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.

翻译：解决非线性变异问题的古典 Ka\v{c}anov 方案可以被解释为固定点迭代方法,通过解决每个步骤的线性问题来更新某一近似。 基于这一观察,我们引入了经修改的Ka\v{c}nov 方法,允许(调整)阻塞,从而在更一般性的假设和更广泛的应用下得出新的趋同分析。例如,在准线性扩散模型的具体背景下,我们的新办法不再要求非线性扩散系数的标准单一性条件维持下去。此外,我们提出了两种不同的适应战略,以实际选择有关的阻隔参数。