On the convergence of the Bermúdez-Moreno algorithm with constant parameters

Bermúdez-Moreno [5] presents a duality numerical algorithm for solving variational inequalities of the second kind. The performance of this algorithm strongly depends on the choice of two constant parameters. Assuming a further hypothesis of the inf-suptype, we present here a convergence theorem that improves on the one presented in [5]: we prove that the convergence is linear, and we give the expression of the asymptotic error constant and the explicit form of the optimal parameters, as a function of some constants related to the variational inequality. Finally, we present some numerical examples that confirm the theoretical results.

[1]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[2]  A. Bermúdez,et al.  Numerical solution of cavitation problems in lubrication , 1989 .

[3]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[4]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[5]  J. Durany,et al.  Numerical computation of free boundary problems in elastohydrodynamic lubrication , 1996 .

[6]  H. Brezis.  Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert , 1973 .

[7]  A. Pazy,et al.  On the asymptotic behavior of iterates of nonexpansive mappings in Hilbert space , 1977 .

[8]  M. A. Vilar,et al.  Solving Shallow Water Equations by a Mixed Implicit Finite Element Method , 1991 .

[9]  Carlos Parés,et al.  Improvement and generalization of a finite element shallow-water solver to multi-layer systems , 1999 .

[10]  Olivier Pironneau.  Méthodes des éléments finis pour les fluides , 1989 .

[11]  Numerical solution of steady-state flow through a porous dam , 1988 .

[12]  R. Glowinski.  Lectures on Numerical Methods for Non-Linear Variational Problems , 1981 .

[13]  A. Bermúdez,et al.  Duality methods for solving variational inequalities , 1981 .

[14]  Carlos Par,et al.  Duality methods with an automatic choice of parameters Application to shallow water equations in conservative form , 2001 .

[15]  I. Babuska.  The finite element method with Lagrangian multipliers , 1973 .