Numerical Algorithms

, Volume 75, Issue 4, pp 879–908 | Cite as

Grid solution of problem with unilateral constraints

Original Paper


The present study deals with the solution of a problem, defined in a three-dimensional domain, arising in fluid mechanics. Such problem is modelled with unilateral constraints on the boundary. Then, the problem to solve consists in minimizing a functional in a closed convex set. The characterization of the solution leads to solve a time-dependent variational inequality. An implicit scheme is used for the discretization of the time-dependent part of the operator and so we have to solve a sequence of stationary elliptic problems. For the solution of each stationary problem, an equivalent form of a minimization problem is formulated as the solution of a multivalued equation, obtained by the perturbation of the previous stationary elliptic operator by a diagonal monotone maximal multivalued operator. The spatial discretization of such problem by appropriate scheme leads to the solution of large scale algebraic systems. According to the size of these systems, parallel iterative asynchronous and synchronous methods are carried out on distributed architectures; in the present study, methods without and with overlapping like Schwarz alternating methods are considered. The convergence of the parallel iterative algorithms is analysed by contraction approaches. Finally, the parallel experiments are presented.


Variational inequality Parallel iterative algorithms Asynchronous iterations Unilateral constraints problem Grid computing Fluid mechanics 

Mathematics Subject Classification (2010)

68U10 65F10 65Y05 65N22 65C20 65Y20 65M12 


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.IRT SystemXPalaiseauFrance
  2. 2.Faculté des Sciences, Département de MathématiquesUniversité d’Annaba, Laboratoire LANOSAnnabaAlgérie
  3. 3.IRIT - ENSEEIHTToulouseFrance
  4. 4.UVSQ-PRISMVersailles CedexFrance

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