# Robust linear domain decomposition schemes for reduced non-linear fracture flow models

## TL;DR

This paper introduces two robust linear domain decomposition algorithms for simulating non-linear fracture flow in porous media, utilizing the L-scheme for linearization and multiscale flux basis for computational efficiency, with proven stability and convergence.

## Contribution

The paper develops two new algorithms, MoLDD and ItLDD, that efficiently handle non-linear fracture flow problems using linearization and precomputed flux bases, advancing computational methods in this field.

## Key findings

- Algorithms are stable and convergent under optimized parameters.
- Extensive numerical tests confirm theoretical predictions.
- Pre-computed flux basis reduces computational complexity.

## Abstract

In this work, we consider compressible single-phase flow problems in a porous media containing a fracture. In the latter, a non-linear pressure-velocity relation is prescribed. Using a non-overlapping domain decomposition procedure, we reformulate the global problem into a non-linear interface problem. We then introduce two new algorithms that are able to efficiently handle the non-linearity and the coupling between the fracture and the matrix, both based on linearization by the so-called L-scheme. The first algorithm, named MoLDD, uses the L-scheme to resolve the non-linearity, requiring at each iteration to solve the dimensional coupling via a domain decomposition approach. The second algorithm, called ItLDD, uses a sequential approach in which the dimensional coupling is part of the linearization iterations. For both algorithms, the computations are reduced only to the fracture by pre-computing, in an offline phase, a multiscale flux basis (the linear Robin-to-Neumann co-dimensional map), that represent the flux exchange between the fracture and the matrix. We present extensive theoretical findings and in particular, the stability and the convergence of both schemes are obtained, where user given parameters are optimized to minimise the number of iterations. Examples on two important fracture models are computed with the library PorePy and agree with the developed theory.