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