Space-Time Domain Decomposition Methods for Diffusion Problems in Mixed Formulations

TL;DR

This paper develops and analyzes space-time domain decomposition methods for diffusion problems in mixed formulations, introducing approaches based on the Steklov-Poincaré operator and OSWR, with proofs of well-posedness and convergence, supported by numerical experiments.

Contribution

It introduces two novel global-in-time domain decomposition methods for mixed diffusion problems, including a convergence proof for OSWR and handling of different time scales.

Findings

Methods are well-posed and convergent.

Numerical results show effectiveness in heterogeneous 2D problems.

Approaches adapt to different time scales in subdomains.

Abstract

This paper is concerned with global-in-time, nonoverlapping domain decomposition methods for the mixed formulation of the diffusion problem. Two approaches are considered: one uses the time-dependent Steklov-Poincar\'e operator and the other uses Optimized Schwarz Waveform Relaxation (OSWR) based on Robin transmission conditions. For each method, a mixed formulation of an interface problem on the space-time interfaces between subdomains is derived, and different time grids are employed to adapt to different time scales in the subdomains. Demonstrations of the well-posedness of the subdomain problems involved in each method and a convergence proof of the OSWR algorithm are given for the mixed formulation. Numerical results for 2D problems with strong heterogeneities are presented to illustrate the performance of the two methods.