Volume 5, Issue 2
Quasi-Optimal Convergence Rate of an AFEM for Quasi-Linear Problems of Monotone Type

Numer. Math. Theor. Meth. Appl., 5 (2012), pp. 131-156.

Published online: 2012-05

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• Abstract

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 $H^1$ 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 Cascón et. al. to a class of nonlinear problems which stem from strongly monotone and Lipschitz operators.

• Keywords

Quasilinear elliptic equations, adaptive finite element methods, optimality.

35J62, 65N30, 65N12

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@Article{NMTMA-5-131, author = {}, title = {Quasi-Optimal Convergence Rate of an AFEM for Quasi-Linear Problems of Monotone Type}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2012}, volume = {5}, number = {2}, pages = {131--156}, abstract = {

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 $H^1$ 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 Cascón et. al. to a class of nonlinear problems which stem from strongly monotone and Lipschitz operators.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2012.m1023}, url = {http://global-sci.org/intro/article_detail/nmtma/5932.html} }
TY - JOUR T1 - Quasi-Optimal Convergence Rate of an AFEM for Quasi-Linear Problems of Monotone Type JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 131 EP - 156 PY - 2012 DA - 2012/05 SN - 5 DO - http://doi.org/10.4208/nmtma.2012.m1023 UR - https://global-sci.org/intro/article_detail/nmtma/5932.html KW - Quasilinear elliptic equations, adaptive finite element methods, optimality. AB -

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 $H^1$ 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 Cascón et. al. to a class of nonlinear problems which stem from strongly monotone and Lipschitz operators.

Eduardo M. Garau, Pedro Morin & Carlos Zuppa. (2020). Quasi-Optimal Convergence Rate of an AFEM for Quasi-Linear Problems of Monotone Type. Numerical Mathematics: Theory, Methods and Applications. 5 (2). 131-156. doi:10.4208/nmtma.2012.m1023
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