Abstract
In this paper algebraic two-level and multilevel preconditioning algorithms for second order elliptic boundary value problems are constructed, where the discretization is done using Rannacher-Turek non-conforming rotated trilinear finite elements. 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 to enable us to fit the general framework for the construction of two-level preconditioners for conforming finite elements and to generalize the method 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. The major contribution of the paper is the derived estimates of the constant γ in the strengthened CBS inequality which is shown to allow the efficient multilevel extension of the related two-level preconditioners. Representative numerical tests well illustrate the optimal complexity of the resulting iterative solver.
The authors gratefully acknowledge the support by the Austrian Academy of Sciences, and by EC INCO Grant BIS-21++ 016639/2005. The first and third authors were also partially supported by Bulgarian NSF Grant VU-MI-202/2006.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Axelsson, O., Vassilevski, P.: Algebraic Multilevel Preconditioning Methods I. Numer. Math. 56, 157–177 (1989)
Axelsson, O., Vassilevski, P.: Algebraic Multilevel Preconditioning Methods II. SIAM J. Numer. Anal. 27, 1569–1590 (1990)
Axelsson, O., Vassilevski, P.: Variable-step multilevel preconditioning methods, I: self-adjoint and positive definite elliptic problems. Num. Lin. Alg. Appl. 1, 75–101 (1994)
Blaheta, R.: Algebraic multilevel methods with aggregations: an overview. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds.) LSSC 2005. LNCS, vol. 3743, pp. 3–14. Springer, Heidelberg (2006)
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)
Georgiev, I., Margenov, S.: MIC(0) Preconditioning of Rotated Trilinear FEM Elliptic Systems. In: Margenov, S., Waśniewski, J., Yalamov, P. (eds.) LSSC 2001. LNCS, vol. 2179, pp. 95–103. Springer, Heidelberg (2001)
Kraus, J.: An algebraic preconditioning method for M-matrices: linear versus nonlinear multilevel iteration. Num. Lin. Alg. Appl. 9, 599–618 (2002)
Margenov, S., Vassilevski, P.: Two-level preconditioning of non-conforming FEM systems. In: Griebel, M., et al. (eds.) Large-Scale Scientific Computations of Engineering and Environmental Problems, Vieweg. Notes on Numerical Fluid Mechanics, vol. 62, pp. 239–248 (1998)
Notay, Y.: Robust parameter-free algebraic multilevel preconditioning. Num. Lin. Alg. Appl. 9, 409–428 (2002)
Rannacher, R., Turek, S.: Simple non-conforming quadrilateral Stokes Element. Numerical Methods for Partial Differential Equations 8(2), 97–112 (1992)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Georgiev, I., Kraus, J., Margenov, S. (2008). Multilevel Preconditioning of Rotated Trilinear Non-conforming Finite Element Problems. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds) Large-Scale Scientific Computing. LSSC 2007. Lecture Notes in Computer Science, vol 4818. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78827-0_8
Download citation
DOI: https://doi.org/10.1007/978-3-540-78827-0_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-78825-6
Online ISBN: 978-3-540-78827-0
eBook Packages: Computer ScienceComputer Science (R0)