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| HAL: hal-00410471, version 1 |
| DOI: 10.1016/j.cma.2010.03.024 |
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| Computer Methods in Applied Mechanics and Engineering 200, 37-40 (2011) 2782-2795 |
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| Guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems |
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| Linda El Alaoui 1Alexandre Ern 2 |
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| (2011-09) |
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| We derive a posteriori error estimates for a class of second-order monotone quasi-linear diffusion-type problems approximated by piecewise affine, continuous finite elements. Our estimates yield a guaranteed and fully computable upper bound on the error measured by the dual norm of the residual, as well as a global error lower bound, up to a generic constant independent of the nonlinear operator. They are thus fully robust with respect to the nonlinearity, thanks to the choice of the error measure. They are also locally efficient, albeit in a different norm, and hence suitable for adaptive mesh refinement. Moreover, they allow to distinguish, estimate separately, and compare the discretization and linearization errors. Hence, the iterative (Newton--Raphson, quasi-Newton) linearization can be stopped whenever the linearization error drops to the level at which it does not affect significantly the overall error. This can lead to important computational savings, as performing an excessive number of unnecessary linearization iterations can be avoided. Numerical experiments for the $p$-Laplacian illustrate the theoretical developments. |
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| 1: | Laboratoire Analyse, Géométrie et Application (LAGA) |
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CNRS : UMR7539 – Université Paris XIII - Paris Nord – Université Paris VIII - Vincennes Saint-Denis |
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| 2: | Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique (CERMICS) |
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INRIA – Ecole des Ponts ParisTech |
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| 3: | Laboratoire Jacques-Louis Lions (LJLL) |
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CNRS : UMR7598 – Université Pierre et Marie Curie [UPMC] - Paris VI |
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| Subject | : | Mathematics/Numerical Analysis |
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| a posteriori error estimate – monotone nonlinear problem – linearization – balancing errors – guaranteed upper bound – robustness |
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| hal-00410471, version 1 | |
| http://hal.archives-ouvertes.fr/hal-00410471 | |
| oai:hal.archives-ouvertes.fr:hal-00410471 | |
| From: Martin Vohralík | |
| Submitted on: Thursday, 20 August 2009 21:06:15 | |
| Updated on: Thursday, 22 December 2011 11:07:42 | |
