Anisotropic h p -adaptive method based on interpolation error estimates in the H 1 -seminorm

Vít Dolejší

Applications of Mathematics (2015)

  • Volume: 60, Issue: 6, page 597-616
  • ISSN: 0862-7940

Abstract

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We develop a new technique which, for the given smooth function, generates the anisotropic triangular grid and the corresponding polynomial approximation degrees based on the minimization of the interpolation error in the broken H 1 -seminorm. This technique can be employed for the numerical solution of boundary value problems with the aid of finite element methods. We present the theoretical background of this approach and show several numerical examples demonstrating the efficiency of the proposed anisotropic adaptive strategy in comparison with other adaptive approaches.

How to cite

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Dolejší, Vít. "Anisotropic $hp$-adaptive method based on interpolation error estimates in the $H^1$-seminorm." Applications of Mathematics 60.6 (2015): 597-616. <http://eudml.org/doc/271815>.

@article{Dolejší2015,
abstract = {We develop a new technique which, for the given smooth function, generates the anisotropic triangular grid and the corresponding polynomial approximation degrees based on the minimization of the interpolation error in the broken $H^1$-seminorm. This technique can be employed for the numerical solution of boundary value problems with the aid of finite element methods. We present the theoretical background of this approach and show several numerical examples demonstrating the efficiency of the proposed anisotropic adaptive strategy in comparison with other adaptive approaches.},
author = {Dolejší, Vít},
journal = {Applications of Mathematics},
keywords = {$hp$-methods; anisotropic mesh adaptation; interpolation error estimates; -methods; anisotropic mesh adaptation; interpolation error estimates},
language = {eng},
number = {6},
pages = {597-616},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Anisotropic $hp$-adaptive method based on interpolation error estimates in the $H^1$-seminorm},
url = {http://eudml.org/doc/271815},
volume = {60},
year = {2015},
}

TY - JOUR
AU - Dolejší, Vít
TI - Anisotropic $hp$-adaptive method based on interpolation error estimates in the $H^1$-seminorm
JO - Applications of Mathematics
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 6
SP - 597
EP - 616
AB - We develop a new technique which, for the given smooth function, generates the anisotropic triangular grid and the corresponding polynomial approximation degrees based on the minimization of the interpolation error in the broken $H^1$-seminorm. This technique can be employed for the numerical solution of boundary value problems with the aid of finite element methods. We present the theoretical background of this approach and show several numerical examples demonstrating the efficiency of the proposed anisotropic adaptive strategy in comparison with other adaptive approaches.
LA - eng
KW - $hp$-methods; anisotropic mesh adaptation; interpolation error estimates; -methods; anisotropic mesh adaptation; interpolation error estimates
UR - http://eudml.org/doc/271815
ER -

