# Convergence Rate Analysis of a Multiplicative Schwarz Method for Variational Inequalities

@article{Badea2003ConvergenceRA, title={Convergence Rate Analysis of a Multiplicative Schwarz Method for Variational Inequalities}, author={Lori Badea and Xuecheng Tai and Junping Wang}, journal={SIAM J. Numer. Anal.}, year={2003}, volume={41}, pages={1052-1073} }

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 overlappingâ€¦Â

## 70 Citations

### Convergence rate of a multiplicative Schwarz method for strongly nonlinear variational inequalities

- MathematicsAnalysis and Optimization of Differential Systems
- 2002

The convergence and estimate the error of a general algorithm for the minimization of non-quadratic functionals over a convex set in a reflexive Banach space is proved, and it is proved that the introduced assumption holds if the conveX set is defined by constraints on the function values almost everywhere in the domain.

### Convergence Rate of a Schwarz Multilevel Method for the Constrained Minimization of Nonquadratic Functionals

- Mathematics, Computer ScienceSIAM J. Numer. Anal.
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It is proved that the convergence of a subspace correction method applied to the constrained minimization of a functional in a general reflexive Banach space has been proved, provided that the convex set verifies a certain assumption.

### On the linear convergence of additive Schwarz methods for the p-Laplacian

- Mathematics, Computer ScienceArXiv
- 2022

This paper presents a new abstract convergence theory of additive Schwarz methods written in terms of a quasi-norm version of the Poincar'{e}--Friedrichs inequality, which is essential for deriving a semi-norm stable decomposition for a two-level domain decomposition setting.

### On the convergence of generalized Schwarz algorithms for solving obstacle problems with elliptic operators

- Computer Science, MathematicsMath. Methods Oper. Res.
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Compared with the classical Schwarz algorithms, the generalized Schwarz algorithms use Robin conditions with parameters as the transmission conditions on the interface boundaries, so the convergence rate can be speeded up by choosing Robin parameters properly.

### Pseudo-linear Convergence of an Additive Schwarz Method for Dual Total Variation Minimization

- Computer Science, MathematicsArXiv
- 2019

The $O(1/n)$-energy convergence of the proposed method is proven, and it is shown that such the particular value depends on the overlapping width $\delta$, and the proposedmethod becomes as efficient as linearly convergent algorithms if $\ delta$ is large.

### Multilevel Schwarz method for the minimization with constraints of non-quadratic functionals.

- Computer Science
- 2004

The main goal of this paper is to check up the dependence of this convergence rate on the mesh and overlapping parameters by numerical tests concerning the solution of the two-obstacle problem of a nonlinear elastic membrane.

### Global convergence rate of a standard multigrid method for variational inequalities

- Computer Science, Mathematics
- 2014

A multigrid algorithm for variational inequalities whose constraints are of the two-obstacle type and the method is introduced as a subspace correction algorithm in a reflexive Banach space, proving its global convergence and estimating the error making some assumptions.

### Convergence rate of some hybrid multigrid methods for variational inequalities

- Mathematics, Computer ScienceJ. Num. Math.
- 2015

This algorithm together with other three algorithms, which are combinations of additive or multiplicative iterations on levels with additive or multiplier 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.

### Multigrid methods for some quasi-variational inequalities

- Mathematics, Computer Science
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It is proved that the assumption made in the general convergence theory holds for the one-obstacle problems, and the convergence rate depending on the number of level meshes is written.

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