In this contribution we report on work done in continuation of [1, 2] where additive multilevel methods for the construction of preconditioners for the stiffness matrix of the Ritz- Galerkin procedure were considered with emphasis on the model problem —∇ω∇u = f with a scalar weight ω.
We present an new approach leading to a preconditioner based on a modification of the construction in [4] using weighted scalar products thereby improving that one in [2]. Further we prove an upper bound in the underlying norm equivalencies which is up to a fixed level completely independent of the weight ω, whereas the lower bound involves an assumption about the local variation the coefficient function which is still weaker than in [1]. More details will be presented in a forthcoming paper.
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
M. Griebel, K. Scherer, and M.A. Schweitzer. Robust norm-equivalencies for diffusion problems. Math. Comp., 76:1141–1161, 2007.
K. Scherer. Weighted norm-equivalences for preconditioning. In Domain Decomposition Methods in Science and Engineering, volume 40 of Lect. Notes Comput. Sci. Eng., pages 405–413. Springer, Berlin, 2005.
J. Xu. Iterative methods by space decomposition and subspace correction. SIAM Rev., 34(4):581–613, 1992.
H. Yserentant. Two preconditioners based on the multi-level splitting of finite element spaces. Numer. Math., 58(2):163–184, 1990.
H. Yserentant. Old and new convergence proofs for multigrid methods. In Acta Numerica, 1993, pages 285–326. Cambridge Univ. Press, Cambridge, 1993.
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
Scherer, K. (2008). Robust Norm Equivalencies and Preconditioning. 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. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75199-1_46
Download citation
DOI: https://doi.org/10.1007/978-3-540-75199-1_46
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-75198-4
Online ISBN: 978-3-540-75199-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)