A stabilized three fields formulation for Discrete Fracture Networks

TL;DR

The paper addresses efficient DFN flow simulation by introducing a stabilized, hybridized three-field formulation that allows independent discretization of each fracture while coupling through a trace unknown on fracture intersections. It develops two stabilization strategies—the natural-norm residual stabilization and a mesh-dependent Barbosa-Hughes-like stabilization—proves stability and a priori error estimates for both, and provides a practical implementation via an auxiliary-space-based stabilization and a simplified mesh-dependent variant. Numerical experiments on non-conforming meshes validate convergence and conditioning results, showing that appropriate stabilization yields robust performance across well-separated and near-touching traces, with the natural-norm scheme often preferred when traces are not well separated. The approach offers flexible meshing, potential for effective preconditioning, and scalability to large DFN problems, with future work focusing on extending theory to challenging trace configurations and advancing preconditioned solvers for large networks.

Abstract

We propose a hybridized domain decomposition formulation of the discrete fracture network model, allowing for independent discretization of the individual fractures. A natural norm stabilization, obtained by penalizing the residual measured in the norm for the space where it naturally lives, is added to the local problem in the individual fracture so that no compatibility condition of inf-sup type is required between the Lagrange multiplier and the primal unknown, which can then be discretized independently of each other. Optimal stability and error estimates are proven, which are confirmed by numerical tests.

Figures (14)

• Figure 1: Triangulations $\mathcal{T}_\delta$ (blue) and $\widehat{\mathcal{T}}$ (red). $F_{i,m}$ is the union of the red triangles. The red dots mark the degrees of freedom for $W_\delta^{i,m}$ (which coincide with the dofs for $\Phi_\delta^{i,m}$), while in the nodes marked with the green dots $\widehat{w}_\delta = 0$.
• Figure 2: test1: Domain with a mesh ($\delta=0.1$) non conforming with fracture intersection.
• Figure 3: test1: Detail of mesh $\hat{\mathcal{T}}$ on fracture $F_1$, $\delta=0.1$.
• Figure 4: test1: $L^2$ error for different values of the stabilization parameter $\alpha$, for both the natural norm stabilization (solid line) and the mesh dependent stabilization (dashed line).
• Figure 5: test1: $H^1$ error for different values of the stabilization parameter $\alpha$, for both the natural norm stabilization (solid line) and the mesh dependent stabilization (dashed line).

Theorems & Definitions (11)

• Remark 3.1
• Theorem 3.3
• proof
• Corollary 3.4
• Remark 3.5
• Lemma 4.1
• proof
• Theorem 5.2
• Corollary 5.3
• Lemma 6.1