Cut Finite Element Methods for Convection-Diffusion in Mixed-Dimensional Domains

TL;DR

This paper introduces a novel cut finite element method (CutFEM) for solving convection-diffusion problems on complex mixed-dimensional domains, such as fractured porous media, using a fixed background mesh and weak coupling.

Contribution

The paper develops a new CutFEM approach for mixed-dimensional convection-diffusion problems, providing error analysis and demonstrating convergence on hierarchical manifolds.

Findings

The method achieves optimal convergence rates in energy and L2 norms.

Numerical experiments confirm theoretical error estimates.

The approach handles solutions with reduced regularity $u \,\in\, H^s$, $1<s\le2$.

Abstract

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.