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.

翻译：用于半线性抛物线问题时间整合的道格拉斯方法等ADI型方法最大标准无条件趋同的结果很难取得,主要是当空间维度超过2百万美元时,如果强加时间独立的迪里赫莱边界条件,这种结果是在PDE问题相当普遍的条件下取得的,为了获得这些界限,在某种意义上保证独立于空间和时间决议的稳定功能的权力约束的理论得到了证明。