A transient global-local generalized FEM for parabolic and hyperbolic PDEs with multi-space/time scales

This paper presents a novel Generalized Finite Element Method with global-local enrichment (GFEM^gl) to solve time-dependent parabolic and hyperbolic problems with subscale features in both space and time. Compared with currently available GFEM^gl, the proposed method requires a solution of local problems to solve transient PDEs, possibly with a different time integrator from the global problem or other local problems. The fine-scale solution can be easily retrieved. The proposed method is tested for the solution of the heat, advection, and advection-diffusion equations whose accuracy, stability, scalability, and convergence are thoroughly analyzed. Compared to direct analysis, numerical results show that the proposed method has the following properties 1) the accuracy closely matches direct analysis result with a fine mesh; 2) critical time step size is loosened; 3) optimal convergence rate is achieved. Moreover, the proposed method allows local problems to be turned off if their solutions are not helpful for the current global time step, which can further improve its efficiency.