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Quasi-Optimal Convergence Rate of an AFEM for Quasi-Linear Problems of Monotone Type

Published online by Cambridge University Press:  28 May 2015

Eduardo M. Garau ,
Pedro Morin  and
Carlos Zuppa
Eduardo M. Garau*
Affiliation:
Departamento de Matemática, Facultad de Ingeniería Química and Instituto de Matemática Aplicada del Litoral (Universidad Nacional del Litoral and Consejo Nacional de Investigaciones Científicas y Técnicas). Güemes 3450. S3000GLN Santa Fe, Argentina
Pedro Morin*
Affiliation:
Departamento de Matemática, Facultad de Ingeniería Química and Instituto de Matemática Aplicada del Litoral (Universidad Nacional del Litoral and Consejo Nacional de Investigaciones Científicas y Técnicas). Güemes 3450. S3000GLN Santa Fe, Argentina
Carlos Zuppa*
Affiliation:
Departamento de Matemática, Facultad de Ciencias Físico Matemáticas y Naturales, Universidad Nacional de San Luis, Chacabuco 918, D5700BWT San Luis, Argentina
*
Corresponding author.Email address:egarau@santafe-conicet.gov.ar
Corresponding author.Email address:pmorin@santafe-conicet.gov.ar
Corresponding author.Email address:zuppa@unsl.edu.ar
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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 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.

Keywords

35J6265N3065N12quasilinear elliptic equationsadaptive finite element methodsoptimality
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 5 , Issue 2 , May 2012 , pp. 131 - 156
Copyright
Copyright © Global Science Press Limited 2012

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