A posteriori error estimates for hierarchical mixed-dimensional elliptic equations

Abstract

Mixed-dimensional elliptic equations exhibiting a hierarchical structure are commonly used to model problems with high aspect ratio inclusions, such as flow in fractured porous media. We derive general abstract estimates based on the theory of functional a posteriori error estimates, for which guaranteed upper bounds for the primal and dual variables and two-sided bounds for the primal-dual pair are obtained. We improve on the abstract results obtained with the functional approach by proposing four different ways of estimating the residual errors based on the extent the approximate solution has conservation properties, i.e.: (1) no conservation, (2) subdomain conservation, (3) grid-level conservation, and (4) exact conservation. This treatment results in sharper and fully computable estimates when mass is conserved either at the grid level or exactly, with a comparable structure to those obtained from grid-based a posteriori techniques. We demonstrate the practical effectiveness of our theoretical results through numerical experiments using four different discretization methods for synthetic problems and applications based on benchmarks of flow in fractured porous media.

A model for saturated–unsaturated flow with fractures acting as capillary barriers

“…To extend the model of a single fracture to a more general setting, we first need to introduce the mixed-dimensional geometric decomposition of the domain. The formulation follows the one originally proposed by Boon et al (2018) and more closely follows the one recently employed by Varela et al (2023). First, consider a domain 𝑌 ⊂ ℝ 2 , decomposed into 𝑚 subdomains Ω 𝑖 of dimension 𝑑 𝑖 = 𝑑(𝑖) so that 𝑌 = ∪ 𝑖∈𝐼 Ω 𝑖 , with 𝐼 = {1, … , 𝑚}.…”

“…For our first numerical example, we perform a convergence analysis for the case of a single vertical line Ω 1 fully embedded in a unit square Ω 2 , coupled via the interfaces Γ 1 and Γ 2 to the left and right of the fracture, respectively. Following closely Varela et al (2023), we assume the existence of a piecewise hydraulic head distribution in the bulk ℎ 2 (we refer to Appendix C for the derivation of all relevant quantities for this analysis), from where all other variables of interest can be derived. In the left panel of Figure 5, we show the geometric setup and the exact hydraulic head distribution in the matrix for the final simulation time.…”

A Model for Saturated-Unsaturated Flow with Fractures Acting as Capillary Barriers

“…To extend the model of a single fracture to a more general setting, we first need to introduce the mixed-dimensional geometric decomposition of the domain. The formulation follows the one originally proposed by Boon et al (2018) and more closely the one recently employed by Varela et al (2022). First, consider a domain 𝑌 ⊂ ℝ !…”

“…Following closely Varela et al (2022), we assume the existence of a piecewise hydraulic quantities for this analysis), from where all other variables of interest can be derived. In the left panel of Figure 4, we show the geometric setup and the exact solution hydraulic head distribution in the matrix for the final simulation time.…”

“…To derive the manufactured solution from Section 4.1, we follow closely the procedure presented by Varela et al (2022). The mixed-dimensional domain consists of a matrix and a single vertical fracture of length 0.5 fully embedded in the middle of the domain.…”