References

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  1. Ait-Ali-Yahia, D., Baruzzi, G., Habashi, W. G., Fortin, M., Dompierre, J., Vallet, M.-G., 10.1002/fld.356, Int. J. Numer. Methods Fluids 39 (2002), 657-673. (2002) Zbl1101.76350MR1911881DOI10.1002/fld.356
  2. Aubry, R., Löhner, R., 10.1002/nme.2446, Int. J. Numer. Methods Eng. 77 (2009), 1247-1289. (2009) Zbl1156.76432DOI10.1002/nme.2446
  3. Babuška, I., Suri, M., 10.1137/1036141, SIAM Rev. 36 (1994), 578-632. (1994) Zbl0813.65118MR1306924DOI10.1137/1036141
  4. Cao, W., 10.1137/050634700, SIAM J. Sci. Comput. 29 (2007), 756-781. (2007) Zbl1136.65100MR2306267DOI10.1137/050634700
  5. Cao, W., 10.1090/S0025-5718-07-01981-3, Math. Comput. 77 (2008), 265-286. (2008) Zbl1149.65010MR2353953DOI10.1090/S0025-5718-07-01981-3
  6. Clavero, C., Gracia, J. L., Jorge, J. C., 10.1093/imanum/dri029, IMA J. Numer. Anal. 26 (2006), 155-172. (2006) MR2193974DOI10.1093/imanum/dri029
  7. Dawson, C., Sun, S., Wheeler, M. F., 10.1016/j.cma.2003.12.059, Comput. Methods Appl. Mech. Eng. 193 (2004), 2565-2580. (2004) Zbl1067.76565MR2055253DOI10.1016/j.cma.2003.12.059
  8. Demkowicz, L., Rachowicz, W., Devloo, P., 10.1023/A:1015192312705, J. Sci. Comput. 17 (2002), 117-142. (2002) Zbl0999.65121MR1910555DOI10.1023/A:1015192312705
  9. Dolejší, V., 10.1007/s007910050015, Comput. Vis. Sci. 1 (1998), 165-178. (1998) Zbl0917.68214MR1839196DOI10.1007/s007910050015
  10. Dolejší, V., ANGENER---software package, Charles University Prague, Faculty of Mathematics and Physics, 2000. www.karlin.mff.cuni.cz/ {dolejsi/angen/angen.htm}. 
  11. Dolejší, V., 10.1016/j.cam.2007.10.055, J. Comput. Appl. Math. 222 (2008), 251-273. (2008) Zbl1165.65055MR2474628DOI10.1016/j.cam.2007.10.055
  12. Dolejší, V., 10.1016/j.matcom.2013.03.001, Math. Comput. Simul. 87 (2013), 87-118. (2013) MR3046879DOI10.1016/j.matcom.2013.03.001
  13. Dolejší, V., 10.1016/j.apnum.2014.03.003, Appl. Numer. Math. 82 (2014), 80-114. (2014) Zbl1291.65340MR3212381DOI10.1016/j.apnum.2014.03.003
  14. Dolejší, V., Felcman, J., 10.1002/num.10104, Numer. Methods Partial Differ. Equations 20 (2004), 576-608. (2004) Zbl1060.65125MR2060780DOI10.1002/num.10104
  15. Dolejší, V., Roos, H.-G., BDF-FEM for parabolic singularly perturbed problems with exponential layers on layers-adapted meshes in space, Neural Parallel Sci. Comput. 18 (2010), 221-235. (2010) MR2722205
  16. Frey, P. J., Alauzet, F., 10.1016/j.cma.2004.11.025, Comput. Methods Appl. Mech. Eng. 194 (2005), 5068-5082. (2005) Zbl1092.76054MR2163554DOI10.1016/j.cma.2004.11.025
  17. John, V., Knobloch, P., 10.1016/j.cma.2006.11.013, Comput. Methods Appl. Mech. Eng. 196 (2007), 2197-2215. (2007) Zbl1173.76342MR2302890DOI10.1016/j.cma.2006.11.013
  18. Knopp, T., Lube, G., Rapin, G., 10.1016/S0045-7825(02)00222-0, Comput. Methods Appl. Mech. Eng. 191 (2002), 2997-3013. (2002) Zbl1001.76058MR1903196DOI10.1016/S0045-7825(02)00222-0
  19. Laug, P., Borouchaki, H., BL2D-V2: isotropic or anisotropic 2D mesher. INRIA, 2002. https://www.rocq.inria.fr/gamma/Patrick.Laug/logiciels/bl2d-v2/INDEX.html, . 
  20. Loseille, A., Alauzet, F., 10.1137/090754078, SIAM J. Numer. Anal. 49 (2011), 38-60. (2011) MR2764420DOI10.1137/090754078
  21. Loseille, A., Alauzet, F., 10.1137/10078654X, SIAM J. Numer. Anal. 49 (2011), 61-86. (2011) MR2764421DOI10.1137/10078654X
  22. Mirebeau, J.-M., 10.1007/s00365-010-9090-y, Constr. Approx. 32 (2010), 339-383. (2010) Zbl1202.65015MR2677884DOI10.1007/s00365-010-9090-y
  23. Mirebeau, J.-M., 10.1007/s00211-011-0412-1, Numer. Math. 120 (2012), 271-305. (2012) Zbl1238.65116MR2874967DOI10.1007/s00211-011-0412-1
  24. Schwab, C., p - and h p -Finite Element Methods. Theory and Applications in Solid and Fluid Mechanics, Numerical Mathematics and Scientific Computation Clarendon Press, Oxford (1998). (1998) Zbl0910.73003MR1695813
  25. Šolín, P., Partial Differential Equations and the Finite Element Method, Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs, and Tracts John Wiley & Sons, Hoboken (2006). (2006) MR2180081
  26. Šolín, P., Demkowicz, L., 10.1016/j.cma.2003.09.015, Comput. Methods Appl. Mech. Eng. 193 (2004), 449-468. (2004) Zbl1044.65082MR2033961DOI10.1016/j.cma.2003.09.015
  27. Sun, S., Discontinuous Galerkin methods for reactive transport in porous media, Ph.D. thesis, The University of Texas, Austin (2003). (2003) MR2705499
  28. Vejchodský, T., Šolín, P., Zítka, M., 10.1016/j.matcom.2007.02.001, Math. Comput. Simul. 76 (2007), 223-228. (2007) Zbl1157.78356MR2392482DOI10.1016/j.matcom.2007.02.001
  29. Zienkiewicz, O. C., Wu, J., 10.1002/nme.1620371304, Int. J. Numer. Methods Eng. 37 (1994), 2189-2210. (1994) Zbl0810.76045MR1285070DOI10.1002/nme.1620371304

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