# Robust Algebraic Multilevel Methods and Algorithms (Radon by Johannes Kraus

By Johannes Kraus

This ebook bargains with algorithms for the answer of linear platforms of algebraic equations with large-scale sparse matrices, with a spotlight on difficulties which are acquired after discretization of partial differential equations utilizing finite point tools. offers a scientific presentation of the new advances in strong algebraic multilevel tools. can be utilized for complicated classes at the subject.

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Robust Algebraic Multilevel Methods and Algorithms (Radon Series on Computational and Applied Mathematics)

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Extra resources for Robust Algebraic Multilevel Methods and Algorithms (Radon Series on Computational and Applied Mathematics)

Sample text

However, the assumptions in [19] result in quite pessimistic estimates for the convergence rate of the nonlinear AMLI method. As it has been shown in [96] the bound on the local decrease of the error in GCG-type iterations can be improved considerably if the nonlinear preconditioner becomes close to a linear operator. 23, see [72]. 10. `/ is assumed to be an SPD matrix. k/ /. Proof. k/ are SPD. k/ 1 arbitrary nonzero vector of dimension Nk . / .. 0/ D 0. / . k/  The following lemma provides the key to the convergence analysis as presented in [72].

Then every application of the recursively defined preconditioner at a given level k with a given number Nk of DOF involves  applications of the preconditioner at level k 1 where the number Nk 1 of DOF is smaller by some factor say %. `/ factorization is used at this point. N` C  N` D c N` C : : : C  ` N0 /  2  ` ! 1    1C C C ::: C D c N` % % % 1 1   `C1 % :  % Since the number of DOF at level k 1 is (assumed to be) 1=% times the number of DOF at level k, each visit of level k must induce less than % visits of level k 1 (at least in average).

63]), so that it is possible to work with the matrix C itself. , solving the system C 2 Ax D C 2 b, which is consistent with Ax D b. 47) as well. 5 Under the assumption that C is symmetric and positive definite, the preconditioned system matrix AQ is SPD as well. 6 Its steps can be rewritten avoiding an explicit reference to the matrix C 21 . , [25]). 21. 20 are generated by the preconditioned matrix, that is, Wn? C 1 A/. 0/ q? , where q? denotes an arbitrary element of the related Krylov space Wn?