Multilevel Preconditioning of 2D Rannacher-Turek FE Problems; Additive and Multiplicative Methods

  • Ivan Georgiev
  • Johannes Kraus
  • Svetozar Margenov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4310)


In the present paper we concentrate on algebraic two-level and multilevel preconditioners for symmetric positive definite problems arising from discretization by Rannacher-Turek non-conforming rotated bilinear finite elements on quadrilaterals. An important point to make is that in this case the finite element spaces corresponding to two successive levels of mesh refinement are not nested (in general). To handle this, a proper two-level basis is required in order to fit the general framework for the construction of two-level preconditioners for conforming finite elements and to generalize the methods to the multilevel case.

The proposed variants of hierarchical two-level basis are first introduced in a rather general setting. Then, the involved parameters are studied and optimized. As will be shown, the obtained bounds – in particular – give rise to optimal order AMLI methods of additive type. The presented numerical tests fully confirm the theoretical estimates.


Hierarchical Basis Coarse Partition Multiplicative Variant Nodal Basis Function Multilevel Precondition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Axelsson, O.: Stabilization of algebraic multilevel iteration methods; additive methods. Numerical Algorithms 21, 23–47 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Axelsson, O.: Iterative solution methods. Cambridge University Press, Cambridge (1994)zbMATHGoogle Scholar
  3. 3.
    Axelsson, O., Gustafsson, I.: Preconditioning and two-level multigrid methods of arbitrary degree of approximations. Math. Comp. 40, 219–242 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Axelsson, O., Padiy, A.: On the additive version of the algebraic multilevel iteration method for anisotropic elliptic problems. SIAM J. Sci. Comput. 20, 1807–1830 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Axelsson, O., Vassilevski, P.S.: Algebraic Multilevel Preconditioning Methods I. Numer. Math. 56, 157–177 (1989)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Axelsson, O., Vassilevski, P.S.: Algebraic Multilevel Preconditioning Methods II. SIAM J. Numer. Anal. 27, 1569–1590 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Axelsson, O., Vassilevski, P.S.: Variable-step multilevel preconditioning methods, I: self-adjoint and positive definite elliptic problems. Num. Lin. Alg. Appl. 1, 75–101 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bank, R., Dupont, T.: An Optimal Order Process for Solving Finite Element Equations. Math. Comp. 36, 427–458 (1981)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Blaheta, R., Margenov, S., Neytcheva, M.: Uniform estimate of the constant in the strengthened CBS inequality for anisotropic non-conforming FEM systems. Numerical Linear Algebra with Applications 11(4), 309–326 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Eijkhout, V., Vassilevski, P.S.: The Role of the Strengthened Cauchy-Bunyakowski-Schwarz Inequality in Multilevel Methods. SIAM Review 33, 405–419 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    I. Georgiev, J. Kraus, S. Margenov: Multilevel preconditioning of rotated bilinear non-conforming FEM problems, submitted. Also available as RICAM-Report 2006-3, RICAM, Linz, Austria, (2006)Google Scholar
  12. 12.
    Rannacher, R., Turek, S.: Simple non-conforming quadrilateral Stokes Element. Numerical Methods for Partial Differential Equations 8(2), 97–112 (1992)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Ivan Georgiev
    • 1
  • Johannes Kraus
    • 2
  • Svetozar Margenov
    • 3
  1. 1.Institute of Mathematics and Informatics and Institute for Parallel Processing, Bulgarian Academy of Sciences, Acad. G. Bonchev Bl. 25A, 1113 SofiaBulgaria
  2. 2.Johann Radon Institute for Computational and Applied Mathematics, Altenbergerstraße 69, A-4040 LinzAustria
  3. 3.Institute for Parallel Processing, Bulgarian Academy of Sciences, Acad. G. Bonchev Bl. 25A, 1113 SofiaBulgaria

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