11th GAMM-Workshop on

Multigrid and Hierarchic Solution Techniques

 

     
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  A. Almendral  
  M. Bader  
  R. Bank  
  M. Bebendorf  
  S. Beuchler  
  D. Braess  
  C. Douglas  
  L. Grasedyck  
  B. Khoromskij  
  R. Kornhuber  
  B. Krukier  
  U. Langer  
  C. Oosterlee  
  G. Pöplau  
  A. Reusken  
  J. Schöberl  
  M.A. Schweitzer  
  S. Serra Capizzano  
  B. Seynaeve  
  D. Smits  
  O. Steinbach  
  R. Stevenson  
  M. Wabro  
  R. Wienands  
 
     
  Rob Stevenson : An optimal adaptive finite element method


We present an adaptive finite element method for solving second order elliptic equations which is (quasi-)optimal in the following sense: If the solution is such that for some s>0, the errors in energy norm of the best continuous piecewise linear approximations subordinate to any partition with N triangles are O(N^{-s}), then given a tol>0, the adaptive method produces an approximation with an error less than tol subordinate to a partition with O(tol^{-1/s}) triangles, taking only O(tol^{-1/s})operations. Our method is based on ideas from [Binev, Dahmen, DeVore '02], in which a coarsening routine was added to the method from [Morin, Nochetto, Siebert '00]. Differences are that we employ non-conforming partitions, our coarsening routine is based on a transformation to a wavelet basis, all our results are valid uniformly in the size of possible jumps of the diffusion coefficients, and that we allow more general right-hand sides. All tolerances in our adaptive method depend on a posteriori estimate of the current error instead an a priori one, which can be expected to give quantitative advantages.
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