In this paper, we propose offline and online adaptive enrichment algorithms for the generalized multiscale approximation of a mixed finite element method with velocity elimination to solve the subsurface flow problem in high-contrast and heterogeneous porous media. In the offline adaptive method, we first derive an a-posteriori error indicator based on one weighted L²-norm of the local residual operator, where the weighted L²-norm is related to the pressure fields of the local snapshot space. Then we enrich the multiscale space by increasing the number of offline basis functions iteratively on coarse elements where the error indicator takes large values. While in the online adaptive method, we add online basis functions on selected coarse elements based on another weighted L²-norm of the local residual operator to enrich the multiscale space, here the weighted L²-norm is associated with the velocity fields of the local snapshot space. Online basis functions are constructed in the online stage depending on the solution of the previous iteration and some optimal estimates. We give theoretical analyses for the convergences of these two adaptive methods, which show that sufficient initial basis functions (belong to the offline space) lead to faster convergence rates. A series of numerical examples are provided to highlight the performances of both these two adaptive methods and also validate the theoretical analyses. Both offline and online adaptive methods are effective that can reduce the relative error substantially. In addition, the online adaptive method generally performs better than the offline adaptive method as online basis functions contain important global information such as distant effects that cannot be captured by offline basis functions. The numerical results also show that with a suitable initial multiscale space that includes all offline basis functions corresponding to relative smaller eigenvalues of each local spectral decomposition in the offline stage, the convergence rate of the online enrichment is independent of the permeability contrast.

在本文中，我们提出了离线和在线自适应富集算法，用于具有速度消除的混合有限元方法的广义多尺度近似，以解决高对比度和非均质多孔介质中的地下流动问题。在离线自适应方法中，我们首先基于局部残差算子的一个加权L ^{2 -}范数导出后验误差指标，其中加权L ²-norm 与本地快照空间的压力场有关。然后我们通过在误差指标取大值的粗元素上迭代地增加离线基函数的数量来丰富多尺度空间。而在线自适应方法中，我们根据局部残差算子的另一个加权L ^{2 -}范数在选定的粗元素上添加在线基函数以丰富多尺度空间，这里是加权L ²-norm 与本地快照空间的速度场相关联。在线基函数是在在线阶段根据前一次迭代的解和一些最优估计构建的。我们对这两种自适应方法的收敛性进行了理论分析，表明足够的初始基函数（属于离线空间）导致更快的收敛速度。提供了一系列数值例子来突出这两种自适应方法的性能，并验证理论分析。离线和在线自适应方法都是有效的，可以大大减少相对误差。此外，在线自适应方法通常比离线自适应方法表现更好，因为在线基函数包含重要的全局信息，例如离线基函数无法捕获的远距离效应。数值结果还表明，在一个合适的初始多尺度空间包括所有离线基函数对应于离线阶段每个局部谱分解的相对较小的特征值时，在线富集的收敛速度与渗透率对比无关。