A space-time multiscale mortar mixed finite element method for parabolic equations

TL;DR

This paper introduces a novel space-time mortar mixed finite element method for parabolic equations, enabling flexible domain decomposition with non-matching grids and asynchronous time steps, ensuring stability and accurate error estimates.

Contribution

It develops a new space-time variational formulation combining mixed finite elements and discontinuous Galerkin methods, with a domain decomposition approach for efficient parallel computation.

Findings

Method achieves stability and convergence with a priori error estimates.

Numerical experiments confirm theoretical results and flexibility.

Parallel interface iteration converges efficiently, enabling localized modeling.

Abstract

We develop a space-time mortar mixed finite element method for parabolic problems. The domain is decomposed into a union of subdomains discretized with non-matching 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 non-overlapping 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.