Convergence rate of some hybrid multigrid methods for variational inequalities

Abstract

In [L. Badea, Global convergence rate of a standard multigrid method for variational inequalities, IMA J. Numer. Anal., 34 (2014), No. 1, 197-216], a global convergence rate of the standard monotone multigrid method for variational inequalities is derived. This algorithm can be also viewed as performing multiplicative iterations on each level and also multiplicative iterations over the levels. In the present paper, this algorithm together with other three algorithms,which are combinations of additive or multiplicative iterations on levels with additive or multiplicative iterations over the levels, are analyzed in a unitary manner and in a more general framework which allow us to consider problems in the Sobolev space W^{1, σ}, 1 < σ < ∞, not only in the usual H¹. The algorithms are given for the constrained minimization of functionals where the convex set is of two-obstacle type and have an optimal computing complexity of the iteration steps. The estimations of the global convergence rate are written in function of the number of levels.