A posteriori error estimates for hierarchical mixed-dimensional elliptic equations

This paper focuses on obtaining a posteriori error estimates for mixed-dimensional elliptic equations exhibiting a hierarchical structure. 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. However, unlike standard results obtained with the functional approach, we propose four different ways of estimating the residual errors based on the level of accuracy available for their approximations, i.e.: (1) no conservation, (2) subdomain conservation, (3) local conservation, and (4) exact conservation. This treatment results in sharper and fully computable estimates when mass is conserved either locally 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 on matching and nonmatching grids for synthetic problems and benchmarks of flow in fractured porous media.