Comptes Rendus
Numerical Analysis
Simultaneous directions parallel methods for elliptic and parabolic systems
Comptes Rendus. Mathématique, Volume 339 (2004) no. 2, pp. 145-150.

Adapting some previous ideas by Lu, Neittaanmäki and Tai, we use simultaneous directions methods to solve linear second-order elliptic and parabolic equations. The task is reduced to solving a (large) family of independent one-dimensional linear systems. We indicate some convergence results. We also present some numerical experiments.

On utilise des méthodes parallèles de type directions simultanées pour résoudre des équations linéaires elliptiques et paraboliques du second ordre. On adapte quelques idées de Lu, Neittaanmäki and Tai, ce qui conduit à une (grande) famille de systèmes différentiels ordinaires qui sont indépendants. Quelques résultats de convergence et stabilité sont indiqués. On présente aussi quelques expériences numériques.

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Published online:
DOI: 10.1016/j.crma.2004.05.007
José R. Galo 1; Isidoro I. Albarreal 2; M. Carmen Calzada 1; José Luis Cruz 1; Enrique Fernández-Cara 2; Mercedes Marín 1

1 Dpto. Informática y Análisis Numérico, Universidad de Córdoba, Campus de Rabanales, Ed. C2-3, 14071 Córdoba, Spain
2 Dpto. E.D.A.N., Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain
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José R. Galo; Isidoro I. Albarreal; M. Carmen Calzada; José Luis Cruz; Enrique Fernández-Cara; Mercedes Marín. Simultaneous directions parallel methods for elliptic and parabolic systems. Comptes Rendus. Mathématique, Volume 339 (2004) no. 2, pp. 145-150. doi : 10.1016/j.crma.2004.05.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.05.007/

[1] I. Albarreal; M.C. Calzada; J.L. Cruz; E. Fernández-Cara; J.R. Galo; M. Marín Convergence analysis and error estimates for a parallel algorithm for solving the Navier–Stokes equations, Numer. Math., Volume 93 (2002), pp. 201-221

[2] J.R. Galo, PhD Thesis, Univ. de Sevilla, 2002

[3] J.R. Galo, I. Albarreal, M.C. Calzada, J.L. Cruz, E. Fernández-Cara, M. Marín, Convergence and optimization of the parallel method of simultaneous directions for the solution of elliptic problems, in press

[4] J.R. Galo, I. Albarreal, M.C. Calzada, J.L. Cruz, E. Fernández-Cara, M. Marín, Stability and convergence of a parallel fractional step method for the solution of linear parabolic problems, in press

[5] T. Lu; P. Neittaanmäki; X.C. Tai A parallel splitting-up method for partial differential equations and its applications to Navier–Stokes equations, RAIRO–Math. Model. Num., Volume 26 (1992) no. 6, pp. 673-708

[6] G.I. Marchuk Splitting and alternating direction methods, Handbook of Numerical Analysis, I, North-Holland, 1990, pp. 197-462

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