Multigrid methods for variational inequalities
نویسنده
چکیده
In this paper we introduce four multigrid algorithms for the constrained minimization of non-quadratic functionals. The algorithm introduced in [L. Badea, Convergence rate of a Schwarz multilevel method for the constrained minimization of non-quadratic functionals, SIAM J. Numer. Anal., 44, 2, 2006, p. 449-477], has a sub-optimal computing complexity because the convex set, which is defined on the finest mesh, is used in the smoothing steps on the coarse levels. The first algorithm we introduce in this paper is a standard V-cycle multigrid iteration which improves the algorithm in the above cited paper, having an optimal computing complexity. This algorithm can be also viewed as performing a multiplicative iteration on each level and a multiplicative one over the levels, too. The three other proposed algorithms are combinations of additive or multiplicative iterations on levels with additive or multiplicative ones over the levels. These algorithms are given for the constrained minimization of non-quadratic functionals where the convex set is of two-obstacle type and have an optimal computing complexity. We give estimations of the global convergence rate in function of the number of levels, and compare our results with the estimations of the asymptotic convergence rate existing in the literature for complementary problems.
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