Convergence Rate Analysis of a Multiplicative Schwarz Method for Variational Inequalities

Abstract

This paper derives a linear convergence for the Schwarz overlapping domain decomposition method when applied to constrained minimization problems. The convergence analysis is based on a minimization approach to the corresponding functional over a convex set. A general framework of convergence is established for some multiplicative Schwarz algorithm. The abstract theory is particularly applied to some obstacle problems, which yields a linear convergence for the corresponding Schwarz overlapping domain decomposition method of one and two levels. Numerical experiments are presented to confirm the convergence estimate derived in this paper.

Multigrid Methods for Elliptic Obstacle Problems on 2D Bisection Grids

“…This technique has also been applied to develop domain decomposition and multigrid methods for obstacle problems in Badea et al (2003); . Furthermore, a constraint decomposition method (CDM) was introduced and proved to have a contraction factor which is almost independent of mesh size by Tai (2003).…”

Two-level domain decomposition algorithm for a nonlinear inverse DOT problem

“…Badea and Tai et al [15][16][17][18][19][20][21][22] have previously applied domain decomposition techniques to a constrained convex minimization problem coming from variational inequalities using space decomposition method. It has been shown that multigrid can be viewed as a special case of such space decomposition.…”

“…Local convergence of the inverse solver, when the variance of the optical coefficients are small, is proved using the abstract convergence theorem presented in. 15 It is shown that the two level approach is indispensable in the algorithm due to the properties of local convergence result.…”

“…considering that V is a convex closed subspace of reflexive Banach space L 2 (Ω), we need only to show that assumptions (3.1), (3,2) and (3,3) for Theorem 3.1 in, 15 holds for our case to prove the above theorem. These assumption can easily be shown using the local strong convexity convexity (3.3), with the aid of Taylor expansion of F ′ , mutual disjointness of V l , l = 1, · · · , d, and Cauchy-Schwarz inequality.…”

Nonsmooth Newton Methods for Set-Valued Saddle Point Problems

“…It is motivated by the fact that a variety of practically relevant nonlinearities F can be either inverted in closed form or can be efficiently inverted by multigrid methods. This includes, e.g., the nonlinearities mentioned above [3,4,26,30,32,33].…”