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Numerische Mathematik

A cut finite element method for a model of pressure in fractured media Erik Burman1 · Peter Hansbo2 · Mats G. Larson3 Received: 2 June 2020 / Revised: 16 October 2020 / Accepted: 16 October 2020 © The Author(s) 2020

Abstract We develop a robust cut finite element method for a model of diffusion in fractured media consisting of a bulk domain with embedded cracks. The crack has its own pressure field and can cut through the bulk mesh in a very general fashion. Starting from a common background bulk mesh, that covers the domain, finite element spaces are constructed for the interface and bulk subdomains leading to efficient computations of the coupling terms. The crack pressure field also uses the bulk mesh for its representation. The interface conditions are a generalized form of conditions of Robin type previously considered in the literature which allows the modeling of a range of flow regimes across the fracture. The method is robust in the following way: (1) Stability of the formulation in the full range of parameter choices; and (2) Not sensitive to the location of the interface in the background mesh. We derive an optimal order a priori error estimate and present illustrating numerical examples. Mathematics Subject Classification 65N30 · 65N12 · 65N15

1 Introduction The numerical modelling of flow in fractured porous media is important both in environmental science and in industrial applications. It is therefore not surprising that it is currently receiving increasing attention from the scientific computing community.

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Peter Hansbo [email protected] Erik Burman [email protected] Mats G. Larson [email protected]

1

Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK

2

Department of Mechanical Engineering, Jönköping University, 551 11 Jönköping, Sweden

3

Department of Mathematics and Mathematical Statistics, Umeå University, 901 87 Umeå, Sweden

123

E. Burman et al.

Here we are interested in models where the fractures are modelled as embedded surfaces of dimension d − 1 in a d dimensional bulk domain. Models on this type of geometries of mixed dimension are typically obtained by averaging the flow equations across the width of the fracture and introducing suitable coupling conditions for the modelling of the interaction with the bulk flow. Such reduced models have been derived for instance in [1,25,29]. The coupling conditions in these models typically take the form of a Robin type condition. The physical properties of the coupling enters as parameters in this interface condition. The size of these parameters can vary with several orders of magnitude depending on the physical properties of the crack and of the material in the porous matrix. This makes it challenging to derive methods that both are flexible with respect to mesh geometries and robust with respect to coupling conditions. A wide variety of different strategies for the discretisation of fractured porous media flow has been proposed in the literature. One approach is to use a me