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Multigrid methods for some quasi-variational inequalities

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  • We introduce four variants of a multigrid method for quasi-variational inequalities composed by a term arising from the minimization of a functional and another one given by an operator. The four variants of the method differ from one to another by the argument of the operator. The method assume that the closed convex set is decomposed as a sum of closed convex level subsets. These methods are first introduced as subspace correction algorithms in a general reflexive Banach space. Under an assumption on the level decomposition of the closed convex set of the problem, we prove that the algorithms are globally convergent if a certain convergence condition is satisfied, and estimate the global convergence rate. These general algorithms become multilevel or multigrid methods if we use finite element spaces associated with the level meshes of the domain and with the domain decompositions on each level. In this case, the methods are multigrid $V$-cycles, but the results hold for other iteration types, the $W$-cycle iterations, for instance. We prove that the assumption we made in the general convergence theory holds for the one-obstacle problems, and write the convergence rate depending on the number of level meshes. The convergence condition in the theorem imposes a upper bound of the number of level meshes we can use in algorithms.
    Mathematics Subject Classification: Primary: 65N55, 65K15; Secondary: 65J15.


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