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    STUDIA MATHEMATICA - Issue no. 2 / 2011  
         
  Article:   REMARKS ON THE STATE OF THE ART OF A POSTERIORI ERROR CONTROL OF ELLIPTIC PDES IN ENERGY NORMS IN PRACTISE.

Authors:  .
 
       
         
  Abstract:  

Five classes of up to 9 a posteriori error estimators compete in three second-order model problems, namely the conforming and nonconforming first-order approximation of the Poisson-Problem plus some conforming obstacle problem. Our numerical results provide sufficient evidence that guaranteed error control in the energy norm is indeed possible with efficiency indices between one and three. The five classes of error estimator consist of the standard residual-based error estimators, averaging error estimators, equilibration error estimators, e.g. the ones of Braess or Luce and Wohlmuth, least-square error estimators and the localisation error estimator of Carstensen and Funken. For the error control for obstacle problems, Braess considers Lagrange multipliers and some resulting auxiliary equation to view the a posteriori error control of the error in the obstacle problem as computable terms plus errors and residuals in the auxiliary equation. Hence all the former a posteriori error estimators apply to this benchmark as well and lead to surprisingly accurate guaranteed upper error bounds. This approach allows an extension to more general boundary conditions and a discussion of efficiency for the affine benchmark examples. The Luce-Wohlmuth and the leastsquare error estimators win the competition in several computational benchmark problems. Novel equilibration of nonconsistency residuals and novel conforming averaging error estimators win the competition for Crouzeix-Raviart nonconforming finite element methods. Furthermore, accurate error control is slightly more expensive but pays off in all applications under consideration while adaptive mesh-refinement is sufficiently pleasant as accurate when based on explicit residual-based error estimates.

Mathematics Subject Classification (2010): 65N30, 65R20, 73C50.

Keywords: A posteriori error estimators, finite element methods.

 
         
     
         
         
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