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

Eduardo M. Garau, Pedro Morin, Carlos Zuppa
2012 Numerical Mathematics: Theory, Methods and Applications  
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 [6] and equivalent to the total error defined by Cascón et al. [2] . This
more » ... al. [2] . 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 [6] and extend the theory from [2] to a class of nonlinear problems which stem from strongly monotone and Lipschitz operators.
doi:10.4208/nmtma.2012.m1023 fatcat:jqdhm2imizcopa6hvoisnkfs2a