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The classical Ka\v{c}anov scheme for the solution of nonlinear variationalproblems can be interpreted as a fixed point iteration method that updates agiven approximation by solving a linear problem in each step. Based on thisobservation, we introduce a modified Ka\v{c}anov method, which allows for(adaptive) damping, and, thereby, to derive a new convergence analysis undermore general assumptions and for a wider range of applications. For instance,in the specific context of quasilinear diffusion models, our new approach doesno longer require a standard monotonicity condition on the nonlinear diffusioncoefficient to hold. Moreover, we propose two different adaptive strategies forthe practical selection of the damping parameters involved.

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