scholarly journals A New Approach to Asymptotic Behavior for a Finite Element Approximation in Parabolic Variational Inequalities

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Salah Boulaaras ◽  
Mohamed Haiour
Keyword(s):  
Finite Element ◽  
Asymptotic Behavior ◽  
Variational Inequalities ◽  
Finite Element Approximation ◽  
Uniform Norm ◽  
Element Approximation ◽  
New Approach ◽  
Elliptic Variational Inequalities ◽  
Energy Behavior ◽  
Spatial Approximation

The paper deals with the theta time scheme combined with a finite element spatial approximation of parabolic variational inequalities. The parabolic variational inequalities are transformed into noncoercive elliptic variational inequalities. A simple result to time energy behavior is proved, and a new iterative discrete algorithm is proposed to show the existence and uniqueness. Moreover, its convergence is established. Furthermore, a simple proof to asymptotic behavior in uniform norm is given.

scholarly journals Finite Element Approximation of Variational Inequalities: An Algorithmic Approach

2018 ◽  
Vol 22 (2) ◽  
pp. 114
Author(s):  
Messaoud Boulbrachene
Keyword(s):  
Finite Element ◽  
Variational Inequalities ◽  
Finite Element Approximation ◽  
Optimal Error Estimates ◽  
Element Approximation ◽  
True Solution ◽  
Algorithmic Approach ◽  
Elliptic Variational Inequalities ◽  
Standard Finite Element Method

In this paper, we introduce a new method to analyze the convergence of the standard finite element method for elliptic variational inequalities with noncoercive operators (VI). The method consists of combining the so-called Bensoussan-Lions algorithm with the characterization of the solution, in both the continuous and discrete contexts, as fixed point of contraction. Optimal error estimates are then derived, first between the continuous algorithm and its finite element counterpart, and then between the true solution and the approximate solution.

New Mixed Finite Element Formulations for Transient and Steady State Creep Problems

1989 ◽  
Vol 42 (11S) ◽  
pp. S150-S156
Author(s):  
Abimael F. D. Loula ◽  
Joa˜o Nisan C. Guerreiro
Keyword(s):  
Finite Element ◽  
Steady State ◽  
Mixed Finite Element ◽  
Finite Element Approximation ◽  
Steady State Creep ◽  
Element Approximation ◽  
New Approach ◽  
Finite Element Approximations ◽  
Transient Problems ◽  
Transient And Steady State

We apply the mixed Petrov–Galerkin formulation to construct finite element approximations for transient and steady-state creep problems. With the new approach we recover stability, convergence, and accuracy of some Galerkin unstable approximations. We also present the main results on the numerical analysis and error estimates of the proposed finite element approximation for the steady problem, and discuss the asymptotic behavior of the continuum and discrete transient problems.

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