Fully discrete heterogeneous multiscale method for parabolic problems with multiple spatial and temporal scales (Q6161580)

Problems with macroscopic and microscopic spatial and temporal scales occur in various phenomena and materials. They are challenging from a numerical point of view. In this work, the Heterogeneous Multiscale Method (HMM) is used for numerical homogenization of a parabolic problem with several temporal and spatial scales. A suitable discretization of the cell problems is proposed and rigorous error estimates are derived for the resulting fully discrete HMM, whereas the order of the mesh size and the time step on the one hand and the period on the other hand are balanced. The convergence rate with respect to the time step and mesh width is investigated for the macrodiscretization as well as for the microdiscretization in numerical experiments.