Robust linear domain decomposition schemes for reduced nonlinear fracture flow models
Abstract
In this work, we consider compressible singlephase flow problems in a porous media containing a fracture. In the latter, a nonlinear pressurevelocity relation is prescribed. Using a nonoverlapping domain decomposition procedure, we reformulate the global problem into a nonlinear interface problem. We then introduce two new algorithms that are able to efficiently handle the nonlinearity and the coupling between the fracture and the matrix, both based on linearization by the socalled Lscheme. The first algorithm, named MoLDD, uses the Lscheme to resolve the nonlinearity, 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 precomputing, in an offline phase, a multiscale flux basis (the linear RobintoNeumann codimensional 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2019
 DOI:
 10.48550/arXiv.1906.05831
 arXiv:
 arXiv:1906.05831
 Bibcode:
 2019arXiv190605831A
 Keywords:

 Mathematics  Numerical Analysis