In this paper,we combine the generalized multiscale finite element method(GMsFEM)with the balanced truncation(BT)method to address a parameterdependent elliptic problem.Basically,in progress of a model reduction we try to obtain accurate solutions with less computational resources.It is realized via a spectral decomposition from the dominant eigenvalues,that is used for an enrichment of multiscale basis functions in the GMsFEM.The multiscale bases computations are localized to specified coarse neighborhoods,and follow an offline-online process in which eigenvalue problems are used to capture the underlying system behaviors.In the BT on reduced scales,we present a local-global strategy where it requires the observability and controllability of solutions to a set of Lyapunov equations.As the Lyapunov equations need expensive computations,the efficiency of our combined approach is shown to be readily flexible with respect to the online space and an reduced dimension.Numerical experiments are provided to validate the robustness of our approach for the parameter-dependent elliptic model.
We apply the multiscale basis functions for the singularly perturbed reaction-diffusion problem on adaptively graded meshes,which can provide a good balance between the numerical accuracy and computational cost.The multiscale space is built through standard finite element basis functions enriched with multiscale basis functions.The multiscale basis functions have abilities to capture originally perturbed information in the local problem,as a result our method is capable of reducing the boundary layer errors remarkably on graded meshes,where the layer-adapted meshes are generated by a given parameter.Through numerical experiments we demonstrate that the multiscale method can acquire second order convergence in the L^(2)norm and first order convergence in the energy norm on graded meshes,which is independent ofε.In contrast with the conventional methods,our method is much more accurate and effective.
