Direct and iterative methods for numerical homogenization

Abstract

Elliptic problems with oscillating coefficients can be approximated up to arbitrary accuracy by using sufficiently fine meshes, i.e., by resolving the fine scale. Well-known multiscale finite elements (Henning et al. ESAIM: Math. Model. Numer. Anal. 48:1331-1349, 2014; Målqvist and Peterseim, Math. Comput. 83:2583-2603, 2014) can be regarded as direct numerical homogenization methods in the sense that they provide approximations of the corresponding (unfeasibly) large linear systems by much smaller systems while preserving the fine-grid discretization accuracy (model reduction). As an alternative, we present iterative numerical homogenization methods that provide approximations up to fine-grid discretization accuracy and discuss differences and commonalities.