Space-time Domain Decomposition and Mixed Formulation for solving reduced fracture models

TL;DR

This paper develops and analyzes two global-in-time domain decomposition methods for modeling flow in porous media with fractures, using mixed formulations and optimized Schwarz waveform relaxation to improve computational efficiency.

Contribution

It introduces a mixed formulation for interface problems and analyzes convergence for OSWR methods with fractures, enhancing modeling of fractured porous media.

Findings

Both methods effectively model flow in fractured media.

Numerical results demonstrate good convergence and performance.

Optimized parameters improve the efficiency of the OSWR method.

Abstract

In this paper we are interested in the "fast path" fracture and we aim to use global-in-time, nonoverlapping domain decomposition methods to model flow and transport problems in a porous medium containing such a fracture. We consider a reduced model in which the fracture is treated as an interface between the two subdomains. Two domain decomposition methods are considered: one uses the time-dependent SteklovPoincar{\'e} operator and the other uses optimized Schwarz waveform relaxation (OSWR) based on Ventcell 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 and in the fracture. Demonstrations of the well-posedness of the Ventcell subdomain problems is given for the mixed formulation. An analysis for the convergence factor of the OSWR algorithm is given in the case with fractures to compute the optimized parameters. Numerical results for two-dimensional problems with strong heterogeneities are presented to illustrate the performance of the two methods.