A modified Ka\v{c}anov iteration scheme with application to quasilinear diffusion models

TL;DR

This paper introduces a modified Kačanov iteration with adaptive damping for solving nonlinear variational problems, expanding applicability to quasilinear diffusion models without requiring standard monotonicity conditions.

Contribution

The paper presents a new modified Kačanov scheme with adaptive damping, providing a broader convergence analysis applicable to more general quasilinear diffusion problems.

Findings

Convergence analysis under more general assumptions.

No need for standard monotonicity in diffusion coefficients.

Two adaptive strategies for damping parameter selection.

Abstract

The classical Ka\v{c}anov 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 Ka\v{c}anov 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.