Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms

Efendiev, Yalchin; Galvis, Juan; Lazarov, Raytcho; Willems, Joerg

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 5, pp. 1175-1199.

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.

Classification : 65F10, 65N20, 65N22, 65N30, 65N55
Mots clés : domain decomposition, robust additive Schwarz preconditioner, spectral coarse spaces, high contrast, Brinkman's problem, multiscale problems

[1] R.A. Adams, Sobolev Spaces, 1st edition. Pure Appl. Math. Academic Press, Inc. (1978). | Zbl

[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. | MR

[3] J.H. Bramble, Multigrid Methods, 1st edition. Longman Scientific & Technical, Essex (1993). | MR | Zbl

[4] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, 2nd edition. Springer (2002). | MR | Zbl

[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. | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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). | MR | Zbl

[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. | MR | Zbl

[10] J. Galvis and Y. Efendiev, Domain decomposition preconditioners for multiscale flows in high-contrast media. Multiscale Model. Simul. 8 (2010) 1461-1483. | MR | Zbl

[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. | MR

[12] V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Comput. Math. Theory and Algorithms 5 (1986). | MR | Zbl

[13] I.G. Graham, P.O. Lechner and R. Scheichl, Domain decomposition for multiscale PDEs. Numer. Math. 106 (2007) 589-626. | MR | Zbl

[14] P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics. Pitman Advanced Publishing Program, Boston, MA 24 (1985). | MR | Zbl

[15] W. Hackbusch, Multi-Grid Methods and Applications, 2nd edition. Springer Series in Comput. Math. Springer, Berlin (2003). | Zbl

[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. | MR | Zbl

[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). | MR | Zbl

[18] J. Mandel and M. Brezina, Balancing domain decomposition for problems with large jumps in coefficients. Math. Comp. 65 (1996) 1387-1401. | MR | Zbl

[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). | MR | Zbl

[20] S.V. Nepomnyaschikh, Mesh theorems on traces, normalizations of function traces and their inversion. Sov. J. Numer. Anal. Math. Modelling 6 (1991) 151-168. | MR | Zbl

[21] C. Pechstein and R. Scheichl, Analysis of FETI methods for multiscale PDEs. Numer. Math. 111 (2008) 293-333. | MR | Zbl

[22] C. Pechstein and R. Scheichl, Analysis of FETI methods for multiscale PDEs - Part II : interface variation. To appear in Numer. Math. | MR

[23] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV : Analysis of Operators. Academic Press, New York (1978). | MR | Zbl

[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] 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. | MR | Zbl

[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). | MR | Zbl

[27] A. Toselli and O. Widlund, Domain Decomposition Methods - Algorithms and Theory. Springer Series in Comput. Math. (2005). | MR | Zbl

[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. | MR | Zbl

[29] P.S. Vassilevski, Multilevel block-factrorization preconditioners. Matrix-based analysis and algorithms for solving finite element equations. Springer-Verlag, New York (2008). | MR | Zbl

[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. | MR | Zbl

[31] J. Willems, Numerical Upscaling for Multiscale Flow Problems. Ph.D. thesis, University of Kaiserslautern (2009).

[32] J. Xu and L.T. Zikatanov, On an energy minimizing basis for algebraic multigrid methods. Comput. Visualisation Sci. 7 (2004) 121-127. | MR | Zbl