Cut Finite Elements for Convection in Fractured Domains
This work provides a numerical method for simulating convection in porous media with embedded fractures, which is important for geoscience applications.
The paper develops a cut finite element method for convection in fractured domains, which are unions of manifolds of different dimensions. The method uses a fixed background mesh with weak enforcement of coupling conditions and stabilization, and proves a priori error estimates with numerical examples.
We develop a cut finite element method (CutFEM) for the convection problem in a so called fractured domain which is a union of manifolds of different dimensions such that a $d$ dimensional component always resides on the boundary of a $d+1$ dimensional component. This type of domain can for instance be used to model porous media with embedded fractures that may intersect. The convection problem can be formulated in a compact form suitable for analysis using natural abstract directional derivative and divergence operators. The cut finite element method is based on using a fixed background mesh that covers the domain and the manifolds are allowed to cut through a fixed background mesh in an arbitrary way. We consider a simple method based on continuous piecewise linear elements together with weak enforcement of the coupling conditions and stabilization. We prove a priori error estimates and present illustrating numerical examples.