%0 Journal Article %1 cascon2008quasioptimal %A Cascón, J. Manuel %A Kreuzer, Christian %A Nochetto, Ricardo H. %A Siebert, Kunibert G. %D 2008 %J SIAM Journal on Numerical Analysis %K imported vorlaeufig %N 5 %P 2524-2550 %R 10.1137/07069047X %T Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method %U http://dx.doi.org/10.1137/07069047X %V 46 %X We analyze the simplest and most standard adaptive finite ele- ment method (AFEM), with any polynomial degree, for general second order linear, symmetric elliptic operators. As it is customary in practice, AFEM marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property. We prove that AFEM is a contraction for the sum of energy error and scaled error estimator, be- tween two consecutive adaptive loops. This geometric decay is instrumental to derive optimal cardinality of AFEM. We show that AFEM yields a decay rate of energy error plus oscillation in terms of number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.

@article{cascon2008quasioptimal, abstract = {We analyze the simplest and most standard adaptive finite ele- ment method (AFEM), with any polynomial degree, for general second order linear, symmetric elliptic operators. As it is customary in practice, AFEM marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property. We prove that AFEM is a contraction for the sum of energy error and scaled error estimator, be- tween two consecutive adaptive loops. This geometric decay is instrumental to derive optimal cardinality of AFEM. We show that AFEM yields a decay rate of energy error plus oscillation in terms of number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.}, added-at = {2018-07-20T10:54:15.000+0200}, author = {Casc\'on, J. Manuel and Kreuzer, Christian and Nochetto, Ricardo H. and Siebert, Kunibert G.}, biburl = {https://puma.ub.uni-stuttgart.de/bibtex/255db640050e1e3a497ebfd72922bb090/mhartmann}, doi = {10.1137/07069047X}, interhash = {2a58bf695583dcb4f29acdc55cacb97a}, intrahash = {55db640050e1e3a497ebfd72922bb090}, journal = {SIAM Journal on Numerical Analysis}, keywords = {imported vorlaeufig}, number = 5, owner = {kohlsk}, pages = {2524-2550}, timestamp = {2018-07-20T08:54:15.000+0200}, title = {Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method}, url = {http://dx.doi.org/10.1137/07069047X}, volume = 46, year = 2008 }