An adaptive space and time method in partially explicit splitting scheme for multiscale flow problems

We propose a space and time adaptive framework for a partially explicit splitting scheme of flow problems in multiscale media. We consider the parabolic equation with high-contrast multiscale coefficients. It is known that sufficient number of multiscale basis are needed to capture non-local features and information across all scales. Moreover, the time step size in an explicit discretization of the problem is affected by the magnitude of the coefficient. To improve computational efficiency, in this work, we introduce an adaptive algorithm to refine temporal size and enrich multiscale spaces. We first utilize a stable temporal splitting scheme with an appropriate construction of multiscale spaces to ensure that the time step size is independent of the contrast. The multiscale subspaces are then constructed to handle the fast-flow and slow-flow parts separately, and implicit and explicit discretization is adopted for the degrees of freedom with respect to the two multiscale subspaces, respectively. A multirate time stepping is introduced for the two parts considering that the flow rates vary significantly in different regions due to the high-contrast features of the multiscale coefficients. We derive both temporal and spatial error estimators to identify local regions for two components of the solutions. Start with coarser time step sizes and a few important multiscale basis functions in each subspace, we can adaptively refine time stepping and add additional basis with constraint energy minimizing properties based on the derived error indicators. The stability and convergence of the method are studied, and several numerical tests are presented to demonstrate the performance of the proposed method.