On conditioning of Schur complements of H-TFETI clusters for 2D problems governed by Laplacian

Petr Vodstrčil; Jiří Bouchala; Marta Jarošová; Zdeněk Dostál

Applications of Mathematics (2017)

  • Volume: 62, Issue: 6, page 699-718
  • ISSN: 0862-7940

Abstract

top
Bounds on the spectrum of the Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients in the analysis of many domain decomposition methods. Here we are interested in the analysis of floating clusters, i.e. subdomains without prescribed Dirichlet conditions that are decomposed into still smaller subdomains glued on primal level in some nodes and/or by some averages. We give the estimates of the regular condition number of the Schur complements of the clusters arising in the discretization of problems governed by 2D Laplacian. The estimates depend on the decomposition and discretization parameters and gluing conditions. We also show how to plug the results into the analysis of H-TFETI methods and compare the estimates with numerical experiments. The results are useful for the analysis and implementation of powerful massively parallel scalable algorithms for the solution of variational inequalities.

How to cite

top

Vodstrčil, Petr, et al. "On conditioning of Schur complements of H-TFETI clusters for 2D problems governed by Laplacian." Applications of Mathematics 62.6 (2017): 699-718. <http://eudml.org/doc/294689>.

@article{Vodstrčil2017,
abstract = {Bounds on the spectrum of the Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients in the analysis of many domain decomposition methods. Here we are interested in the analysis of floating clusters, i.e. subdomains without prescribed Dirichlet conditions that are decomposed into still smaller subdomains glued on primal level in some nodes and/or by some averages. We give the estimates of the regular condition number of the Schur complements of the clusters arising in the discretization of problems governed by 2D Laplacian. The estimates depend on the decomposition and discretization parameters and gluing conditions. We also show how to plug the results into the analysis of H-TFETI methods and compare the estimates with numerical experiments. The results are useful for the analysis and implementation of powerful massively parallel scalable algorithms for the solution of variational inequalities.},
author = {Vodstrčil, Petr, Bouchala, Jiří, Jarošová, Marta, Dostál, Zdeněk},
journal = {Applications of Mathematics},
keywords = {two-level domain decomposition; hybrid FETI; Schur complement; bounds on the spectrum},
language = {eng},
number = {6},
pages = {699-718},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On conditioning of Schur complements of H-TFETI clusters for 2D problems governed by Laplacian},
url = {http://eudml.org/doc/294689},
volume = {62},
year = {2017},
}

TY - JOUR
AU - Vodstrčil, Petr
AU - Bouchala, Jiří
AU - Jarošová, Marta
AU - Dostál, Zdeněk
TI - On conditioning of Schur complements of H-TFETI clusters for 2D problems governed by Laplacian
JO - Applications of Mathematics
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 6
SP - 699
EP - 718
AB - Bounds on the spectrum of the Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients in the analysis of many domain decomposition methods. Here we are interested in the analysis of floating clusters, i.e. subdomains without prescribed Dirichlet conditions that are decomposed into still smaller subdomains glued on primal level in some nodes and/or by some averages. We give the estimates of the regular condition number of the Schur complements of the clusters arising in the discretization of problems governed by 2D Laplacian. The estimates depend on the decomposition and discretization parameters and gluing conditions. We also show how to plug the results into the analysis of H-TFETI methods and compare the estimates with numerical experiments. The results are useful for the analysis and implementation of powerful massively parallel scalable algorithms for the solution of variational inequalities.
LA - eng
KW - two-level domain decomposition; hybrid FETI; Schur complement; bounds on the spectrum
UR - http://eudml.org/doc/294689
ER -

References

top
  1. Brenner, S. C., 10.1007/s002110050446, Numer. Math. 83 (1999), 187-203. (1999) Zbl0936.65141MR1712684DOI10.1007/s002110050446
  2. Dostál, Z., Neto, F. A. M. Gomes, Santos, S. A., 10.1016/S0377-0427(99)00368-4, J. Comput. Appl. Math. 126 397-415 (2000). (2000) Zbl0970.65074MR1806768DOI10.1016/S0377-0427(99)00368-4
  3. Dostál, Z., Horák, D., Kučera, R., 10.1002/cnm.881, Commun. Numer. Methods Eng. 22 (2006), 1155-1162. (2006) Zbl1107.65104MR2282408DOI10.1002/cnm.881
  4. Dostál, Z., Kozubek, T., Sadowská, M., Vondrák, V., 10.1007/978-1-4939-6834-3, Advances in Mechanics and Mathematics 36, Springer, New York (2016). (2016) Zbl06658822MR3586594DOI10.1007/978-1-4939-6834-3
  5. Farhat, C., Lesoinne, M., Pierson, K., 10.1002/1099-1506(200010/12)7:7/8<687::AID-NLA219>3.0.CO;2-S, Numer. Linear Algebra Appl. 7 (2000), 687-714. (2000) Zbl1051.65119MR1802366DOI10.1002/1099-1506(200010/12)7:7/8<687::AID-NLA219>3.0.CO;2-S
  6. Farhat, C., Mandel, J., Roux, F.-X., 10.1016/0045-7825(94)90068-X, Comput. Methods Appl. Mech. Eng. 115 (1994), 365-385. (1994) MR1285024DOI10.1016/0045-7825(94)90068-X
  7. Farhat, C., Roux, F.-X., 10.1002/nme.1620320604, Int. J. Numer. Methods Eng. 32 (1991), 1205-1227. (1991) Zbl0758.65075MR3618550DOI10.1002/nme.1620320604
  8. Farhat, C., Roux, F.-X., 10.1137/0913020, SIAM J. Sci. Stat. Comput. 13 (1992), 379-396. (1992) Zbl0746.65086MR1145192DOI10.1137/0913020
  9. Klawonn, A., Rheinbach, O., 10.1002/pamm.200810841, Proc. Appl. Math. Mech. 8 (2008), 10841-10843. (2008) DOI10.1002/pamm.200810841
  10. Klawonn, A., Rheinbach, O., 10.1002/zamm.200900329, ZAMM, Z. Angew. Math. Mech. 90 (2010), 5-32. (2010) Zbl1355.65169MR2603676DOI10.1002/zamm.200900329
  11. Lee, J., 10.1137/100783753, SIAM J. Sci. Comput. 35 (2013), A1350--A1375. (2013) Zbl1276.65037MR3055241DOI10.1137/100783753
  12. Lukáš, D., Bouchala, J., Vodstrčil, P., Malý, L., 10.1007/s10492-015-0095-5, Appl. Math., Praha 60 (2015), 265-283. (2015) Zbl1363.65215MR3419962DOI10.1007/s10492-015-0095-5
  13. Pechstein, C., 10.1007/978-3-642-23588-7, Lecture Notes in Computational Science and Engineering 90, Springer, Berlin (2013). (2013) Zbl1272.65100MR3013465DOI10.1007/978-3-642-23588-7
  14. Toselli, A., Widlund, O. B., Domain Decomposition Methods---Algorithms and Theory, Springer Series on Computational Mathematics 34, Springer, Berlin (2005). (2005) Zbl1069.65138MR2104179

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.