A High-Order Conservative Cut Finite Element Method for Problems in Time-Dependent Domains

TL;DR

This work develops a mass-conserving, high-order unfitted space-time CutFEM for convection-diffusion on evolving domains by leveraging Reynolds' transport theorem to achieve natural mass conservation. A time-dependent macroelement stabilization strategy is introduced to stabilize the cut elements efficiently, reducing the stabilization footprint and improving matrix sparsity and conditioning. The method supports both bulk and coupled bulk-surface problems, with high-order accuracy demonstrated through comprehensive numerical experiments, and demonstrates mass conservation up to machine precision for the conservative formulation. The framework relies on Gauss-Lobatto time quadrature and Sayé-based high-order space quadrature on cut elements, and it is extensible to nonlinear surface coupling models such as Langmuir adsorption of surfactants. Overall, the approach provides a robust, high-order, mass-conserving unfitted discretization for moving-domain problems with practical implications for surfactant transport and related applications.

Abstract

A mass-conservative high-order unfitted finite element method for convection-diffusion equations in evolving domains is proposed. The space-time method presented in [P. Hansbo, M. G. Larson, S. Zahedi, Comput. Methods Appl. Mech. Engrg. 307 (2016)] is extended to naturally achieve mass conservation by utilizing Reynold's transport theorem. Furthermore, by partitioning the time-dependent domain into macroelements, a more efficient stabilization procedure for the cut finite element method in time-dependent domains is presented. Numerical experiments illustrate that the method fulfills mass conservation, attains high-order convergence, and the condition number of the resulting system matrix is controlled while sparsity is increased. Problems in bulk domains as well as coupled bulk-surface problems are considered.

Figures (20)

• Figure 1: A domain sketch.
• Figure 2: Background domain in white and elements in the active meshes $\mathcal{T}_{h}^n$ (left) and $\mathcal{T}_{h,\Gamma}^n$ (right) in grey. The boundary $\Gamma(t)$ is visualized at time instances $t=t_{n-1}$ (green) and $t=t_n$ (blue).
• Figure 3: Macroelement stabilization (left) versus full stabilization (right). The surface is here visualized at three time instances, $Q_n = \{t_{n-1}, (t_{n-1}+t_n)/2, t_n\} \subset I_n$ which are used to construct the macroelement partition. Black faces mark external borders of macroelements $M\in\mathcal{M}_{h}^n$, and the yellow faces mark the internal faces $F\in \mathcal{F}_{h}^{\mathcal{M},n}$ in the left panel and $F\in \mathcal{F}_h^{n}$ in the right panel. The parameter $\delta$ is chosen as $\delta = 0.3$.
• Figure 4: Quadrature nodes produced by the algorithm presented in saye for approximating an integral on a cut element $\Omega(t)\cap K$. The nodes are distributed as a tensor product between two one-dimensional quadrature rules with five quadrature nodes. This results in an order of accuracy of $10$.
• Figure 5: Example \ref{['subsection:ex1']}: Numerical solution using $h=0.025$. The active mesh is illustrated in grey.