Nonconforming P1 elements on distorted triangulations: Lower bounds for the discrete energy norm error

Peter Oswald

Applications of Mathematics (2017)

  • Volume: 62, Issue: 5, page 433-457
  • ISSN: 0862-7940

Abstract

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Compared to conforming P1 finite elements, nonconforming P1 finite element discretizations are thought to be less sensitive to the appearance of distorted triangulations. E.g., optimal-order discrete H 1 norm best approximation error estimates for H 2 functions hold for arbitrary triangulations. However, the constants in similar estimates for the error of the Galerkin projection for second-order elliptic problems show a dependence on the maximum angle of all triangles in the triangulation. We demonstrate on an example of a special family of distorted triangulations that this dependence is essential, and due to the deterioration of the consistency error. We also provide examples of sequences of triangulations such that the nonconforming P1 Galerkin projections for a Poisson problem with polynomial solution do not converge or converge at arbitrarily low speed. The results complement analogous findings for conforming P1 finite elements.

How to cite

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Oswald, Peter. "Nonconforming P1 elements on distorted triangulations: Lower bounds for the discrete energy norm error." Applications of Mathematics 62.5 (2017): 433-457. <http://eudml.org/doc/294094>.

@article{Oswald2017,
abstract = {Compared to conforming P1 finite elements, nonconforming P1 finite element discretizations are thought to be less sensitive to the appearance of distorted triangulations. E.g., optimal-order discrete $H^1$ norm best approximation error estimates for $H^2$ functions hold for arbitrary triangulations. However, the constants in similar estimates for the error of the Galerkin projection for second-order elliptic problems show a dependence on the maximum angle of all triangles in the triangulation. We demonstrate on an example of a special family of distorted triangulations that this dependence is essential, and due to the deterioration of the consistency error. We also provide examples of sequences of triangulations such that the nonconforming P1 Galerkin projections for a Poisson problem with polynomial solution do not converge or converge at arbitrarily low speed. The results complement analogous findings for conforming P1 finite elements.},
author = {Oswald, Peter},
journal = {Applications of Mathematics},
keywords = {nonconforming P1 element; lowest order Raviart-Thomas element; discrete energy norm estimate; divergence of finite element method; maximum angle condition; distorted triangulation},
language = {eng},
number = {5},
pages = {433-457},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Nonconforming P1 elements on distorted triangulations: Lower bounds for the discrete energy norm error},
url = {http://eudml.org/doc/294094},
volume = {62},
year = {2017},
}

TY - JOUR
AU - Oswald, Peter
TI - Nonconforming P1 elements on distorted triangulations: Lower bounds for the discrete energy norm error
JO - Applications of Mathematics
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 5
SP - 433
EP - 457
AB - Compared to conforming P1 finite elements, nonconforming P1 finite element discretizations are thought to be less sensitive to the appearance of distorted triangulations. E.g., optimal-order discrete $H^1$ norm best approximation error estimates for $H^2$ functions hold for arbitrary triangulations. However, the constants in similar estimates for the error of the Galerkin projection for second-order elliptic problems show a dependence on the maximum angle of all triangles in the triangulation. We demonstrate on an example of a special family of distorted triangulations that this dependence is essential, and due to the deterioration of the consistency error. We also provide examples of sequences of triangulations such that the nonconforming P1 Galerkin projections for a Poisson problem with polynomial solution do not converge or converge at arbitrarily low speed. The results complement analogous findings for conforming P1 finite elements.
LA - eng
KW - nonconforming P1 element; lowest order Raviart-Thomas element; discrete energy norm estimate; divergence of finite element method; maximum angle condition; distorted triangulation
UR - http://eudml.org/doc/294094
ER -

References

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  1. Acosta, G., Durán, R. G., 10.1137/S0036142997331293, SIAM J. Numer. Anal. 37 (1999), 18-36. (1999) Zbl0948.65115MR1721268DOI10.1137/S0036142997331293
  2. Babuška, I., Aziz, A. K., 10.1137/0713021, SIAM J. Numer. Anal. 13 (1976), 214-226. (1976) Zbl0324.65046MR0455462DOI10.1137/0713021
  3. Braess, D., 10.1017/CBO9780511618635, Cambridge University Press, Cambridge (2007). (2007) Zbl1118.65117MR2322235DOI10.1017/CBO9780511618635
  4. Braess, D., 10.1007/s10092-009-0003-z, Calcolo 46 (2009), 149-155. (2009) Zbl1192.65142MR2520373DOI10.1007/s10092-009-0003-z
  5. Brenner, S. C., 10.1137/S0036142902401311, SIAM J. Numer. Anal. 41 (2003), 306-324. (2003) Zbl1045.65100MR1974504DOI10.1137/S0036142902401311
  6. Brenner, S. C., Scott, L. R., 10.1007/978-0-387-75934-0, Texts in Applied Mathematics 15, Springer, New York (2008). (2008) Zbl0804.65101MR2373954DOI10.1007/978-0-387-75934-0
  7. Carstensen, C., Gedicke, J., Rim, D., 10.4208/jcm.1108-m3677, J. Comput. Math. 30 (2012), 337-353. (2012) Zbl1274.65290MR2965987DOI10.4208/jcm.1108-m3677
  8. Carstensen, C., Peterseim, D., Schedensack, M., 10.1137/110845707, SIAM J. Numer. Anal. 50 (2012), 2803-2823. (2012) Zbl1261.65115MR3022243DOI10.1137/110845707
  9. Crouzeix, M., Raviart, P.-A., 10.1051/m2an/197307R300331, Rev. Franc. Automat. Inform. Rech. Operat. 7 (1973), 33-76. (1973) Zbl0302.65087MR0343661DOI10.1051/m2an/197307R300331
  10. Hannukainen, A., Korotov, S., Křížek, M., 10.1007/s00211-011-0403-2, Numer. Math. 120 (2012), 79-88. (2012) Zbl1255.65196MR2885598DOI10.1007/s00211-011-0403-2
  11. Jamet, P., 10.1051/m2an/197610r100431, Rev. Franc. Automat. Inform. Rech. Operat. 10, Analyse Numer. 10 (1976), 43-60. (1976) Zbl0346.65052MR0455282DOI10.1051/m2an/197610r100431
  12. Kučera, V., On necessary and sufficient conditions for finite element convergence, arXiv:1601.02942 (2016). (2016) MR3700195
  13. Marini, L. D., 10.1137/0722029, SIAM J. Numer. Anal. 22 (1985), 493-496. (1985) Zbl0573.65082MR0787572DOI10.1137/0722029
  14. Oswald, P., 10.1007/s10492-015-0107-5, Appl. Math., Praha 60 (2015), 473-484. (2015) Zbl1363.65202MR3396476DOI10.1007/s10492-015-0107-5
  15. Raviart, P.-A., Thomas, J. M., 10.1007/bfb0064470, Mathematical Aspects of Finite Element Method I. Galligani, E. Magenes Proc. Conf., Rome, 1975, Lect. Notes Math. 606, Springer, New York (1977), 292-315. (1977) Zbl0362.65089MR0483555DOI10.1007/bfb0064470
  16. Schwarz, H. A., Sur une définition erroneé de l'aire d'une surface courbe, Gesammelte Mathematische Abhandlungen 2 Springer, Berlin (1890), 309-311, 369-370. (1890) 
  17. Vohralík, M., 10.1080/01630560500444533, Numer. Funct. Anal. Optimization 26 (2005), 925-952. (2005) Zbl1089.65124MR2192029DOI10.1080/01630560500444533

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