Volume 56, Number 2, March-April 2022
|Page(s)||433 - 450|
|Published online||18 February 2022|
A modified Kačanov iteration scheme with application to quasilinear diffusion models
Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
2 Mathematics Institute, University of Bern, CH-3012 Bern, Switzerland
* Corresponding author: email@example.com
Accepted: 13 January 2022
The classical Kačanov scheme for the solution of nonlinear variational problems can be interpreted as a fixed point iteration method that updates a given approximation by solving a linear problem in each step. Based on this observation, we introduce a modified Kačanov method, which allows for (adaptive) damping, and, thereby, to derive a new convergence analysis under more general assumptions and for a wider range of applications. For instance, in the specific context of quasilinear diffusion models, our new approach does no longer require a standard monotonicity condition on the nonlinear diffusion coefficient to hold. Moreover, we propose two different adaptive strategies for the practical selection of the damping parameters involved.
Mathematics Subject Classification: 35J62 / 47J25 / 47H05 / 47H10 / 65J15 / 65N12
Key words: Quasilinear elliptic PDE / strongly monotone problems / fixed point iterations / Kačanov method / quasi-Newtonian fluids / shear-thickening fluids
© The authors. Published by EDP Sciences, SMAI 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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