Numerical homogenization beyond scale separation

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.