Adaptive generalized multiscale approximation of a mixed finite element method with velocity elimination

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.