In this paper, the unconditional stability and mass‐preserving splitting domain decomposition method (S‐DDM) for solving three‐dimensional parabolic equations is analyzed. At each time step level, three steps (x‐direction, y‐direction, and z‐direction) are proposed to compute the solutions on each sub‐domains. The interface fluxes are first predicted by the semi‐implicit flux schemes. Second, the interior solutions and fluxes are computed by the splitting implicit solution and flux coupled schemes. Last, we recompute the interface fluxes by the explicit schemes. Due to the introduced z‐directional splitting and domain decomposition, the analysis of stability and convergence is scarcely evident and quite difficult. By some mathematical technique and auxiliary lemmas, we prove strictly our scheme meet unconditional stability and give the error estimates in L²‐norm. Numerical experiments are presented to illustrate the theoretical analysis.

中文翻译：

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解三维抛物方程的无条件稳定性和保质量S-DDM方案

本文分析了求解三维抛物方程的无条件稳定性和保质量分裂域分解方法（S‐DDM）。在每个时间步级别，提出了三个步骤（x方向，y方向和z方向）来计算每个子域上的解。界面通量首先通过半隐式通量方案进行预测。其次，通过分裂隐式解和通量耦合方案来计算内部解和通量。最后，我们通过显式方案重新计算界面通量。由于引入了z方向拆分和域分解，稳定性和收敛性的分析几乎没有，而且非常困难。通过一些数学技术和辅助引理，我们严格证明了我们的方案满足无条件稳定性，并在L ²范数中给出了误差估计。数值实验表明了理论分析。