A SPACE-TIME MULTISCALE MORTAR MIXED FINITE ELEMENT METHOD FOR PARABOLIC EQUATIONS

We develop a space-time mortar mixed finite element method for parabolic problems. The domain is decomposed into a union of subdomains discretized with nonmatching spatial grids and asynchronous time steps. The method is based on a space-time variational formulation that couples mixed finite elements in space with discontinuous Galerkin in time. Continuity of flux (mass conservation) across space-time interfaces is imposed via a coarse-scale space-time mortar variable that approximates the primary variable. Uniqueness, existence, and stability as well as a priori error estimates for the spatial and temporal errors are established. A space-time nonoverlapping domain decomposition method is developed that reduces the global problem to a space-time coarse-scale mortar interface problem. Each interface iteration involves solving in parallel space-time subdomain problems. The spectral properties of the interface operator and the convergence of the interface iteration are analyzed. Numerical experiments are provided that illustrate the theoretical results and the flexibility of the method for modeling problems with features that are localized in space and time.

Automated summary

We propose a space-time mortar mixed finite element method for parabolic problems, which handles nonmatching spatial grids and asynchronous time steps. This method combines mixed finite elements and discontinuous Galerkin method, and enforces continuity of flux across space-time interfaces via a coarse-scale mortar variable. We establish uniqueness, existence, stability, and error estimates for the spatial and temporal errors. Additionally, we develop a space-time nonoverlapping domain decomposition method that reduces the global problem to a coarse-scale mortar interface problem, solving parallel subdomain problems at each interface iteration. The numerical experiments demonstrate the theoretical results and the flexibility of the method in modeling localized space-time features.