A posteriori error estimates for mixed-dimensional Darcy flow using non-matching grids

TL;DR

This paper develops guaranteed a posteriori error estimators for mixed-dimensional Darcy flow problems on non-matching grids, extending previous work to more flexible grid configurations with proven reliability and effectiveness.

Contribution

It introduces transfer grids and stable projection operators to extend a posteriori error estimates to non-matching mixed-dimensional grids, maintaining guaranteed and computable bounds.

Findings

Estimators are reliable for non-matching grids.

Effectivity comparable to matching grid estimators.

Validated through 3D benchmark problems.

Abstract

In this article, we extend the a posteriori error estimates for hierarchical mixed-dimensional elliptic equations developed in [Varela et al., J. Numer. Math., 48 (2023), pp. 247-280] to the setting of non-matching mixed-dimensional grids. The extension is achieved by introducing transfer grids between the planar subdomain and interface grids, together with stable discrete projection operators for primal (potential) and dual (flux) variables. The proposed non-matching estimators remain fully guaranteed and computable. Numerical experiments, including three-dimensional problems based on community benchmarks for incompressible Darcy flow in fractured porous media, demonstrate reliable performance of the estimators for the non-matching grids and effectivity that is comparable to the estimators for matching grids.