In this research, we propose an online basis enrichment strategy within the framework of a recently developed constraint energy minimizing generalized multiscale discontinuous Galerkin method (CEM-GMsDGM). Combining the technique of oversampling, one makes use of the information of the current residuals to adaptively construct basis functions in the online stage to reduce the error of multiscale approximation. A complete analysis of the method is presented, which shows the proposed online enrichment leads to a fast convergence from multiscale approximation to the fine-scale solution. The error reduction can be made sufficiently large by suitably selecting oversampling regions and the number of oversampling layers. Further, the convergence rate of the enrichment algorithm depends on a factor of exponential decay regarding the number of oversampling layers and a user-defined parameter. Numerical results are provided to demonstrate the effectiveness and efficiency of the proposed online adaptive algorithm.

中文翻译：

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约束能量最小化的广义多尺度不连续Galerkin方法在线自适应算法

在这项研究中，我们在最近开发的约束能量最小化广义多尺度不连续伽勒金方法（CEM-GMsDGM）的框架内提出了一种在线基础富集策略。结合过采样技术，利用当前残差的信息在在线阶段自适应地构造基函数，以减少多尺度逼近的误差。提出了对该方法的完整分析，表明所提出的在线浓缩可导致从多尺度近似到精细尺度解决方案的快速收敛。通过适当地选择过采样区域和过采样层的数量，可以使误差减小足够大。进一步，富集算法的收敛速度取决于关于过采样层数和用户定义参数的指数衰减因子。数值结果表明了所提出的在线自适应算法的有效性和效率。