A posteriori error estimates for hierarchical mixed-dimensional elliptic equations

This paper focuses on deriving a posteriori error estimates for mixed-dimensional elliptic equations exhibiting a hierarchical structure. After careful construction of a mixed-dimensional functional analysis framework, 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.: no conservation, subdomain conservation, local conservation, and exact conservation. This treatment not only results in sharper estimates but also in fully computable ones (e.g., with no undetermined constants) when mass is conserved either locally or exactly. Abstract estimates are then made concrete using local mass-conservative approximations. Our estimators are tested with four numerical methods on matching and nonmatching grids for both synthetic and benchmarks of flow in fractured porous media. The numerical results support our theoretical findings with satisfactory results.