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Domain Decomposition Preconditioners for Multiscale Flows in High-Contrast Media

Domain Decomposition Preconditioners for Multiscale Flows in High-Contrast Media,10.1137/090751190,Multiscale Modeling & Simulation,Juan Galvis,Yalchi

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Domain Decomposition Preconditioners for Multiscale Flows in High-Contrast Media   (Citations: 8)
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In this paper, we study domain decomposition preconditioners for multiscale ∞ows in high contrast media. Our problems are motivated by porous media applications where low conductivity regions play an important role in determin- ing ∞ow patterns. We consider ∞ow equations governed by elliptic equations in heterogeneous media with large contrast between high and low conductivity regions. This contrast brings an additional small scale (in addition to small spatial scales) into the problem expressed as the ratio between low and high conductivity values. Using weighted coarse interpolation, we show that the condition number of the preconditioned systems using domain decomposition methods is independent of the contrast. For this purpose, Poincare inequalities for weighted norms are proved in the paper. The results are further generalized by employing extension theorems from homogenization theory. Our numerical observations conflrm the theoretical results.
Journal: Multiscale Modeling & Simulation - MULTISCALE MODEL SIMUL , vol. 8, no. 4, pp. 1461-1483, 2010
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    • ...Note however that other weighted Poincar e type inequalities have been proved by Xu and Zhu [42] (based on [4]) and in a recent article by Galvis and Efendiev [12]...
    • ...A very similar weighted Poincare inequality to the one we gave here (with one functional) was also recently proved by Galvis and Efendiev [12] and used in the analysis of two-level overlapping Schwarz...

    Clemens Pechstein, et al. Analysis of FETI methods for multiscale PDEs. Part II: interface varia...

    • ...The proof of this has been achieved very recently in 28, and it requires weighted Poincaré inequalities of a similar type than (<0013" ref-type="disp-formula">13)...

    Clemens Pechstein, et al. Scaling up through domain decomposition

    • ...To obtain sharper bounds in some of these cases, it is possible to refine the standard analyses and use Poincar´ inequalities on annulus type boundary layers of each subdomain [5, 8, 10, 12, 15], or weighted Poincar´ type inequalities [4, 9, 13]...
    • ...In this short note we want to collect and expand on the results in [4, 9] and present a new class of weighted Poincar´ inequalities for a rather general class of piecewise constant coefficients...
    • ...should be compared with one of the main results in [4], where a similar Poincar´ inequality is proved with a constant depending on the number of inclusions...

    Clemens Pechstein, et al. Weighted Poincaré Inequalities and Applications in Domain Decompositio...

    • ...We apply a recently proposed [5] robust overlapping Schwarz method with a certain spectral construction of the coarse space in the setting of element agglomeration algebraic multigrid methods (or agglomeration AMGe) for elliptic problems with highcontrast coefficients...
    • ...The purpose of this paper is to present some preliminary results on the performance of recently proposed overlapping Schwarz methods [5] for elliptic equations with high-contrast coefficients...
    • ...In this paper, we extend the methods and results of [5] to the multilevel case...
    • ...The approach proposed in [5] for the two-level case, which we extend in the present contribution to the multilevel case, needs only to identify the “vertices” of the agglomerates; no additional topological relations are required (assuming that we have somehow come up with an agglomeration algorithm or when geometric meshes are simply used as in our present experiments)...
    • ...The two-level version of the method was proven in [5 ]t o be robust with respect to the contrast...
    • ...where the second sum runs over all subdomains at level , =1 ,..., L. See [5] for a two-level version of this method...
    • ...where the second sum runs over all subdomains at level , =1 ,..., L. See [5] for a two-level version of this method. In [5], it is proved that Cond(M −1...
    • ...We note that the multilevel extension of [5] would require the solution of fine triangulation eigenvalue problems in each subdomain at every level...
    • ...We have the following results which proof uses tools developed in [2, 3, 5 ], see also [4]...
    • ...T = mink λk >ρ ; for =0 ,..., L, where ρ is chosen based on the first large (order one) jump of the eigenvalues (c.f., [2, 5, 6])...

    Yalchin Efendiev, et al. Spectral Element Agglomerate Algebraic Multigrid Methods for Elliptic ...

    • ...The proof of this has only been achieved very recently in [12], and it requires weighted Poincar e inequalities of a similar type than (13) below...
    • ...The key tools for a theoretical analysis of this robustness are new weighted Poincar e and discrete Sobolev inequalities which we have proved in [31] (see also [12])...

    Clemens Pechstein, et al. Scaling Up through Domain Decomposition

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