Unified analysis of discontinuous Galerkin approximations of flows in fractured porous media on polygonal and polyhedral grids

Paola F. Antonietti; Chiara Facciolà; Marco Verani; Paola F. Antonietti; Chiara Facciolà; Marco Verani

doi:10.3934/mine.2020017

Mathematics in Engineering

2020, Volume 2, Issue 2: 340-385. doi: 10.3934/mine.2020017

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Research article

Unified analysis of discontinuous Galerkin approximations of flows in fractured porous media on polygonal and polyhedral grids

MOX- Laboratory for Modeling and Scientific Computing, Department of Mathematics, Politecnico of Milan, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

Received: 21 March 2019 Accepted: 08 January 2020 Published: 19 February 2020

We propose a unified formulation based on discontinuous Galerkin methods on polygonal/polyhedral grids for the simulation of flows in fractured porous media. We adopt a model for single-phase flows where the fracture is modeled as a (d-1)-dimensional interface in a d-dimensional bulk domain, and model the flow in the porous medium and in the fracture by means of the Darcy's law. The two problems are then coupled through physically consistent conditions. We focus on the numerical approximation of the coupled bulk-fracture problem and present and analyze, in an unified setting, all the possible combinations of primal-primal, mixed-primal, primal-mixed and mixed-mixed formulations for the bulk and fracture problems, respectively. For all the possible combinations, we prove their well-posedness and derive a priori hp-version error estimates in a suitable (mesh-dependent) energy norm. Finally, preliminary numerical experiments assess the theoretical error estimates and accuracy of the proposed formulations.
- discontinuous Galerkin,
- fractured porous media,
- polytopic grids
Citation: Paola F. Antonietti, Chiara Facciolà, Marco Verani. Unified analysis of discontinuous Galerkin approximations of flows in fractured porous media on polygonal and polyhedral grids[J]. Mathematics in Engineering, 2020, 2(2): 340-385. doi: 10.3934/mine.2020017

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Abstract

We propose a unified formulation based on discontinuous Galerkin methods on polygonal/polyhedral grids for the simulation of flows in fractured porous media. We adopt a model for single-phase flows where the fracture is modeled as a (d-1)-dimensional interface in a d-dimensional bulk domain, and model the flow in the porous medium and in the fracture by means of the Darcy's law. The two problems are then coupled through physically consistent conditions. We focus on the numerical approximation of the coupled bulk-fracture problem and present and analyze, in an unified setting, all the possible combinations of primal-primal, mixed-primal, primal-mixed and mixed-mixed formulations for the bulk and fracture problems, respectively. For all the possible combinations, we prove their well-posedness and derive a priori hp-version error estimates in a suitable (mesh-dependent) energy norm. Finally, preliminary numerical experiments assess the theoretical error estimates and accuracy of the proposed formulations.

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