Adaptive Least-Squares Mixed Generalized Multiscale Finite Element Methods

In this paper, we present two kinds of adaptive least-squares mixed generalized multiscale finite element methods (GMsFEMs) for solving an elliptic problem in highly heterogeneous porous media. An offline adaptive method is developed through iteratively enriching the local velocity and pressure multiscale basis functions based on residual-based error indicators. In addition, an online adaptive method is also proposed to create the new multiscale basis functions in the online stage for both velocity and pressure. The enriched basis functions are computed by the residual and maximizing the reduction in error. The offline adaptive method attempts to provide a good approximation space based on the heterogeneity of the coefficient, and the online adaptive method aims at providing a good approximation space by taking the given source into account. Both of the adaptive methods can achieve a better approximation than the uniform enrichment method using the same number of basis functions. Convergence analysis is carried out for the adaptive least-squares GMsFEMs. The analysis suggests that, by choosing a suitable number of initial basis functions for velocity and pressure, the online adaptive method can render a faster convergence rate compared with the offline adaptive method and the uniform enrichment. A few numerical results are presented to confirm the analysis and the performance of the presented adaptive multiscale methods.