Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms

Yalchin Efendiev; Juan Galvis; Raytcho Lazarov; Joerg Willems

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2012)

  • Volume: 46, Issue: 5, page 1175-1199
  • ISSN: 0764-583X

Abstract

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An abstract framework for constructing stable decompositions of the spaces corresponding to general symmetric positive definite problems into “local” subspaces and a global “coarse” space is developed. Particular applications of this abstract framework include practically important problems in porous media applications such as: the scalar elliptic (pressure) equation and the stream function formulation of its mixed form, Stokes’ and Brinkman’s equations. The constant in the corresponding abstract energy estimate is shown to be robust with respect to mesh parameters as well as the contrast, which is defined as the ratio of high and low values of the conductivity (or permeability). The derived stable decomposition allows to construct additive overlapping Schwarz iterative methods with condition numbers uniformly bounded with respect to the contrast and mesh parameters. The coarse spaces are obtained by patching together the eigenfunctions corresponding to the smallest eigenvalues of certain local problems. A detailed analysis of the abstract setting is provided. The proposed decomposition builds on a method of Galvis and Efendiev [Multiscale Model. Simul. 8 (2010) 1461–1483] developed for second order scalar elliptic problems with high contrast. Applications to the finite element discretizations of the second order elliptic problem in Galerkin and mixed formulation, the Stokes equations, and Brinkman’s problem are presented. A number of numerical experiments for these problems in two spatial dimensions are provided.

How to cite

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Efendiev, Yalchin, et al. "Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 46.5 (2012): 1175-1199. <http://eudml.org/doc/273336>.

@article{Efendiev2012,
abstract = {An abstract framework for constructing stable decompositions of the spaces corresponding to general symmetric positive definite problems into “local” subspaces and a global “coarse” space is developed. Particular applications of this abstract framework include practically important problems in porous media applications such as: the scalar elliptic (pressure) equation and the stream function formulation of its mixed form, Stokes’ and Brinkman’s equations. The constant in the corresponding abstract energy estimate is shown to be robust with respect to mesh parameters as well as the contrast, which is defined as the ratio of high and low values of the conductivity (or permeability). The derived stable decomposition allows to construct additive overlapping Schwarz iterative methods with condition numbers uniformly bounded with respect to the contrast and mesh parameters. The coarse spaces are obtained by patching together the eigenfunctions corresponding to the smallest eigenvalues of certain local problems. A detailed analysis of the abstract setting is provided. The proposed decomposition builds on a method of Galvis and Efendiev [Multiscale Model. Simul. 8 (2010) 1461–1483] developed for second order scalar elliptic problems with high contrast. Applications to the finite element discretizations of the second order elliptic problem in Galerkin and mixed formulation, the Stokes equations, and Brinkman’s problem are presented. A number of numerical experiments for these problems in two spatial dimensions are provided.},
author = {Efendiev, Yalchin, Galvis, Juan, Lazarov, Raytcho, Willems, Joerg},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {domain decomposition; robust additive Schwarz preconditioner; spectral coarse spaces; high contrast; Brinkman’s problem; multiscale problems; Brinkman's problem; porous media; elliptic (pressure) equation; finite element; second-order elliptic problem; Stokes equation; numerical experiments},
language = {eng},
number = {5},
pages = {1175-1199},
publisher = {EDP-Sciences},
title = {Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms},
url = {http://eudml.org/doc/273336},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Efendiev, Yalchin
AU - Galvis, Juan
AU - Lazarov, Raytcho
AU - Willems, Joerg
TI - Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2012
PB - EDP-Sciences
VL - 46
IS - 5
SP - 1175
EP - 1199
AB - An abstract framework for constructing stable decompositions of the spaces corresponding to general symmetric positive definite problems into “local” subspaces and a global “coarse” space is developed. Particular applications of this abstract framework include practically important problems in porous media applications such as: the scalar elliptic (pressure) equation and the stream function formulation of its mixed form, Stokes’ and Brinkman’s equations. The constant in the corresponding abstract energy estimate is shown to be robust with respect to mesh parameters as well as the contrast, which is defined as the ratio of high and low values of the conductivity (or permeability). The derived stable decomposition allows to construct additive overlapping Schwarz iterative methods with condition numbers uniformly bounded with respect to the contrast and mesh parameters. The coarse spaces are obtained by patching together the eigenfunctions corresponding to the smallest eigenvalues of certain local problems. A detailed analysis of the abstract setting is provided. The proposed decomposition builds on a method of Galvis and Efendiev [Multiscale Model. Simul. 8 (2010) 1461–1483] developed for second order scalar elliptic problems with high contrast. Applications to the finite element discretizations of the second order elliptic problem in Galerkin and mixed formulation, the Stokes equations, and Brinkman’s problem are presented. A number of numerical experiments for these problems in two spatial dimensions are provided.
LA - eng
KW - domain decomposition; robust additive Schwarz preconditioner; spectral coarse spaces; high contrast; Brinkman’s problem; multiscale problems; Brinkman's problem; porous media; elliptic (pressure) equation; finite element; second-order elliptic problem; Stokes equation; numerical experiments
UR - http://eudml.org/doc/273336
ER -

