Abstract In the numerical solution of elliptic equations, multiscale methods typically involve two steps: the solution of families of local solutions or multiscale basis functions (an embarrassingly parallel task) associated with subdomains of a domain decomposition of the original domain, followed by the solution of a global problem. In the solution of multiphase flow problems approximated by an operator splitting method one has to solve an elliptic equation every time step of a simulation, that would require that all multiscale basis functions be recomputed. In this work, we focus on the development of a novel method that replaces a full update of local solutions by reusing multiscale basis functions that are computed at an earlier time of a simulation. The procedure is based on classical perturbation theory. It can take advantage of both an offline stage (where multiscale basis functions are computed at the initial time of a simulation) as well as of a good initial guess for velocity and pressure. The formulation of the method is carefully explained and several numerical studies are presented and discussed. They provide an indication that the proposed procedure can be of value in speeding-up the solution of multiphase flow problems by multiscale methods.

中文翻译：

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二阶椭圆方程的多尺度微扰法

摘要 在椭圆方程的数值求解中，多尺度方法通常包括两个步骤：局部解族或多尺度基函数族的求解（一个令人尴尬的并行任务）与原始域的域分解的子域相关，然后是解一个全球性的问题。在求解由算子分裂方法近似的多相流问题时，必须在模拟的每个时间步长求解椭圆方程，这将需要重新计算所有多尺度基函数。在这项工作中，我们专注于开发一种新方法，该方法通过重用在模拟早期计算的多尺度基函数来代替局部解的完全更新。该过程基于经典微扰理论。它可以利用离线阶段（在模拟的初始时间计算多尺度基函数）以及对速度和压力的良好初始猜测。仔细解释了该方法的公式，并介绍和讨论了一些数值研究。它们表明，所提出的程序在通过多尺度方法加速多相流问题的求解方面具有价值。