Weak scalability of domain decomposition methods for discrete fracture networks

TL;DR

This paper provides a theoretical analysis demonstrating that Optimized Schwarz Methods can achieve weak scalability when applied to Discrete Fracture Networks, under certain assumptions, offering insights for efficient domain decomposition in fractured media modeling.

Contribution

It offers the first theoretical proof of weak scalability for OSMs on DFNs, extending understanding of DD methods in complex fractured media.

Findings

OSMs can be weakly scalable under certain assumptions

The analysis offers heuristics for practical computational efficiency

Methodology can be generalized to other DD methods for DFNs

Abstract

Discrete Fracture Networks (DFNs) are complex three-dimensional structures characterized by the intersections of planar polygonal fractures, and are used to model flows in fractured media. Despite being suitable for Domain Decomposition (DD) techniques, there are relatively few works on the application of DD methods to DFNs. In this manuscript, we present a theoretical study of Optimized Schwarz Methods (OSMs) applied to DFNs. Interestingly, we prove that the OSMs can be weakly scalable (that is, they converge to a given tolerance in a number of iterations independent of the number of fractures) under suitable assumptions on the domain decomposition. This contribution fits in the renewed interest on the weak scalability of DD methods after recent works showed weak scalability of DD methods for specific geometric configurations, even without coarse spaces. Despite simplifying assumptions which may be violated in practice, our analysis provides heuristics to minimize the computational efforts in realistic settings. Finally, we emphasize that the methodology proposed can be straightforwardly generalized to study other classical DD methods applied to DFNs.