Brandts, Chen and Yang between 2004 and 2006. Since the new method leads to a non-symmetric system matrix, its application seems however more expensive than applying the least-squares mixed finite element method.

How to cite

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Brandts, Jan. "Analysis of a non-standard mixed finite element method with applications to superconvergence." Applications of Mathematics 54.3 (2009): 225-235. <http://eudml.org/doc/37817>.

@article{Brandts2009,
abstract = {We show that a non-standard mixed finite element method proposed by Barrios and Gatica in 2007, is a higher order perturbation of the least-squares mixed finite element method. Therefore, it is also superconvergent whenever the least-squares mixed finite element method is superconvergent. Superconvergence of the latter was earlier investigated by Brandts, Chen and Yang between 2004 and 2006. Since the new method leads to a non-symmetric system matrix, its application seems however more expensive than applying the least-squares mixed finite element method.},
author = {Brandts, Jan},
journal = {Applications of Mathematics},
keywords = {least-squares mixed finite element method; non-standard mixed finite element method; superconvergence; supercloseness; least-squares mixed finite element method; non-standard mixed finite element method; supercloseness},
language = {eng},
number = {3},
pages = {225-235},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Analysis of a non-standard mixed finite element method with applications to superconvergence},
url = {http://eudml.org/doc/37817},
volume = {54},
year = {2009},
}

TY - JOUR
AU - Brandts, Jan
TI - Analysis of a non-standard mixed finite element method with applications to superconvergence
JO - Applications of Mathematics
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 3
SP - 225
EP - 235
AB - We show that a non-standard mixed finite element method proposed by Barrios and Gatica in 2007, is a higher order perturbation of the least-squares mixed finite element method. Therefore, it is also superconvergent whenever the least-squares mixed finite element method is superconvergent. Superconvergence of the latter was earlier investigated by Brandts, Chen and Yang between 2004 and 2006. Since the new method leads to a non-symmetric system matrix, its application seems however more expensive than applying the least-squares mixed finite element method.
LA - eng
KW - least-squares mixed finite element method; non-standard mixed finite element method; superconvergence; supercloseness; least-squares mixed finite element method; non-standard mixed finite element method; supercloseness
UR - http://eudml.org/doc/37817
ER -

References

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  1. Barrios, T. P., Gatica, G. N., 10.1016/j.cam.2006.01.017, J. Comput. Appl. Math. 200 (2007), 653-676. (2007) Zbl1112.65106MR2289241DOI10.1016/j.cam.2006.01.017
  2. Bochev, P. B., Gunzburger, M. D., 10.1137/S0036144597321156, SIAM Rev. 40 (1998), 789-837. (1998) Zbl0914.65108MR1659689DOI10.1137/S0036144597321156
  3. Brandts, J. H., 10.1007/s002110050064, Numer. Math. 68 (1994), 311-324. (1994) Zbl0823.65103MR1313147DOI10.1007/s002110050064
  4. Brandts, J. H., Křížek, M., 10.1093/imanum/23.3.489, IMA J. Numer. Anal. 23 (2003), 489-505. (2003) Zbl1042.65081MR1987941DOI10.1093/imanum/23.3.489
  5. Brandts, J. H., Křížek, M., Superconvergence of tetrahedral quadratic finite elements, J. Comput. Math. 23 (2005), 27-36. (2005) Zbl1072.65137MR2124141
  6. Brandts, J. H., Chen, Y. P., 10.1007/978-3-642-18775-9_14, In: Numerical Mathematics and Advanced Applications M. Feistauer, V. Dolejší, P. Knobloch, K. Najzar Springer (2004), 169-175. (2004) Zbl1056.65110MR2121365DOI10.1007/978-3-642-18775-9_14
  7. Brandts, J. H., Chen, Y. P., Superconvergence of least-squares mixed finite elements, Int. J. Numer. Anal. Model. 3 (2006), 303-310. (2006) Zbl1096.65108MR2237884
  8. 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
  9. Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Franc. Automat. Inform. Rech. Operat. 8, R-2 (1974), 129-151. (1974) Zbl0338.90047MR0365287
  10. Brezzi, F., Fortin, M., Mixed and hybrid finite element methods, Springer Berlin-Heidelberg-New York (1991). (1991) Zbl0788.73002MR1115205
  11. Cai, Z., Lazarov, R., Manteuffel, T. A., McCormick, S. F., 10.1137/0731091, SIAM J. Numer. Anal. 31 (1994), 1785-1799. (1994) MR1302685DOI10.1137/0731091
  12. Carey, G. F., Pehlivanov, A. I., 10.1016/S0045-7825(97)00098-4, Comput. Methods Appl. Mech. Eng. 150 (1997), 125-131. (1997) Zbl0907.65101MR1487940DOI10.1016/S0045-7825(97)00098-4
  13. Carey, G. F., Pehlivanov, A. I., Shen, Y., Bose, A., Wang, K. C., 10.1002/(SICI)1097-0363(199801)27:1/4<97::AID-FLD652>3.0.CO;2-2, Int. J. Numer. Methods Fluids 27 (1998), 97-107. (1998) Zbl0904.76043MR1602155DOI10.1002/(SICI)1097-0363(199801)27:1/4<97::AID-FLD652>3.0.CO;2-2
  14. Ciarlet, P., The Finite Element Methods for Elliptic Problems. Classics in Applied Mathematics 40. 2nd Edition, SIAM Philadelphia (2002). (2002) MR1930132
  15. Křížek, M., Neittaanmäki, P., Stenberg, R., Finite element methods: superconvergence, post-processing and a posteriori estimates. Proc. Conf. Univ. of Jyväskylä, 1996. Lect. Notes Pure Appl. Math., 96, Marcel Dekker New York (1998). (1998) MR1602809
  16. 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
  17. 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
  18. Pehlivanov, A. I., Carey, G. F., Vassilevski, P. S., 10.1007/s002110050179, Numer. Math. 72 (1996), 501-522. (1996) Zbl0878.65096MR1376110DOI10.1007/s002110050179
  19. Raviart, P. A., Thomas, J. M., 10.1007/BFb0064470, Lect. Notes Math. 606 (1977), 292-315. (1977) Zbl0362.65089MR0483555DOI10.1007/BFb0064470
  20. Wahlbin, L. B., Superconvergence in Galerkin Finite Element Methods. Lect. Notes Math. 1605, Springer Berlin (1995). (1995) MR1439050

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