A posteriori error estimates for hierarchical mixed-dimensional elliptic equations

TL;DR

This paper develops and compares new a posteriori error estimates for hierarchical mixed-dimensional elliptic equations, improving accuracy and computability for modeling flow in fractured porous media.

Contribution

It introduces four novel residual error estimation methods based on conservation properties, enhancing the sharpness and practicality of error bounds in mixed-dimensional elliptic problems.

Findings

Sharper error estimates when mass is conserved at grid level or exactly

Effective application across four discretization methods

Validated through numerical experiments on synthetic and benchmark problems

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.