Results on unconditional convergence in the Maximum norm for ADI-type methods, such as the Douglas method, applied to the time integration of semilinear parabolic problems are quite difficult to get, mainly when the number of space dimensions $m$ is greater than two. Such a result is obtained here under quite general conditions on the PDE problem in case that time-independent Dirichlet boundary conditions are imposed. To get these bounds, a theorem that guarantees, in some sense, power-boundeness of the stability function independently of both the space and time resolutions is proved.

中文翻译：

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抛物线问题的ADI型方法的最大范数收敛

ADI型方法（例如道格拉斯方法）的最大范数的无条件收敛的结果很难应用于半线性抛物线问题的时间积分，主要是在空间维数$ m $大于2的情况下。在强加非时间依赖Dirichlet边界条件的情况下，在相当普通的条件下，在PDE问题上可以获得这样的结果。为了获得这些界限，证明了一个定理，从某种意义上说，该定理可以保证稳定函数的幂有界性，而与空间和时间分辨率无关。