Numerical approximation usually aims at modifications of standard finite element approximations of partial differential equations with highly oscillatory coefficients that preserve the accuracy known in the smooth case. Using classical homogenization as a guideline, these modifications are obtained from local auxiliary problems (Abdulle et al., 2012; Efendiev and Hou, 2009; Hughes et al., 1998). The error analysis for these kinds of methods is typically restricted to coefficients with separated scales and often requires periodicity (Abdulle, 2011; Abdulle et al., 2012; Hou et al., 1999). These restrictions were overcome in a recent paper by Malqvist and Peterseim (2014) that provides quasioptimal energy and L~2 error estimates without any additional assumptions on periodicity and scale separation (Henning et al., 2014; Malqvist and Peterseim, 2014). While their approach relies on (approximate) orthogonal subspace decomposition, alternative decompositions into a coarse space and local fine-grid spaces associated with low and high frequencies has been recently considered by Kornhuber and Yserentant (2015). Here, we review these two decomposition techniques providing direct (Malqvist and Peterseim, 2014) and iterative methods (Kornhuber and Yserentant, 2015) for numerical homogenization in order to better understand conceptual similarities and differences. We also illustrate the performance of the iterative variant by first numerical experiments in d = 3 space dimensions.
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