Cut Finite Elements for Convection in Fractured Domains

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

Dual virtual element method for discrete fractures networks

Discrete fracture networks is a key ingredient in the simulation of physical processes which involve fluid flow in the underground, when the surrounding rock matrix is considered impervious. In this paper we present two different models to compute the pressure field and Darcy velocity in the system. The first allows a normal flow out of a fracture at the intersections, while the second grants also a tangential flow along the intersections. For the numerical discretization, we use the mixed virtual finite element method as it is known to handle grid elements of, almost, any arbitrary shape. The flexibility of the discretization allows us to loosen the requirements on grid construction, and thus significantly simplify the flow discretization compared to traditional discrete fracture network models. A coarsening algorithm, from the algebraic multigrid literature, is also considered to further speed up the computation. The performance of the method is validated by numerical experiments