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

Eduardo M. Garau, Pedro Morin & Carlos Zuppa

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 D"orfler'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 Casc'on 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 Casc'on 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

  • AMS Subject Headings

35J62 65N30 65N12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-5-131, author = {Eduardo M. Garau, Pedro Morin and Carlos Zuppa}, 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 D"orfler'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 Casc'on 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 Casc'on 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 AU - Eduardo M. Garau, Pedro Morin & Carlos Zuppa JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 131 EP - 156 PY - 2012 DA - 2012/05 SN - 5 DO - http://dor.org/10.4208/nmtma.2012.m1023 UR - https://global-sci.org/intro/article_detail/nmtma/5932.html KW - quasilinear elliptic equations KW - adaptive finite element methods KW - 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 D"orfler'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 Casc'on 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 Casc'on et. al.  to a class of nonlinear problems which stem from strongly monotone and Lipschitz operators.

Eduardo M. Garau, Pedro Morin & Carlos Zuppa. (1970). 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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