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

EDUARDO MARIO GARAU; PEDRO MORIN; CARLOS ZUPPA

NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS

GLOBAL SCIENCE PRESS

Lugar: Hong Kong; Año: 2012 vol. 5 p. 131 - 156

1004-8979

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 Dörfler´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 and equivalent to the total error defined by Cascón, Kreuzer, Nochetto and Siebert. 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 the paper by Diening and Kreuzer and extend the theory from Cascón et. al. to a class of nonlinear problems which stem from strongly monotone and Lipschitz operators.