A posteriori error estimates for hierarchical mixed-dimensional elliptic equations

Jhabriel Varela, Elyes Ahmed, Eirik Keilegavlen, Jan Martin Nordbotten, Florin Adrian Radu

Published 2021-01-20Version 1

In this paper, we derive a posteriori error estimates for mixed-dimensional elliptic equations exhibiting a hierarchical structure. Exploiting the exterior calculus perspective of such equations, we introduce mixed-dimensional variables and operators, which, together with careful construction of the functional spaces, allow us to recast the set of partial differential equations as a regular linear elliptic problem structure-wise. We therefrom apply the well-established theory of functional a posteriori error estimates to our model to derive guaranteed abstract as well as fully computable upper bounds. Our estimators are tested using three different families of locally-mass conservative methods on synthetic problems and verification benchmarks of flow in fractured porous media. The numerical results support our theoretical findings while showcasing satisfactory effectivity indices.