Inexact Uzawa algorithms for variational inequalities of the second kind

In this paper we discuss inexact Uzawa algorithms and inexact non-linear Uzawa algorithms to solve discretized variational inequalities of the second kind. We prove convergence results for the algorithms. Numerical examples are included to show the effectiveness of the algorithms.

[1]  R. Glowinski,et al.  Numerical Analysis of Variational Inequalities , 1981 .

[2]  R. Bank,et al.  A class of iterative methods for solving saddle point problems , 1989 .

[3]  Xiao-Liang Cheng,et al.  On the Nonlinear Inexact Uzawa Algorithm for Saddle-Point Problems , 2000, SIAM J. Numer. Anal..

[4]  Jun Zou,et al.  An Iterative Method with Variable Relaxation Parameters for Saddle-Point Problems , 2001, SIAM J. Matrix Anal. Appl..

[5]  P. Panagiotopoulos Inequality problems in mechanics and applications , 1985 .

[6]  Jean E. Roberts,et al.  Mixed and hybrid finite element methods , 1987 .

[7]  R. Glowinski,et al.  Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics , 1987 .

[8]  Apostol T. Vassilev,et al.  Analysis of the Inexact Uzawa Algorithm for Saddle Point Problems , 1997 .

[9]  J. Schöberl Efficient contact solvers based on domain decomposition techniques , 2001 .

[10]  F. Thomasset Finite element methods for Navier-Stokes equations , 1980 .

[11]  D. Kinderlehrer,et al.  An introduction to variational inequalities and their applications , 1980 .

[12]  W. Han,et al.  Plasticity: Mathematical Theory and Numerical Analysis , 1999 .

[13]  B. V. Dean,et al.  Studies in Linear and Non-Linear Programming. , 1959 .

[14]  C. Baiocchi,et al.  Variational and quasivariational inequalities: Applications to free boundary problems , 1983 .

[15]  A. Friedman Variational principles and free-boundary problems , 1982 .

[16]  W. Queck The convergence factor of preconditioned algorithms of the Arrow-Hurwicz type , 1989 .

[17]  R. Hoppe,et al.  Adaptive multilevel methods for obstacle problems , 1994 .

[18]  J. Rodrigues Obstacle Problems in Mathematical Physics , 1987 .

[19]  J. Oden,et al.  Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods , 1987 .

[20]  R. Kornhuber Monotone multigrid methods for elliptic variational inequalities II , 1996 .

[21]  J. Lions,et al.  Inequalities in mechanics and physics , 1976 .

[22]  R. Glowinski,et al.  Numerical Methods for Nonlinear Variational Problems , 1985 .

[23]  R. Hoppe Multigrid Algorithms for Variational Inequalities , 1987 .

[24]  J. Pasciak,et al.  A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems , 1988 .

[25]  Joseph E. Pasciak,et al.  Uzawa type algorithms for nonsymmetric saddle point problems , 2000, Math. Comput..

[26]  R. Kornhuber Monotone multigrid methods for elliptic variational inequalities I , 1994 .

[27]  Ragnar Winther,et al.  A Preconditioned Iterative Method for Saddlepoint Problems , 1992, SIAM J. Matrix Anal. Appl..

[28]  W. Han,et al.  Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity , 2002 .

[29]  G. Golub,et al.  Inexact and preconditioned Uzawa algorithms for saddle point problems , 1994 .