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.
Highlights
The numerical modelling of flow in fractured porous media is important both in environmental science and in industrial applications
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
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
Summary
The numerical modelling of flow in fractured porous media is important both in environmental science and in industrial applications. The approach is inspired by the work of Stenberg [26] and may be viewed as a version of the Nitsche method that can handle Robin type conditions and which converges to the standard Nitsche method when the Robin parameter tends to infinity Previous applications of this approach in the context of fitted finite elements include [22,37]. Previous related work work on cut finite element methods include the interface problem [21]; overlapping meshes [23]; coupled bulk-surface problems [8, 11,12,20]; mixed dimensional problems [9], and surface partial differential equations [7,32]. The outline of the paper is as follows: In Sect. 2 we introduce the model problem, show an elliptic regularity result which is robust with respect to the critical parameters in the interface condition, and discuss the relation between our formulation of the interface conditions and previous work; in Sect. 3 we formulate the cut finite element method; in Sect. 4 we prove the basic properties of the formulation and in particular an optimal order a priori error estimate which is uniform in the full range of interface parameters; and in Sect. 5 we present numerical results
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
