Numerical homogenization is a methodology for the computational solution of multiscale partial differential equations. It aims at reducing complex large-scale problems to simplified numerical models valid on some target scale of interest, thereby accounting for the impact of features on smaller scales that are otherwise not resolved. While constructive approaches in the mathematical theory of homogenization are restricted to problems with a clear scale separation, modern numerical homogenization methods can accurately handle problems with a continuum of scales. This paper reviews such approaches embedded in a historical context and provides a unified variational framework for their design and numerical analysis. Apart from prototypical elliptic model problems, the class of partial differential equations covered here includes wave scattering in heterogeneous media and serves as a template for more general multi-physics problems.

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超越尺度分离的数值均质化

数值均匀化是一种计算多尺度偏微分方程的方法。它旨在将复杂的大规模问题简化为在某些感兴趣的目标尺度上有效的简化数值模型，从而考虑特征对较小尺度的影响，否则这些问题无法解决。虽然均匀化数学理论中的建设性方法仅限于具有明确尺度分离的问题，但现代数值均匀化方法可以准确地处理具有连续尺度的问题。本文回顾了嵌入在历史背景中的这些方法，并为它们的设计和数值分析提供了一个统一的变分框架。除了原型椭圆模型问题，