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A mixed finite element method for Darcy flow in fracturedporous media with non-matching grids∗

Published online by Cambridge University Press:  19 December 2011

Carlo D’Angelo  and
Anna Scotti
Carlo D’Angelo
Affiliation:
MOX-Dip. di Matematica “F. Brioschi” Politecnico di Milano via Bonardi 9, 20133 Milano, Italy. carlo.dangelo@polimi.it
Anna Scotti
Affiliation:
MOX-Dip. di Matematica “F. Brioschi” Politecnico di Milano via Bonardi 9, 20133 Milano, Italy; anna.scotti@mail.polimi.it
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Abstract

We consider an incompressible flow problem in a N-dimensional fracturedporous domain (Darcy’s problem). The fracture is represented by a(N − 1)-dimensional interface, exchanging fluid with the surroundingmedia. In this paper we consider the lowest-order(ℝ T0, ℙ0) Raviart-Thomas mixed finite elementmethod for the approximation of the coupled Darcy’s flows in the porous media and withinthe fracture, with independent meshes for the respective domains. This is achieved thanksto an enrichment with discontinuous basis functions on triangles crossed by the fractureand a weak imposition of interface conditions. First, we study the stability andconvergence properties of the resulting numerical scheme in the uncoupled case, when theknown solution of the fracture problem provides an immersed boundary condition. We detailthe implementation issues and discuss the algebraic properties of the associated linearsystem. Next, we focus on the coupled problem and propose an iterative porousdomain/fracture domain iterative method to solve for fluid flow in both the porous mediaand the fracture and compare the results with those of a traditional monolithic approach.Numerical results are provided confirming convergence rates and algebraic propertiespredicted by the theory. In particular, we discuss preconditioning and equilibrationtechniques to make the condition number of the discrete problem independent of theposition of the immersed interface. Finally, two and three dimensional simulations ofDarcy’s flow in different configurations (highly and poorly permeable fracture) areanalyzed and discussed.

Keywords

Darcy’s equationfractured porous mediamixed finite elementunfitted meshfictitious domainembedded interfaceextended finite element
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 2 , March 2012 , pp. 465 - 489
Copyright
© EDP Sciences, SMAI 2011

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