Quasi-optimal convergence for an inexact AFEM

PEDRO MORIN

Concepción

Workshop; WONAPDE 2010, Third Chilean Workshop on Numerical Analysis of Partial Differential Equations; 2010

We analyze an adaptive finite element method (AFEM) which \emph{does not require} an exact computation of the \emph{Galerkin approximation} at each mesh. We propose to approximate the Galerkin solution $U_{k+1}$ at step $k+1$ of the adaptive process up to a tolerance dictated by the difference between $\tilde U_k$ and $\tilde U_{k+1}$. Here $\tilde U_k$ denotes the computed approximate solution at step $k$ of the adaptive iteration. The adaptive algorithm is guided by D\"orfler's marking strategy~\cite{Dorfler}, using the residual-type a posteriori error estimators for the \emph{inexact} (computed) solutions $\tilde U_k$. We prove a quasi-optimality result analogous to those of Stevenson~\cite{Stevenson} and Casc\'on et.\ al.~\cite{CKNS-quasi-opt}. This turns out to be a theoretical result of optimality for a very practical and realistic adaptive method, which is computationally cheaper than the usual AFEM. Several numerical tests show that a few iterations of the iterative linear solver are needed to obtain the desired accuracy in each step of the adaptive loop.