A cut finite element method for a model of pressure in fractured media

TL;DR

This paper introduces a robust cut finite element method for modeling diffusion in fractured media, allowing cracks to cut through the mesh in a flexible manner and accurately capturing pressure interactions.

Contribution

It develops a novel cut finite element approach with stability and accuracy for pressure modeling in fractured media with complex crack geometries.

Findings

Method is stable across all parameters.

Achieves optimal order a priori error estimates.

Numerical examples confirm theoretical results.

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.