Week 7,8 (Nov 20,27)

Manel Cascon. Convergence and Optimality of an AFEM for General Second Order Linear Elliptic PDE

ABSTRACT: Despite the overwhelming computational evidence that AFEM lead to optimal meshes, the mathematical theory started very recently with Binev et al [1] and Stevenson [3]. Paper [1] discusses geometric properties of bisection which turn out to be crucial for optimality; The AFEM in [1]
includes a coarsening step which is not necessary in practise for linear elliptic PDE. Stevenson presents in [3] an algorithm with optimal complexity without coarsening, but it does not work with general linear elliptic operator and is unpractical.

In this work, we modify the Morin-Nochetto-Siebert algorithm [2] with a new procedure for oscillation reduction relative to error estimator. We prove that the resulting AFEM is a contraction for the total error between two consecutive adaptive loops. This strict error reduction property then leads to the optimality of AFEM, that is the total error decays in terms of number of degrees of freedom as predicted by the best approximation. This work is joint with R. H. Nochetto.

Bibliography

[1] P. Binev, W. Dahmen, and R. DeVore Adaptive finite element methods with convergence rate, Numer. Math., 97(2), pp.~219-268 (2004).

[2] P. Morin, R.H. Nochetto, and K.G. Siebert, Data Oscillation and Convergence of adaptive FEM. SIAM J. Numer. Anal. 38, pp.~466--488 (2000).

[3] R. Stevenson, Optimality of a standard adaptive finite element method Found. Comput. Math. Online. DOI 10.1007/s10208-005-0183-0, (2006).