Quasi-optimal convergence rate of an AFEM for quasi-linear problems
It provides theoretical justification for the efficiency of adaptive methods in nonlinear settings, which is important for computational scientists and engineers solving such problems.
This paper proves that a standard adaptive finite element method achieves quasi-optimal convergence rates for a class of nonlinear elliptic equations, extending results previously known only for linear problems.
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 Dörfler'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 Cascón 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.