Abstract
In this article, we give a new rigorous condition number estimate of the finite element tearing and interconnecting (FETI) method and a variant thereof, all-floating FETI. We consider a scalar elliptic equation in a two- or three-dimensional domain with a highly heterogeneous (multiscale) diffusion coefficient. This coefficient is allowed to have large jumps not only across but also along subdomain interfaces and in the interior of the subdomains. In other words, the subdomain partitioning does not need to resolve any jumps in the coefficient. Under suitable assumptions, we derive bounds for the condition numbers of one-level and all-floating FETI that are robust with respect to strong variations in the contrast in the coefficient, and that are explicit in some geometric parameters associated with the coefficient variation. In particular, robustness holds for face, edge, and vertex islands in high-contrast media. As a central tool we prove and use new weighted Poincaré and discrete Sobolev type inequalities that are explicit in the weight. Our theoretical findings are confirmed in a series of numerical experiments.
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Adams, R.A., Fournier, J.J.F.: Sobolev spaces. In: Pure and Applied Mathematics, vol. 140, 2nd edn. Academic Press, Amsterdam (2003)
Aksoylu B., Graham I.G., Klie H., Scheichl R.: Towards a rigorously justified algebraic preconditioner for high-contrast diffusion problems. Comput. Visual. Sci. 11(4–6), 319–331 (2008)
Alcouffe R.E., Brandt A., Dendy J.J.E., Painter J.W.: The multi-grid method for the diffusion equation with strongly discontinuous coefficients. SIAM J. Sci. Comput. 2(4), 430–454 (1981)
Bramble J.H., Xu J.: Some estimates for a weighted L 2 projection. Math. Comp. 56(194), 463–476 (1991)
Brenner S.C.: Analysis of two-dimensional FETI-DP preconditioners by the standard additive Schwarz framework. Electron. Trans. Numer. Anal. 16, 165–185 (2003)
Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. In: Texts in Applied Mathematics, vol. 15, 2nd edn. Springer-Verlag, New York (2002)
Chan, T.F., Mathew, T.: Domain decomposition methods. In: Acta Numerica 1994. Cambridge University Press (1994)
Cliffe K.A., Graham I.G., Scheichl R., Stals L.: Parallel computation of flow in heterogeneous media modelled by mixed finite elements. J. Comput. Phys. 164(2), 258–282 (2000)
Dohrmann C.R., Klawonn A., Widlund O.B.: Domain decomposition for less regular subdomains: overlapping Schwarz in two dimensions. SIAM J. Numer. Anal. 46(4), 2153–2168 (2008)
Dostál Z., Horák D., Kučera R.: Total FETI—an easier implementable variant of the FETI method for numerical solution of elliptic PDE. Commun. Numer. Methods Eng. 22(12), 1155–1162 (2006)
Dryja, M., Sarkis, M.: Technical tools for boundary layers and applications to heterogeneous coefficients. In: Huang, Y., Kornhuber, R., Widlund O., Xu, J. (eds.) Decomposition Methods in Science and Engineering XIX. Lecture Notes in Computational Science and Engineering, vol. 78, pp. 205–212. Springer-Verlag, Berlin (2011)
Dryja M., Sarkis M.V., Widlund O.B.: Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions. Numer. Math. 72, 313–348 (1996)
Farhat C., Roux F.-X.: A method of finite element tearing and interconnecting and its parallel solution algorithm. Int. J. Numer. Methods Eng. 32, 1205–1227 (1991)
Galvis J., Efendiev Y.: Domain decomposition preconditioners for multiscale flows in high contrast media. Multiscale Model. Simul. 8(4), 1461–1483 (2010)
Graham I.G., Hagger M.J.: Unstructured additive Schwarz-conjugate gradient method for elliptic problems with highly discontinuous coefficients. SIAM J. Sci. Comput. 20(6), 2041–2066 (1999)
Graham I.G., Lechner P.O., Scheichl R.: Domain decomposition for multiscale PDEs. Numer. Math. 106(4), 589–626 (2007)
Graham I.G., Scheichl R.: Robust domain decomposition algorithms for multiscale PDEs. Numer. Methods Partial Differ. Equ. 23, 859–878 (2007)
Graham I.G., Scheichl R.: Coefficient-explicit condition number bounds for overlapping additive Schwarz. In: Langer, U., Discacciati, M., Keyes, D.E., Widlund, O.B., Zulehner, W. (eds) Domain Decomposition Methods in Science and Engineering XVII. Lecture Notes in Computational Science and Engineering, vol. 60, pp. 365–372. Springer-Verlag, Berlin (2008)
Klawonn A., Rheinbach O.: Robust FETI-DP methods for heterogeneous three dimensional elasticity problems. Comput. Methods Appl. Mech. Eng. 196, 1400–1414 (2007)
Klawonn A., Rheinbach O., Widlund O.B.: An analysis of a FETI-DP algorithm on irregular subdomains in the plane. SIAM J. Numer. Anal. 46(5), 2484–2504 (2008)
Klawonn A., Widlund O.B.: FETI and Neumann-Neumann iterative substructuring methods: connections and new results. Comm. Pure Appl. Math. 54(1), 57–90 (2001)
Klawonn A., Widlund O.B., Dryja M.: Dual-primal FETI methods for three-dimensional elliptic problems with heterogeneous coefficients. SIAM J. Numer. Anal. 40(1), 159–179 (2002)
