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2012 | 10 | 1 | 357-369
Tytuł artykułu

Robust semi-coarsening multilevel preconditioning of biquadratic FEM systems

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
While a large amount of papers are dealing with robust multilevel methods and algorithms for linear FEM elliptic systems, the related higher order FEM problems are much less studied. Moreover, we know that the standard hierarchical basis two-level splittings deteriorate for strongly anisotropic problems. A first robust multilevel preconditioner for higher order FEM systems obtained after discretizations of elliptic problems with an anisotropic diffusion tensor is presented in this paper. We study the behavior of the constant in the strengthened CBS inequality for semi-coarsening mesh refinement which is a quality measure for hierarchical two-level splittings of the considered biquadratic FEM stiffness matrices. The presented new theoretical estimates are confirmed by numerically computed CBS constants for a rich set of parameters (coarsening factor and anisotropy ratio). In the paper we consider also the problem of solving efficiently systems with the pivot block matrices arising in the hierarchical basis two-level splittings. Combining the proven uniform estimates with the theory of the Algebraic MultiLevel Iteration (AMLI) methods we obtain an optimal order multilevel algorithm whose total computational cost is proportional to the size of the discrete problem with a proportionality constant independent of the anisotropy ratio.
Wydawca
Czasopismo
Rocznik
Tom
10
Numer
1
Strony
357-369
Opis fizyczny
Daty
wydano
2012-02-01
online
2011-12-09
Twórcy
Bibliografia
  • [1] Axelsson O., Stabilization of algebraic multilevel iteration methods; additive methods, Numer. Algorithms, 1999, 21(1–4), 23–47 http://dx.doi.org/10.1023/A:1019136808500
  • [2] Axelsson O., Blaheta R., Neytcheva M., Preconditioning of boundary value problems using elementwise Schur complements, SIAM J. Matrix Anal. Appl., 2009, 31(2), 767–789 http://dx.doi.org/10.1137/070679673
  • [3] Axelsson O., Margenov S., On multilevel preconditioners which are optimal with respect to both problem and discretization parameters, Comput. Methods Appl. Math., 2003, 3(1), 6–22
  • [4] Axelsson O., Vassilevski P.S., Algebraic multilevel preconditioning methods. I, Numer. Math., 1989, 56(2–3), 157–177 http://dx.doi.org/10.1007/BF01409783
  • [5] Ayuso de Dios B., Zikatanov L., Uniformly convergent iterative methods for discontinuous Galerkin discretizations, J. Sci. Comput., 2009, 40(1–3), 4–36 http://dx.doi.org/10.1007/s10915-009-9293-1
  • [6] Boyanova P., Margenov S., Robust multilevel methods for elliptic and parabolic problems, In: Efficient Preconditioning Methods for Elliptic Partial Differential Equations, Bentham Science Publishers, 2011 (in press)
  • [7] Faragó I., Karátson J., Numerical Solution of Nonlinear Elliptic Problems via Preconditioning Operators: Theory and Applications, Adv. Comput. Theory Pract., 11, Nova Science Publishers, Hauppauge, 2002
  • [8] Georgiev I., Lymbery M., Margenov S., Analysis of the CBS constant for quadratic finite elements, In: Numerical Methods and Applications, Borovets, August 20–24, 2010, Lecture Notes in Comput. Sci., 6046, Springer, New York, 2011, 412–419
  • [9] Kraus J.K., Algebraic multilevel preconditioning of finite element matrices using local Schur complements, Numer. Linear Algebra Appl., 2006, 13(1), 49–70 http://dx.doi.org/10.1002/nla.462
  • [10] Kraus J., Margenov S., Robust Algebraic Multilevel Methods and Algorithms, Radon Ser. Comput. Appl. Math., 5, Walter de Gruyter, Berlin, 2009
  • [11] Kraus J., Margenov S., Synka J., On the multilevel preconditioning of Crouzeix-Raviart elliptic problems, Numer. Linear Algebra Appl., 2008, 15(5), 395–416 http://dx.doi.org/10.1002/nla.543
  • [12] Kraus J.K., Tomar S.K., A multilevel method for discontinuous Galerkin approximation of three-dimensional anisotropic elliptic problems, Numer. Linear Algebra Appl., 2008, 15(5), 417–438 http://dx.doi.org/10.1002/nla.544
  • [13] Maitre J.F., Musy S., The contraction number of a class of two-level methods; an exact evaluation for some finite element subspaces and model problems, In: Multigrid Methods, Cologne, 1981, Lecture Notes in Math., 960, Springer, Berlin-New York, 1982, 535–544
  • [14] Margenov S.D., Semi-coarsening AMLI algorithms for elasticity problems, Numer. Linear Algebra Appl., 1998, 5(5), 347–362 http://dx.doi.org/10.1002/(SICI)1099-1506(199809/10)5:5<347::AID-NLA137>3.0.CO;2-5
  • [15] Margenov S., Xanthis L., Zikatanov L., On the optimality of the semicoarsening AMLI algorithm, In: 2nd IMACS International Symposium on Iterative Methods in Linear Algebra, Blagoevgrad, June 17–20, 1995, IMACS Series in Computational and Applied Mathematics, 3, IMACS, 1996, 270–279
  • [16] Mense C., Nabben R., On algebraic multilevel methods for non-symmetric systems - convergence results, Electr. Trans. Numer. Anal., 2008, 30, 323–345
  • [17] Mense C., Nabben R., On algebraic multi-level methods for non-symmetric systems - comparison results, Linear Algebra Appl., 2008, 429(10), 2567–2588 http://dx.doi.org/10.1016/j.laa.2008.04.045
  • [18] Neytcheva M., On element-by-element Schur complement approximations, Linear Algebra Appl., 2011, 434(11), 2308–2324 http://dx.doi.org/10.1016/j.laa.2010.03.031
  • [19] Pultarová I., Preconditioning and a posteriori error estimates using h- and p-hierarchical finite elements with rectangular supports, Numer Linear Algebra Appl., 2009, 16(5), 415–430 http://dx.doi.org/10.1002/nla.624
  • [20] Vassilevski P.S., Multilevel Block Factorization Preconditioners, Springer, New York, 2008
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-011-0082-3
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