Domain decomposition methods: Recent advances and new challenges in engineering

Rixen, Daniel; Magoulès, Frédéric

doi:10.1016/j.cma.2006.09.003

Cited by 14 publications

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“…Also the effectiveness of Schwarz methods for these problems, especially those in fluid mechanics, has been demonstrated in many papers. See the proceedings of the annual domain decomposition conference [14] and [20]- [22], [23]- [24], [26]. Moreover, a priori estimates of the errors for stationary problems is given in several papers; see for instance [21], [22] where a variational formulation of the classical Schwarz method is derived.…”

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Asymptotic behavior and a posteriori error estimates in Sobolev space for the generalized overlapping domain decomposition method for evolutionary HJB equation with nonlinear source terms. Part 1

Boulaaras¹

2016

J. Nonlinear Sci. Appl.

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A posteriori error estimates for the generalized overlapping domain decomposition method with Dirichlet boundary conditions on the boundaries for the discrete solutions on subdomains of evolutionary HJB equation with nonlinear source terms are established using the semi-implicit time scheme combined with a finite element spatial approximation. Also the techniques of the residual a posteriori error analysis are used. Moreover, using Benssoussan-Lions' algorithm, an asymptotic behavior in H 1 0 -norm is deduced. Furthermore, the results of some numerical experiments are presented to support the theory.

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“…The method uses new optimized boundary conditions specially designed to take into account the heterogeneity between the subdomains on the boundaries. A recent overview of the current state of the art on domain decomposition methods can be found in [1], [26].…”

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confidence: 99%

Asymptotic behavior and a posteriori error estimates in Sobolev space for the generalized overlapping domain decomposition method for evolutionary HJB equation with nonlinear source terms. Part 1

Boulaaras¹

2016

J. Nonlinear Sci. Appl.

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Towards a multi‐scale computerized bone diagnostic system: 2D micro‐scale finite element analysis

Podshivalov

Holdstein

Fischer

et al. 2008

Commun. Numer. Meth. Engng.

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SUMMARYCurrently, there is major interest within the biomedical community in developing accurate non-invasive means for analyzing bone micro-structure. This paper presents a new approach for a multi-scale finite element (FE) analysis of a trabecular structure. Two domain decomposition approaches are investigated as a basis for computational analysis at the micro-scale level, which is then applied for solving a 2D elasticity FE problem. In addition, a homogenization procedure from micro-to macro-scale level is presented. The proposed new multi-scale FE method has the potential to provide new insights into bone structure and behavior. Moreover, it is expected that the outcomes of this research will develop into a computerized virtual biopsy system.

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A posteriori error estimates for the generalized Schwarz method of a new class of advection‐diffusion equation with mixed boundary condition

Boulaaras

Brahim

Bouzenada

2018

Math Methods in App Sciences

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Domain decomposition methods: Recent advances and new challenges in engineering

Cited by 14 publications

References 0 publications

Asymptotic behavior and a posteriori error estimates in Sobolev space for the generalized overlapping domain decomposition method for evolutionary HJB equation with nonlinear source terms. Part 1

Asymptotic behavior and a posteriori error estimates in Sobolev space for the generalized overlapping domain decomposition method for evolutionary HJB equation with nonlinear source terms. Part 1

Towards a multi‐scale computerized bone diagnostic system: 2D micro‐scale finite element analysis

A posteriori error estimates for the generalized Schwarz method of a new class of advection‐diffusion equation with mixed boundary condition

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