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 pos...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(2008)and equivalent to the total error defined by Cascón et.al.(2008).This contraction implies linear convergence of the discrete solutions to the exact solution in the usual H1 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 Cascón et.al.to a class of nonlinear problems which stem from strongly monotone and Lipschitz operators.展开更多
基金supported by CONICET through Grant PIP 112-200801-02182Universidad Nacional del Litoral through Grant CAI+D PI 062-312+1 种基金Agencia Nacional de Promoción Científica y Tecnológica,through grant PICT-2008-0622by Universidad Nacional de San Luis through Grant 22/F730-FCFMyN(Argentina).
文摘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(2008)and equivalent to the total error defined by Cascón et.al.(2008).This contraction implies linear convergence of the discrete solutions to the exact solution in the usual H1 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 Cascón et.al.to a class of nonlinear problems which stem from strongly monotone and Lipschitz operators.
