  In this work, we propose a high-order multiscale method for an elliptic model
problem with rough and possibly highly oscillatory coefficients. Convergence
rates of higher order are obtained using the regularity of the right-hand side
only. Hence, no restrictive assumptions on the coefficient, the domain, or the
exact solution are required. In the spirit of the Localized Orthogonal
Decomposition, the method constructs coarse problem-adapted ansatz spaces by
solving auxiliary problems on local subdomains. More precisely, our approach is
based on the strategy presented by Maier [SIAM J. Numer. Anal. 59(2), 2021].
The unique selling point of the proposed method is an improved localization
strategy curing the effect of deteriorating errors with respect to the mesh
size when the local subdomains are not large enough. We present a rigorous a
priori error analysis and demonstrate the performance of the method in a series
of numerical experiments.
