A Cut Finite Element Method for Elliptic Bulk Problems with Embedded Surfaces

TL;DR

This paper introduces an unfitted finite element method for modeling flow in fractured porous media, accurately capturing the interface and flow dynamics across complex fracture networks.

Contribution

It presents a novel cut finite element approach with Nitsche mortaring for coupled flow in fractured media, including bifurcating fractures and error analysis.

Findings

Achieves optimal error estimates under regularity assumptions

Successfully models flow in complex fracture networks

Numerical examples validate theoretical results

Abstract

We propose an unfitted finite element method for flow in fractured porous media. The coupling across the fracture uses a Nitsche type mortaring, allowing for an accurate representation of the jump in the normal component of the gradient of the discrete solution across the fracture. The flow field in the fracture is modelled simultaneously, using the average of traces of the bulk variables on the fractured. In particular the Laplace-Beltrami operator for the transport in the fracture is included using the average of the projection on the tangential plane of the fracture of the trace of the bulk gradient. Optimal order error estimates are proven under suitable regularity assumptions on the domain geometry. The extension to the case of bifurcating fractures is discussed. Finally the theory is illustrated by a series of numerical examples.