A Nonconforming Finite Element Approximation for Optimal Control of an Obstacle Problem

2016 ◽  
Vol 16 (4) ◽  
pp. 653-666 ◽  
Author(s):  
Asha K. Dond ◽  
Thirupathi Gudi ◽  
Neela Nataraj
Keyword(s):  
Finite Element ◽  
Optimization Problems ◽  
Finite Element Approximation ◽  
The State ◽  
Element Approximation ◽  
Nonconforming Finite Element ◽  
Nonconforming Finite Elements ◽  
Adjoint Variables ◽  
Variational Discretization ◽  
And Control

AbstractThe article deals with the analysis of a nonconforming finite element method for the discretization of optimization problems governed by variational inequalities. The state and adjoint variables are discretized using Crouzeix–Raviart nonconforming finite elements, and the control is discretized using a variational discretization approach. Error estimates have been established for the state and control variables. The results of numerical experiments are presented.

Edge Patch-Wise Local Projection Stabilized Nonconforming FEM for the Oseen Problem

2019 ◽  
Vol 19 (2) ◽  
pp. 189-214 ◽  
Author(s):  
Rahul Biswas ◽  
Asha K. Dond ◽  
Thirupathi Gudi
Keyword(s):  
Finite Element ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Nonconforming Finite Element ◽  
Polynomial Space ◽  
Element Space ◽  
Piecewise Constant ◽  
Local Projection ◽  
Finite Element Space ◽  
Oseen Problem

AbstractIn finite element approximation of the Oseen problem, one needs to handle two major difficulties, namely, the lack of stability due to convection dominance and the incompatibility between the approximating finite element spaces for the velocity and the pressure. These difficulties are addressed in this article by using an edge patch-wise local projection (EPLP) stabilization technique. The article analyses the EPLP stabilized nonconforming finite element methods for the Oseen problem. For approximating the velocity, the lowest-order Crouzeix–Raviart (CR) nonconforming finite element space is considered; whereas for approximating the pressure, two discrete spaces are considered, namely, the piecewise constant polynomial space and the lowest-order CR finite element space. The proposed discrete weak formulation is a combination of the standard Galerkin method, EPLP stabilization and weakly imposed boundary condition by using Nitsche’s technique. The resulting bilinear form satisfies an inf-sup condition with respect to EPLP norm, which leads to the well-posedness of the discrete problem. A priori error analysis assures the optimal order of convergence in both the cases, that is, order one in the case of piecewise constant approximation and \frac{3}{2} in the case of CR-finite element approximation for pressure. The numerical experiments illustrate the theoretical findings.

AN EFFICIENT NUMERICAL METHOD FOR CAVITATION IN NONLINEAR ELASTICITY

2011 ◽  
Vol 21 (08) ◽  
pp. 1733-1760 ◽  
Author(s):  
XIANMIN XU ◽  
DUVAN HENAO
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Method ◽  
Nonlinear Elasticity ◽  
Elastic Energy ◽  
Finite Element Approximation ◽  
Void Coalescence ◽  
Element Approximation ◽  
Nonconforming Finite Element ◽  
First Time

This paper is concerned with the numerical computation of cavitation in nonlinear elasticity. The Crouzeix–Raviart nonconforming finite element method is shown to prevent the degeneration of the mesh provoked by the conventional finite element approximation of this problem. Upon the addition of a suitable stabilizing term to the elastic energy, the method is used to solve cavitation problems in both radially symmetric and non-radially symmetric settings. While the radially symmetric examples serve to illustrate the efficiency of the method, and for validation purposes, the experiments with non-centered and multiple cavities (carried out for the first time) yield novel observations of situations potentially leading to void coalescence.

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