A Cut Finite Element Method for Elliptic Bulk Problems with Embedded Surfaces
This work provides a rigorous numerical method for simulating flow in fractured porous media, which is important for geoscience applications like reservoir engineering.
The paper develops an unfitted finite element method for flow in fractured porous media, using Nitsche-type mortaring to handle coupling across fractures and including the Laplace-Beltrami operator for fracture transport. Optimal error estimates are proven and validated with numerical examples.
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.