Langer U., Pechstein C.: Coupled finite and boundary element tearing and interconnecting solvers for nonlinear potential problems. ZAMM Z. Angew. Math. Mech. 86(12), 915–931 (2006)
Mandel J., Dohrmann C.R., Tezaur R.: An algebraic theory for primal and dual substructuring methods by constraints. Appl. Numer. Math. 54, 167–193 (2005)
Mandel J., Tezaur R.: Convergence of a substrucuring method with Lagrange multipliers. Numer. Math. 73, 473–487 (1996)
Of, G.: BETI-Gebietszerlegungsmethoden mit schnellen Randelementverfahren und Anwendungen. PhD thesis, Universität Stuttgart, Germany, January 2006 (in German)
Of G., Steinbach O.: The all-floating boundary element tearing and interconnectiong method. J. Numer. Math. 17(4), 277–298 (2009)
Pechstein, C.: Finite and boundary element tearing and interconnecting methods for multiscale elliptic partial differential equations. PhD thesis, Institute of Computational Mathematics, Johannes Kepler University, Linz, Austria, December 2008. http://www.numa.uni-linz.ac.at/Teaching/PhD/Finished/pechstein
Pechstein C.: Boundary element tearing and interconnecting methods in unbounded domains. Appl. Numer. Math. 59(11), 2824–2842 (2009)
Pechstein C., Scheichl R.: Analysis of FETI methods for multiscale PDEs. Numer. Math. 111(2), 293–333 (2008)
Pechstein C., Scheichl R.: Scaling up through domain decomposition. Appl. Anal. 88(10–11), 1589–1608 (2009)
Pechstein C., Scheichl R.: Robust FETI solvers for multiscale elliptic PDEs. In: Roos, J., Costa, L.R.J. (eds) Scientific Computing in Electrical Engineering. Mathematics in Industry, vol. 14, pp. 421–428. Springer-Verlag, Berlin (2010)
Pechstein, C., Scheichl, R.: Weighted Poincaré inequalities. NuMa Report 2010-10, Institute of Computational Mathematics, Johannes Kepler University, Linz, Austria (2010, submitted)
Pechstein C., Scheichl R.: Weighted Poincaré inequalities and applications in domain decomposition. In: Huang, Y., Kornhuber, R., Widlund, O., Xu, J. (eds) Decomposition Methods in Science and Engineering XIX. Lecture Notes in Computational Science and Engineering, vol. 78, pp. 197–204. Springer-Verlag, Berlin (2010)
Rixen, D., Farhat, C.: Preconditioning the FETI method for problems with intra- and inter-subdomain coefficient jumps. In: Bjørstad, P.E., Espedal, M., Keyes, D. (eds.) Ninth International Conference on Domain Decomposition Methods, pp. 472–479 (1997). http://www.ddm.org/DD9/Rixen.pdf
Rixen D., Farhat C.: A simple and efficient extension of a class of substructure based preconditioners to heterogeneous structural mechanics problems. Int. J. Numer. Methods Eng. 44, 489–516 (1999)
Ruge J., Stüben K.: Efficient solution of finite difference and finite element equations by algebraic multigrid (AMG). In: Paddon, D.J., Holstein, H. (eds) Multigrid Methods for Integral and Differential Equations. IMA Conference Series, pp. 169–212. Clarendon Press, Oxford (1985)
Sarkis M.: Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements. Numer. Math. 77(3), 383–406 (1997)
Sarkis, M.V.: Schwarz preconditioners for elliptic problems with discontinuous coefficients using conforming and non-conforming elements. PhD thesis, Department of Computer Science, Courant Institute of Mathematical Sciences, TR-671 (1994)
Scheichl R., Vainikko E.: Additive Schwarz and aggregation-based coarsening for elliptic problems with highly variable coefficients. Computing 80(4), 319–343 (2007)
Schenk O., Gärtner K.: On fast factorization pivoting methods for sparse symmetric indefinite systems. Electron. Trans. Numer. Anal. 23, 158–179 (2006)
Scott L.R., Zhang S.: Finite element interpolation of non-smooth functions satisfying boundary conditions. Math. Comput. 54, 483–493 (1990)
Stein E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Math Series, vol. 30. Princeton University Press, Princeton (1970)
Toselli, A., Widlund, O.: Domain decoposition methods—algorithms and theory. In: Springer Series in Computational Mathematics, vol. 34. Springer-Verlag, Berlin (2005)
Van lent J., Scheichl R., Graham I.G.: Energy minimizing coarse spaces for two-level Schwarz methods for multiscale PDEs. Numer. Linear Algebra Appl. 16(10), 775–799 (2009)
Vanek P., Mandel J., Brezina M.: Algebraic multigrid by smoothed aggregation for 2nd and 4th order elliptic problems. Computing 56(3), 179–196 (1996)
Xu J., Zhu Y.: Uniform convergent multigrid methods for elliptic problems with strongly discontinuous coefficients. Math. Models Methods Appl. Sci. 18(1), 77–105 (2008)
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Pechstein, C., Scheichl, R. Analysis of FETI methods for multiscale PDEs. Part II: interface variation. Numer. Math. 118, 485–529 (2011). https://doi.org/10.1007/s00211-011-0359-2
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DOI: https://doi.org/10.1007/s00211-011-0359-2