Adaptive modelling of variably saturated seepage problems

B. Ashby, C. Bortolozo, A. Lukyanov, T. Pryer

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1 Citation (SciVal)


In this article, we present a goal-oriented adaptive finite element method for a class of subsurface flow problems in porous media, which exhibit seepage faces. We focus on a representative case of the steady state flows governed by a nonlinear Darcy-Buckingham law with physical constraints on subsurface-atmosphere boundaries. This leads to the formulation of the problem as a variational inequality. The solutions to this problem are investigated using an adaptive finite element method based on a dual-weighted a posteriori error estimate, derived with the aim of reducing error in a specific target quantity. The quantity of interest is chosen as volumetric water flux across the seepage face, and therefore depends on an a priori unknown free boundary. We apply our method to challenging numerical examples as well as specific case studies, from which this research originates, illustrating the major difficulties that arise in practical situations. We summarise extensive numerical results that clearly demonstrate the designed method produces rapid error reduction measured against the number of degrees of freedom.

Original languageEnglish
Pages (from-to)55-81
Number of pages27
JournalQuarterly Journal of Mechanics and Applied Mathematics
Issue number1
Publication statusPublished - 1 Feb 2021

Bibliographical note

Funding Information:
This research was conducted during a visit partially supported through the QJMAM Fund for Applied Mathematics. B.A. was supported through a PhD scholarship awarded by the a.oeEPSRC Centre for Doctoral Training in the Mathematics of Planet Earth at Imperial College London and the University of Readinga. EP/L016613/1. T.P. was partially supported through the EPSRC grant EP/P000835/1 and the Newton Fund grant 261865400. CAB acknowledges support from the Conselho Nacional de Desenvolvimento CientA fico e TecnolA3gico (CNPq) for the Research Fellowship Program (grants 300610/2017-3 and 301219/2020-6) and for the Research Grant 433481/2018-8.

Publisher Copyright:
© 2021 The Author.

Copyright 2021 Elsevier B.V., All rights reserved.

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering
  • Applied Mathematics


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