Quasi-Optimal Convergence Rate of an AFEM for Quasi-Linear Problems of Monotone Type

We prove the quasi-optimal convergence of a standard adaptive finite element method (AFEM) for a class of nonlinear elliptic second-order equations of monotone type. The adaptive algorithm is based on residual-type a posteriori error estimators and Dorfler's strategy is assumed for marking. We first prove a contraction property for a suitable definition of total error, analogous to the one used by Diening and Kreuzer (2008) and equivalent to the total error defined by Cascon et. al. (2008). This contraction implies linear convergence of the discrete solutions to the exact solution in the usual H-1 Sobolev norm. Secondly, we use this contraction to derive the optimal complexity of the AFEM. The results are based on ideas from Diening and Kreuzer and extend the theory from Cascon et. al. to a class of nonlinear problems which stem from strongly monotone and Lipschitz operators.