A stabilized Lagrange multiplier finite-element method for flow in porous media with fractures

Markus Köppel (1) , Vincent Martin (2) , Jean E. Roberts (3, 4)

1 IANS - Institut für Angewandte Analysis und Numerische Simulation [Stuttgart]
2 LMAC - Laboratoire de Mathématiques Appliquées de Compiègne
3 Inria de Paris
4 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

In this work we introduce a stabilized, numerical method for a multi-dimensional, discrete-fracture model (DFM) for single-phase Darcy flow in fractured porous media. In the model, introduced in an earlier work, flow in the (n − 1)-dimensional fracture domain is coupled with that in the n-dimensional bulk or matrix domain by the use of Lagrange multipliers. Thus the model permits a finite element discretization in which the meshes in the fracture and matrix domains are independent so that irregular meshing and in particular the generation of small elements can be avoided. In this paper we introduce in the numerical formulation, which is a saddle-point problem based on a primal, variational formulation for flow in the matrix domain and in the fracture system, a consistent stabilizing term which penalizes discontinuities in the Lagrange multipliers. For this penalized scheme we show stability and prove convergence. With numerical experiments we analyze the performance of the method for various choices of the penalization parameter and compare with other numerical DFM's.

Mots clés: Penalization; Nonconforming grids; Stabilized Lagrange multiplier method; Finite element method; Discrete fracture model

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]; Mécanique des fluides [physics.class-ph]

Markus Köppel, Vincent Martin, Jean E. Roberts. A stabilized Lagrange multiplier finite-element method for flow in porous media with fractures. GEM - International Journal on Geomathematics, In press, ⟨10.1007/s13137-019-0117-7⟩. ⟨hal-01761591v3⟩