We study the numerical approximation of advection–diffusion equations with highly oscillatory coefficients and possibly dominant advection terms by means of the Multiscale Finite Element Method (MsFEM). The latter method is a now classical, finite element type method that performs a Galerkin approximation on a problem-dependent basis set, itself precomputed in an offline stage. The approach is implemented here using basis functions that locally resolve both the diffusion and the advection terms. Variants with additional bubble functions and possibly weak inter-element continuity are proposed. Some theoretical arguments and a comprehensive set of numerical experiments allow to investigate and compare the stability and the accuracy of the approaches. The best approach constructed is shown to be adequate for both the diffusion- and advection-dominated regimes, and does not rely on an auxiliary stabilization parameter that would have to be properly adjusted.

中文翻译：

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MsFEM 用于异质介质中以平流为主的问题：通过不一致性变体进行稳定


我们通过多尺度有限元法 （MsFEM） 研究了具有高振荡系数和可能占主导地位的平流项的对流-扩散方程的数值近似。后一种方法是现在经典的有限元类型方法，它在与问题相关的基集上执行 Galerkin 近似，其本身是在离线阶段预先计算的。该方法在这里使用局部解析扩散项和平流项的基函数实现。提出了具有附加气泡函数和可能较弱的元件间连续性的变体。一些理论论点和一套全面的数值实验允许研究和比较方法的稳定性和准确性。事实证明，构建的最佳方法对于以扩散和平流为主的状况都足够，并且不依赖于必须适当调整的辅助稳定参数。