Geodesic
On some variants of domain decomposition methods
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ Vyčislitelʹnaâ matematika i informatika, Tome 3 (2014) no. 2, pp. 5-19

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The paper considers the algorithms for solving large sparse SLAEs arising from grid approximations of boundary value problems. The SLAEs and algorithms are not limited in asense of number of unknowns, computational nodes, processors and/or cores. This problem isreduced to a distributed variant of algebraic 3D-domain decomposition, in which no excessiveload of the root process is present, i.e. all MPI-processes, each of which corresponds to its ownsubdomain, are almost equal. The computational process consists of two main stages. The firststage is the automatic decomposition, based on the analysis of the matrix portrait and theformation of large-block representation of the original SLAE. The second stage implements aKrylov subspace iterative process with FGMRes (flexible generalized minimal residual method)using either exact or approximate inverse of diagonal blocks as a preconditioner. The methodsdescribed are implemented as a part of Krylov, a library of algebraic solvers. The paper presentssome features of current parallel implementation and estimates of resource usage. Efficiency ofthe developed algorithms is illustrated by solving several typical model problems with differentparameters and in different configurations of multiprocessor computer systems.
Keywords: domain decomposition methods, matrix graphs, parallel algorithms, grid equations, sparse linear algebraic systems. 
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     author = {V. P. Il'in and D. V. Perevozkin},
     title = {On some variants of domain decomposition methods},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a Vy\v{c}islitelʹna\^a matematika i informatika},
     pages = {5--19},
     publisher = {mathdoc},
     volume = {3},
     number = {2},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VYURV_2014_3_2_a0/}
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V. P. Il'in; D. V. Perevozkin. On some variants of domain decomposition methods. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ Vyčislitelʹnaâ matematika i informatika, Tome 3 (2014) no. 2, pp. 5-19. http://geodesic.mathdoc.fr/item/VYURV_2014_3_2_a0/