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.

翻译：本文针对具有粗糙且可能高度振荡系数的椭圆模型问题，提出了一种高阶多尺度方法。仅利用右端项的正则性即可获得高阶收敛速率，因此无需对系数、区域或精确解施加限制性假设。该方法遵循局部正交分解的思想，通过求解局部子域上的辅助问题，构建粗尺度问题自适应试探空间。具体而言，本文基于Maier [SIAM J. Numer. Anal. 59(2), 2021] 提出的策略。该方法的核心优势在于改进的局部化策略，可消除局部子域尺寸不足时误差随网格尺寸劣化的效应。我们给出了严格的先验误差分析，并通过系列数值实验验证了方法的性能。