Abstract
We study the construction of subspaces for quadratic FEM orthotropic elliptic problems with a focus on the robustness with respect to mesh and coefficient anisotropy. In the general setting of an arbitrary elliptic operator it is known that standard hierarchical basis (HB) techniques do not result in splittings in which the angle between the coarse space and its (hierarchical) complement is uniformly bounded with respect to the ratio of anisotropy. In this paper we present a robust splitting of the finite element space of continuous piecewise quadratic functions for the orthotropic problem. As a consequence of this result we obtain also a uniform condition number bound for a special sparse Schur complement approximation. Further we construct a uniform preconditioner for the pivot block with optimal order of computational complexity.
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References
Axelsson, O.: Stabilization of algebraic multilevel iteration method; Additive methods. Numerical Algorithms 21(1-4), 23–47 (1999)
Axelsson, O., Blaheta, R., Neytcheva, M.: Preconditioning of boundary value problems using elementwise Schur complements. SIAM J. Matrix Anal. Appl. 31, 767–789 (2009)
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)
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 and Applications 11(4), 309–326 (2004)
Georgiev, I., Lymbery, M., Margenov, S.: Analysis of the CBS Constant for Quadratic Finite Elements. In: Dimov, I., Dimova, S., Kolkovska, N. (eds.) NMA 2010. LNCS, vol. 6046, pp. 412–419. Springer, Heidelberg (2011)
Kraus, J.: Algebraic multilevel preconditioning of finite element matrices using local Schur complements. Numer. Linear Algebra Appl. 13, 49–70 (2006)
Kraus, J., Margenov, S.: Robust Algebraic Multilevel Methods and Algorithms. De Gruyter, Germany (2009)
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. Lect. Notes Math. 960, 535–544 (1982)
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Kraus, J., Lymbery, M., Margenov, S. (2012). On the Robustness of Two-Level Preconditioners for Quadratic FE Orthotropic Elliptic Problems. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds) Large-Scale Scientific Computing. LSSC 2011. Lecture Notes in Computer Science, vol 7116. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29843-1_66
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DOI: https://doi.org/10.1007/978-3-642-29843-1_66
Publisher Name: Springer, Berlin, Heidelberg
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