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A posteriori error estimates for hierarchical mixed-dimensional elliptic equations

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.
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Category

Academic article

Language

English

Author(s)

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

Affiliation

  • SINTEF Digital / Mathematics and Cybernetics
  • University of Bergen

Date

21.10.2022

Year

2022

Published in

Journal of Numerical Mathematics

ISSN

1570-2820

View this publication at Norwegian Research Information Repository