References

top
  1. [1] R.A. Adams, Sobolev Spaces, 1st edition. Pure Appl. Math. Academic Press, Inc. (1978). Zbl0314.46030
  2. [2] W. Bangerth, R. Hartmann and G. Kanschat, deal.II – a general purpose object oriented finite element library. ACM Trans. Math. Softw. 33 (2007) 24/1–24/27. MR2404402
  3. [3] J.H. Bramble, Multigrid Methods, 1st edition. Longman Scientific & Technical, Essex (1993). Zbl0786.65094MR1247694
  4. [4] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, 2nd edition. Springer (2002). Zbl0804.65101MR1894376
  5. [5] H.C. Brinkman, A calculation of the viscouse force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A1 (1947) 27–34. Zbl0041.54204
  6. [6] T. Chartier, R.D. Falgout, V.E. Henson, J. Jones, T. Manteuffel, S. McCormick, J. Ruge and P.S. Vassilevski, Spectral AMGe (AMGe). SIAM J. Sci. Comput.25 (2003) 1–26. Zbl1057.65096MR2047193
  7. [7] M. Dryja, M.V. Sarkis and O.B. Widlund, Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions. Numer. Math.72 (1996) 313–348. Zbl0857.65131MR1367653
  8. [8] Y. Efendiev and T.Y. Hou, Multiscale finite element methods, Theory and applications. Surveys and Tutorials in Appl. Math. Sci. Springer, New York 4 (2009). Zbl1163.65080MR2477579
  9. [9] R.E. Ewing, O. Iliev, R.D. Lazarov, I. Rybak and J. Willems, A simplified method for upscaling composite materials with high contrast of the conductivity. SIAM J. Sci. Comput.31 (2009) 2568–2586. Zbl1202.80025MR2520290
  10. [10] J. Galvis and Y. Efendiev, Domain decomposition preconditioners for multiscale flows in high-contrast media. Multiscale Model. Simul.8 (2010) 1461–1483. Zbl1206.76042MR2718268
  11. [11] J. Galvis and Y. Efendiev, Domain decomposition preconditioners for multiscale flows in high contrast media : reduced dimension coarse spaces. Multiscale Model. Simul.8 (2010) 1621–1644. Zbl05869382MR2728702
  12. [12] V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Comput. Math. Theory and Algorithms 5 (1986). Zbl0585.65077MR851383
  13. [13] I.G. Graham, P.O. Lechner and R. Scheichl, Domain decomposition for multiscale PDEs. Numer. Math.106 (2007) 589–626. Zbl1141.65084MR2317926
  14. [14] P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics. Pitman Advanced Publishing Program, Boston, MA 24 (1985). Zbl0695.35060MR775683
  15. [15] W. Hackbusch, Multi-Grid Methods and Applications, 2nd edition. Springer Series in Comput. Math. Springer, Berlin (2003). Zbl0595.65106
  16. [16] T.Y. Hou, X.-H. Wu and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comp.68 (1999) 913–943. Zbl0922.65071MR1642758
  17. [17] A. Klawonn, O.B. Widlund and M. Dryja, Dual-primal FETI methods for three-dimensional elliptic problems with heterogeneous coefficients. SIAM J. Numer. Anal. 40 (2002) 159–179 (electronic). Zbl1032.65031MR1921914
  18. [18] J. Mandel and M. Brezina, Balancing domain decomposition for problems with large jumps in coefficients. Math. Comp.65 (1996) 1387–1401. Zbl0853.65129MR1351204
  19. [19] T.P.A. Mathew, Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations. Lect. Notes Comput. Sci. Eng. Springer, Berlin Heidelberg (2008). Zbl1147.65101MR2445659
  20. [20] S.V. Nepomnyaschikh, Mesh theorems on traces, normalizations of function traces and their inversion. Sov. J. Numer. Anal. Math. Modelling6 (1991) 151–168. Zbl0816.65097MR1126677
  21. [21] C. Pechstein and R. Scheichl, Analysis of FETI methods for multiscale PDEs. Numer. Math.111 (2008) 293–333. Zbl1170.65097MR2456834
  22. [22] C. Pechstein and R. Scheichl, Analysis of FETI methods for multiscale PDEs – Part II : interface variation. To appear in Numer. Math. Zbl05939381MR2810804
  23. [23] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV : Analysis of Operators. Academic Press, New York (1978). Zbl0242.46001MR493421
  24. [24] M.V. Sarkis, Schwarz Preconditioners for Elliptic Problems with Discontinuous Coefficients Using Conforming and Non-Conforming Elements. Ph.D. thesis, Courant Institute, New York University (1994). 
  25. [25] M.V. Sarkis, Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements. Numer. Math.77 (1997) 383–406. Zbl0884.65119MR1469678
  26. [26] B.F. Smith, P.E. Bjørstad and W.D. Gropp, Domain Decomposition, Parallel Multilevel Methods for Elliptic Partial Differential Equations, 1st edition. Cambridge University Press, Cambridge (1996). Zbl0857.65126MR1410757
  27. [27] A. Toselli and O. Widlund, Domain Decomposition Methods – Algorithms and Theory. Springer Series in Comput. Math. (2005). Zbl1069.65138MR2104179
  28. [28] J. Van Lent, R. Scheichl and I.G. Graham, Energy-minimizing coarse spaces for two-level Schwarz methods for multiscale PDEs. Numer. Linear Algebra Appl.16 (2009) 775–799. Zbl1224.65292MR2567313
  29. [29] P.S. Vassilevski, Multilevel block-factrorization preconditioners. Matrix-based analysis and algorithms for solving finite element equations. Springer-Verlag, New York (2008). Zbl1170.65001MR2427040
  30. [30] J. Wang and X. Ye, New finite element methods in computational fluid dynamics by H(div) elements. SIAM J. Numer. Anal.45 (2007) 1269–1286. Zbl1138.76049MR2318812
  31. [31] J. Willems, Numerical Upscaling for Multiscale Flow Problems. Ph.D. thesis, University of Kaiserslautern (2009). 
  32. [32] J. Xu and L.T. Zikatanov, On an energy minimizing basis for algebraic multigrid methods. Comput. Visualisation Sci.7 (2004) 121–127. Zbl1077.65130MR2097099

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