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. The latter method is a now classical, finite element type method that performs a Galerkin approximation on a problem-dependent basis set, itself pre-computed 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.

翻译：本文研究采用多尺度有限元方法对具有高度振荡系数且可能以平流项为主导的平流-扩散方程进行数值逼近。该方法是一种经典的有限元类方法，在依赖于问题本身的基函数集上进行伽辽金逼近，这些基函数本身在离线阶段预先计算。本文采用局部解析扩散项和平流项的基函数实现该方法。提出了添加气泡函数及可能具有弱单元间连续性的变体形式。通过理论论证和全面的数值实验，研究并比较了各方法的稳定性与精度。所构建的最佳方法被证明适用于扩散主导与平流主导两种状态，且无需依赖需人工调整的辅助稳定参数。