A modified Kačanov iteration scheme with application to quasilinear diffusion models

Heid, Pascal; Wihler, Thomas P.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 2, pp. 433-450

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.

Classification : 35J62, 47J25, 47H05, 47H10, 65J15, 65N12
Keywords: Quasilinear elliptic PDE, strongly monotone problems, fixed point iterations, Kačanov method, quasi-Newtonian fluids, shear-thickening fluids

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