Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A posteriori error estimates for mixed-dimensional Darcy flow using non-matching grids

Jhabriel Varela, Christian E. Schaerer, Eirik Keilegavlen, Inga Berre

TL;DR

The paper addresses the challenge of providing reliable a posteriori error estimates for mixed-dimensional Darcy flow on non-matching grids. It introduces transfer grids and stable discrete projections to couple subdomains and interfaces without grid conformity, yielding fully computable, guaranteed bounds for primal and dual variables. The approach is validated through 3D fracture benchmarks and manufactured tests, demonstrating robust estimators with comparable performance to matching-grid cases. This work enables efficient adaptive refinement in complex, multi-dimensional porous media by accommodating independent discretizations across subdomains and interfaces.

Abstract

In this article, we extend the a posteriori error estimates for hierarchical mixed-dimensional elliptic equations developed in [Varela et al., J. Numer. Math., 48 (2023), pp. 247-280] to the setting of non-matching mixed-dimensional grids. The extension is achieved by introducing transfer grids between the planar subdomain and interface grids, together with stable discrete projection operators for primal (potential) and dual (flux) variables. The proposed non-matching estimators remain fully guaranteed and computable. Numerical experiments, including three-dimensional problems based on community benchmarks for incompressible Darcy flow in fractured porous media, demonstrate reliable performance of the estimators for the non-matching grids and effectivity that is comparable to the estimators for matching grids.

A posteriori error estimates for mixed-dimensional Darcy flow using non-matching grids

TL;DR

The paper addresses the challenge of providing reliable a posteriori error estimates for mixed-dimensional Darcy flow on non-matching grids. It introduces transfer grids and stable discrete projections to couple subdomains and interfaces without grid conformity, yielding fully computable, guaranteed bounds for primal and dual variables. The approach is validated through 3D fracture benchmarks and manufactured tests, demonstrating robust estimators with comparable performance to matching-grid cases. This work enables efficient adaptive refinement in complex, multi-dimensional porous media by accommodating independent discretizations across subdomains and interfaces.

Abstract

In this article, we extend the a posteriori error estimates for hierarchical mixed-dimensional elliptic equations developed in [Varela et al., J. Numer. Math., 48 (2023), pp. 247-280] to the setting of non-matching mixed-dimensional grids. The extension is achieved by introducing transfer grids between the planar subdomain and interface grids, together with stable discrete projection operators for primal (potential) and dual (flux) variables. The proposed non-matching estimators remain fully guaranteed and computable. Numerical experiments, including three-dimensional problems based on community benchmarks for incompressible Darcy flow in fractured porous media, demonstrate reliable performance of the estimators for the non-matching grids and effectivity that is comparable to the estimators for matching grids.

Paper Structure

This paper contains 27 sections, 5 theorems, 52 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Mixed-dimensional setting and the model problem
  3. Mixed-dimensional decomposition
  4. The model problem
  5. Functional framework
  6. Local and global spaces
  7. Continuous transfer operators
  8. Weak forms
  9. Discrete setting
  10. Grid definitions
  11. Discrete projection operators
  12. Discrete projections for the weak primal problem
  13. Discrete projections of potentials for the weak dual problem
  14. Discrete projections of interface fluxes for the weak dual problem
  15. Primal and dual approximations
  16. ...and 12 more sections

Key Result

Lemma 1

For all admissible arguments:

Figures (7)

  • Figure 1: Mixed-dimensional geometric setting. Left: A fracture $\Omega_{{\check\jmath}}$ fully embedded in the matrix $\Omega_{{\hat{\jmath}}}$. A close-up of the upper side of the fracture shows the interface $\Gamma_j$ coupling $\Omega_{{\check\jmath}}$ and $\Omega_{{\hat{\jmath}}}$ through its internal boundary $\partial_j\Omega_{{\hat{\jmath}}}$. There exists a similar coupling on the lower side of the fracture, not shown in the figures. Right: A non-matching coupling between $\mathcal{T}_{\partial_{j}\Omega_{{\hat{\jmath}}}}$, $\mathcal{T}_{\Gamma_{j}}$ and $\mathcal{T}_{\Omega_{{\check\jmath}}}$. Grids are formally defined in Section \ref{['sec:grid']}.
  • Figure 2: Complete geometric decomposition of an mD domain. Left: Subdomains. Two 2D subdomains ($\Omega_1$ and $\Omega_2$) hosting four 1D fractures ($\Omega_3$, $\Omega_4$, $\Omega_5$, $\Omega_6$) that intersect at a common 0D point ($\Omega_7$). Center: Interfaces. Twelve interfaces coupling neighboring subdomains of co-dimension one. Right: Internal and external boundary conditions.
  • Figure 3: Transfer grids $\mathcal{T}_{\Gamma_j, X}$ coupling interface grids $\mathcal{T}_{\Gamma_j}$ with $\mathcal{T}_{X}$, $X\in\{\partial_j\Omega_{{\hat{\jmath}}},\Omega_{{\check\jmath}}\}$. Top: 1D coupling. Bottom: 2D coupling. The shaded quadrilateral corresponds to a macro-element $S_{\Gamma_j, X}$ that was further refined into four simplices using its barycenter coordinate.
  • Figure 4: 3D/2D numerical verification setup. Left: Mixed-dimensional grid showcasing the matrix grid $\mathcal{T}_{\Omega_2}$ and the fully embedded fracture grid $\mathcal{T}_{\Omega_1}$. Right: Exploded view of the non-matching coupling between $\mathcal{T}_{\partial_1\Omega_2}$ (black), $\mathcal{T}_{\Gamma_1}$ (blue), and $\mathcal{T}_{\Omega_1}$ (red) with the corresponding transfer grids $\mathcal{T}_{\Gamma_1, \partial_1\Omega_2}$ and $\mathcal{T}_{\Gamma_1, \Omega_1}$ (triangles are colored for visualization purposes only). Example shown for $h=0.15$ and translation $+ \bm{e}_y$.
  • Figure 5: Regular network. Left: Geometry of the fracture network and boundary conditions. Right: Local diffusive error estimators of 2D interface grids for the finest mesh resolution.
  • ...and 2 more figures

Theorems & Definitions (33)

  • Definition 1: Strong primal form
  • Definition 2: Strong dual form
  • Remark 1
  • Remark 2: $\mathbf{X}_{i}$ vs. $\mathbf{V}_i$
  • Definition 3: Continuous transfer operators
  • Lemma 1: Basic properties of transfer operators
  • Remark 3: Continuous transfers vs. discrete projections
  • Definition 4: Weak primal form
  • Remark 4: Well-posedness of the primal form
  • Definition 5: Dual weak form
  • ...and 23 more