An Optimal Adaptive Finite Element Method for an Obstacle Problem

2015 ◽  
Vol 15 (3) ◽  
pp. 259-277 ◽  
Author(s):  
Carsten Carstensen ◽  
Jun Hu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Obstacle Problem ◽  
Posteriori Error ◽  
Adaptive Finite Element Method ◽  
Error Estimator ◽  
A Posteriori ◽  
Optimal Complexity ◽  
A Posteriori Error ◽  
Element Method

AbstractThis paper provides a refined a posteriori error control for the obstacle problem with an affine obstacle which allows for a proof of optimal complexity of an adaptive algorithm. This is the first adaptive mesh-refining finite element method known to be of optimal complexity for some variational inequality. The result holds for first-order conforming finite element methods in any spacial dimension based on shape-regular triangulation into simplices for an affine obstacle. The key contribution is the discrete reliability of the a posteriori error estimator from [Numer. Math. 107 (2007), 455–471] in an edge-oriented modification which circumvents the difficulties caused by the non-existence of a positive second-order approximation [Math. Comp. 71 (2002), 1405–1419].

A Reliable Residual Based A Posteriori Error Estimator for a Quadratic Finite Element Method for the Elliptic Obstacle Problem

2015 ◽  
Vol 15 (2) ◽  
pp. 145-160 ◽  
Author(s):  
Thirupathi Gudi ◽  
Kamana Porwal
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Lagrange Multiplier ◽  
Obstacle Problem ◽  
Posteriori Error ◽  
Optimal Order ◽  
Error Estimator ◽  
A Posteriori ◽  
Discrete Solution ◽  
Element Method

AbstractA residual based a posteriori error estimator is derived for a quadratic finite element method (FEM) for the elliptic obstacle problem. The error estimator involves various residuals consisting of the data of the problem, discrete solution and a Lagrange multiplier related to the obstacle constraint. The choice of the discrete Lagrange multiplier yields an error estimator that is comparable with the error estimator in the case of linear FEM. Further, an a priori error estimate is derived to show that the discrete Lagrange multiplier converges at the same rate as that of the discrete solution of the obstacle problem. The numerical experiments of adaptive FEM show optimal order convergence. This demonstrates that the quadratic FEM for obstacle problem exhibits optimal performance.

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