Space-time domain decomposition for advection-diffusion problems in mixed formulations

TL;DR

This paper develops and compares two space-time domain decomposition methods for advection-diffusion problems in porous media, enabling different time steps in subdomains to efficiently handle heterogeneities and multiple time scales.

Contribution

It introduces two novel domain-decomposition schemes using Steklov--Poincaré and OSWR methods with mixed formulations for space-time interfaces, allowing adaptive time stepping in heterogeneous porous-media simulations.

Findings

Both methods effectively handle strong heterogeneities in 2D problems.

Numerical results demonstrate the methods' capability for realistic underground storage simulations.

The proposed preconditioner improves convergence for the Steklov--Poincaré approach.

Abstract

This paper is concerned with the numerical solution of porous-media flow and transport problems , i. e. heterogeneous, advection-diffusion problems. Its aim is to investigate numerical schemes for these problems in which different time steps can be used in different parts of the domain. Global-in-time, non-overlapping domain-decomposition methods are coupled with operator splitting making possible the different treatment of the advection and diffusion terms. Two domain-decomposition methods 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 interface is derived, and different time grids are employed to adapt to different time scales in the subdomains. A generalized Neumann-Neumann preconditioner is proposed for the first method. To illustrate the two methods numerical results for two-dimensional problems with strong heterogeneities are presented. These include both academic problems and more realistic prototypes for simulations for the underground storage of nuclear waste.