Cut Finite Elements for Convection in Fractured Domains

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.

References in corpus (3)

• Dual Virtual Element Methods for Discrete Fracture Matrix Models
• A Nitsche Method for Elliptic Problems on Composite Surfaces
• Modeling, Structure and Discretization of Mixed-dimensional Partial Differential Equations

Cited by in corpus (4)

• A hybrid-mixed finite element method for single-phase Darcy flow in fractured porous media
• A hybridizable discontinuous Galerkin method on unfitted meshes for single-phase Darcy flow in fractured porous media
• The hybrid dimensional representation of permeability tensor: a reinterpretation of the discrete fracture model and its extension on nonconforming meshes
• A novel equi-dimensional finite element method for flow and transport in fractured porous media satisfying discrete maximum principle and conservation properties