Cut Finite Elements for Convection in Fractured Domains

TL;DR

This paper introduces a novel cut finite element method (CutFEM) for solving convection problems in complex fractured domains, enabling accurate modeling of porous media with intersecting fractures using fixed background meshes.

Contribution

The paper develops a new CutFEM approach for convection in fractured domains, including error analysis and numerical validation, handling arbitrary mesh intersections.

Findings

Proven a priori error estimates for the method.

Numerical examples demonstrate method effectiveness.

Applicable to porous media with complex fracture networks.

Abstract

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.