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.

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.

A prioriestimates and optimal finite element approximation of the MHD flow in smooth domains

2018 ◽  
Vol 52 (1) ◽  
pp. 181-206 ◽  
Author(s):  
Yinnian He ◽  
Jun Zou
Keyword(s):  
Magnetic Field ◽  
Finite Element ◽  
Error Estimates ◽  
Mhd Flow ◽  
Finite Element Approximation ◽  
Optimal Error Estimates ◽  
Element Approximation ◽  
Optimal Error ◽  
Discrete Velocity ◽  
Mhd System

We study a finite element approximation of the initial-boundary value problem of the 3D incompressible magnetohydrodynamic (MHD) system under smooth domains and data. We first establish several important regularities anda prioriestimates for the velocity, pressure and magnetic field (u,p,B) of the MHD system under the assumption that ∇u∈L4(0,T;L2(Ω)3 × 3) and ∇ ×B∈L4(0,T;L2(Ω)3). Then we formulate a finite element approximation of the MHD flow. Finally, we derive the optimal error estimates of the discrete velocity and magnetic field in energy-norm and the discrete pressure inL2-norm, and the optimal error estimates of the discrete velocity and magnetic field inL2-norm by means of a novel negative-norm technique, without the help of the standard duality argument for the Navier-Stokes equations.

scholarly journals ANISOTROPIC QUASI-WILSON ELEMENT WITH CONFORMING FINITE ELEMENT APPROXIMATION FOR COUPLED CONTINUUM PIPE-FLOW/DARCY MODEL IN KARST AQUIFERS

2016 ◽  
Vol 21 (4) ◽  
pp. 431-449 ◽  
Author(s):  
Wei Liu ◽  
Jintao Cui
Keyword(s):  
Finite Element ◽  
Pipe Flow ◽  
Finite Element Approximation ◽  
Flow In Porous Media ◽  
Optimal Error Estimates ◽  
Element Approximation ◽  
Approximation Solution ◽  
Darcy Model ◽  
Conforming Finite Element ◽  
Wilson Element

This paper presents a numerical method for solving systems of partial differential equations describing flow in porous media with an embedded and inclined conduit pipe. This work considers a coupled continuum pipe-flow/Darcy model. The numerical schemes presented are based on combinations of the quasi-Wilson element on anisotropic mesh and the conforming finite element on regular mesh. The existence and uniqueness of the approximation solution are obtained. Optimal error estimates in both L2 and H1 norms are obtained independent of the regularity condition on the mesh. Numerical examples show the accuracy and efficiency of the proposed scheme.

A mixed-primal finite element approximation of a sedimentation–consolidation system

2016 ◽  
Vol 26 (05) ◽  
pp. 867-900 ◽  
Author(s):  
Mario Alvarez ◽  
Gabriel N. Gatica ◽  
Ricardo Ruiz-Baier
Keyword(s):  
Finite Element ◽  
Fixed Point ◽  
Volume Fraction ◽  
Variable Viscosity ◽  
Finite Element Approximation ◽  
Weak Form ◽  
Variational Problems ◽  
Optimal Error Estimates ◽  
Element Approximation ◽  
Coupled Flow

This paper is devoted to the mathematical and numerical analysis of a strongly coupled flow and transport system typically encountered in continuum-based models of sedimentation–consolidation processes. The model focuses on the steady-state regime of a solid–liquid suspension immersed in a viscous fluid within a permeable medium, and the governing equations consist in the Brinkman problem with variable viscosity, written in terms of Cauchy pseudo-stresses and bulk velocity of the mixture; coupled with a nonlinear advection — nonlinear diffusion equation describing the transport of the solids volume fraction. The variational formulation is based on an augmented mixed approach for the Brinkman problem and the usual primal weak form for the transport equation. Solvability of the coupled formulation is established by combining fixed point arguments, certain regularity assumptions, and some classical results concerning variational problems and Sobolev spaces. In turn, the resulting augmented mixed-primal Galerkin scheme employs Raviart–Thomas approximations of order [Formula: see text] for the stress and piecewise continuous polynomials of order [Formula: see text] for velocity and volume fraction, and its solvability is deduced by applying a fixed-point strategy as well. Then, suitable Strang-type inequalities are utilized to rigorously derive optimal error estimates in the natural norms. Finally, a few numerical tests illustrate the accuracy of the augmented mixed-primal finite element method, and the properties of the model.

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