Quasi-optimal convergence rate of an AFEM for quasi-linear problems

TL;DR

This paper establishes the quasi-optimal convergence rate of an adaptive finite element method for nonlinear elliptic equations, demonstrating linear convergence and optimal resource utilization.

Contribution

It proves the quasi-optimal convergence of AFEM for nonlinear elliptic problems using residual error estimators and contraction analysis.

Findings

Proves linear convergence of AFEM for nonlinear elliptic equations.

Establishes the optimal cardinality of the adaptive algorithm.

Demonstrates quasi-optimal convergence rates.

Abstract

We prove the quasi-optimal convergence of a standard adaptive finite element method (AFEM) for nonlinear elliptic second-order equations of monotone type. The adaptive algorithm is based on residual-type a posteriori error estimators and D\"orfler's strategy is assumed for marking. We first prove a contraction property for a suitable definition of total error, which is equivalent to the total error as defined by Casc\'on et al. (in SIAM J. Numer. Anal. 46 (2008), 2524--2550), and implies linear convergence of the algorithm. Secondly, we use this contraction to derive the optimal cardinality of the AFEM.