scholarly journals AN ADAPTIVE FINITE ELEMENT APPROXIMATION OF A GENERALIZED AMBROSIO–TORTORELLI FUNCTIONAL

2013 ◽  
Vol 23 (09) ◽  
pp. 1663-1697 ◽  
Author(s):  
SIOBHAN BURKE ◽  
CHRISTOPH ORTNER ◽  
ENDRE SÜLI
Keyword(s):  
Finite Element ◽  
Finite Element Approximation ◽  
Energy Functional ◽  
Adaptive Finite Element Method ◽  
Element Approximation ◽  
Adaptive Finite Element ◽  
Local Minimizers ◽  
Constrained Minimization Problem ◽  
Adaptive Finite Element Approximation ◽  
Bound Constrained Minimization

The Francfort–Marigo model of brittle fracture is posed in terms of the minimization of a highly irregular energy functional. A successful method for discretizing the model is to work with an approximation of the energy. In this work a generalized Ambrosio–Tortorelli functional is used. This leads to a bound-constrained minimization problem, which can be posed in terms of a variational inequality. We propose, analyze and implement an adaptive finite element method for computing (local) minimizers of the generalized functional.

scholarly journals Quasi-optimality of an Adaptive Finite Element Method for Cathodic Protection

2019 ◽  
Vol 53 (5) ◽  
pp. 1645-1665
Author(s):  
Guanglian Li ◽  
Yifeng Xu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Cathodic Protection ◽  
Finite Element Approximation ◽  
Adaptive Finite Element Method ◽  
Error Estimator ◽  
Element Approximation ◽  
Adaptive Finite Element ◽  
Energy Error ◽  
Element Method

In this work, we derive a reliable and efficient residual-typed error estimator for the finite element approximation of a 2D cathodic protection problem governed by a steady-state diffusion equation with a nonlinear boundary condition. We propose a standard adaptive finite element method involving the Dörfler marking and a minimal refinement without the interior node property. Furthermore, we establish the contraction property of this adaptive algorithm in terms of the sum of the energy error and the scaled estimator. This essentially allows for a quasi-optimal convergence rate in terms of the number of elements over the underlying triangulation. Numerical experiments are provided to confirm this quasi-optimality.

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