Cut Finite Element Methods for Convection-Diffusion in Mixed-Dimensional Domains
This work addresses modeling challenges in fractured porous media for computational engineering, but it is incremental as it adapts existing CutFEM techniques to a specific domain type.
The authors tackled convection-diffusion problems on mixed-dimensional domains, such as fractured porous media, by developing a cut finite element method (CutFEM) that uses a non-conforming background mesh and proved a priori error estimates with convergence for solutions of reduced regularity, achieving theoretical rates confirmed by numerical experiments.
We develop a cut finite element method (CutFEM) for convection-diffusion problems posed on mixed-dimensional domains, i.e., unions of manifolds of different dimensions arranged in a hierarchical structure where lower-dimensional components form parts of the boundaries of higher-dimensional ones. Such domains arise, for instance, in the modeling of fractured porous media with intersecting fractures. The model problem is formulated in a compact abstract form using mixed-dimensional directional derivative and divergence operators, which allows the problem to be expressed in a way that closely resembles the classical convection-diffusion equation. The proposed CutFEM is based on a fixed background mesh that does not conform to the geometry, with each manifold component represented through its associated active mesh. The method employs continuous piecewise linear elements together with weak enforcement of coupling conditions and suitable stabilization. We prove a priori error estimates in energy and $L^2$ norms and establish convergence, also for solutions with reduced regularity $u \in H^s$, $1 < s \le 2$. Numerical experiments confirm the theoretical convergence rates and illustrate the performance of the method.