Abstract In numerical solving boundary value problems for parabolic equations, two- or three-level implicit schemes are in common use. Their computational implementation is based on solving a discrete elliptic problem at a new time level. For this purpose, various iterative methods are applied to multidimensional problems evaluating an approximate solution with some error. It is necessary to ensure that these errors do not violate the stability of the approximate solution, i.e., the approximate solution must converge to the exact one. In the present paper, these questions are investigated in numerical solving a Cauchy problem for a linear evolutionary equation of first order, which is considered in a finite-dimensional Hilbert space. The study is based on the general theory of stability (well-posedness) of operator-difference schemes developed by Samarskii. The iterative methods used in the study are considered from the same general positions.

中文翻译：

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不完全迭代隐式方案

摘要 在数值求解抛物型方程边值问题中，普遍使用两级或三级隐式格式。他们的计算实现基于在新的时间级别解决离散椭圆问题。为此，将各种迭代方法应用于多维问题，评估具有一定误差的近似解。必须确保这些误差不违反近似解的稳定性，即近似解必须收敛到精确解。在本文中，这些问题在数值求解一阶线性演化方程的柯西问题中进行了研究，该方程在有限维希尔伯特空间中被考虑。该研究基于 Samarskii 开发的算子差分方案的稳定性（适定性）一般理论。研究中使用的迭代方法是从相同的一般立场考虑的。