caa a22 4500 378860127 CHVBK 20180305123344.0 cr unu---uuuuu 161128s2003 xx s 000 0 eng 10.2478/cmam-2003-0002 doi (NATIONALLICENCE)gruyter-10.2478/cmam-2003-0002 On Multilevel Preconditioners which are Optimal with Respect to Both Problem and Discretization Parameters [Elektronische Daten] Preconditioners based on various multilevel extensions of two-level piecewise linear finite element methods lead to iterative methods which have an optimal order computational complexity with respect to the size (or discretization parameter) of the system. The methods can be in block matrix factorized form, recursively extended via certain matrix polynomial approximations of the arising Schur complement matrices or on additive, i.e., block diagonal form using stabilizations of the condition number at certain levels. The resulting spectral equivalence holds uniformly with respect to jumps in the coefficients of the differential operator and for arbitrary triangulations. Such methods were first presented by Axelsson and Vassilevski in the late 1980s. An important part of the algorithm is the treatment of systems with a diagonal block matrix, which arises on each finer level in a recursive refinement method and corresponds to the added degrees of freedom on that level. This block is well-conditioned for model type problems but becomes increasingly ill-conditioned when the coefficient matrix becomes more anisotropic or, equivalently, when the mesh aspect ratio increases. This paper presents some methods for approximating this matrix also leading to a preconditioner with spectral equivalence bounds which hold uniformly with respect to both the problem and the discretization parameters. Therefore, the same holds also for the preconditioner to the global matrix. This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. multilevel preconditioners nationallicence partial differential equations nationallicence hierarchical basis nationallicence optimal order preconditioners nationallicence Axelsson O. Department of Mathematics, University of Nijmegen, Toernooiveld, 6525 ED Nijmegen, The Netherlands. Margenov S. Central Laboratory for Parallel Processing, Bulgarian Academy of Sciences, Acad. G. Bonchev, Bl. 25A, 1113 Sofia, Bulgaria. Computational Methods in Applied Mathematics De Gruyter 3/1(2003), 6-22 1609-4840 3:1<6 2003 3 cmam https://doi.org/10.2478/cmam-2003-0002 text/html Onlinezugriff via DOI 1 research article jats NATIONALLICENCE 856 40 https://doi.org/10.2478/cmam-2003-0002 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Axelsson O. Department of Mathematics, University of Nijmegen, Toernooiveld, 6525 ED Nijmegen, The Netherlands NATIONALLICENCE 700 1- Margenov S. Central Laboratory for Parallel Processing, Bulgarian Academy of Sciences, Acad. G. Bonchev, Bl. 25A, 1113 Sofia, Bulgaria NATIONALLICENCE 773 0- Computational Methods in Applied Mathematics De Gruyter 3/1(2003), 6-22 1609-4840 3:1<6 2003 3 cmam CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-gruyter