Evolving finite elements for advection diffusion with an evolving interface

TL;DR

This work tackles parabolic advection-diffusion on domains with evolving interfaces by developing an ALE-based evolving finite element method that uses fitted, isoparametric meshes to capture the moving boundary with high accuracy. The authors build a rigorous moving-space framework, construct evolving FE spaces with a lift, and formulate a discrete variational problem whose mass, stiffness, and transport terms are defined on time-dependent domains. They prove an optimal a priori error bound in the $L^2$-norm of order $h^{k+1}$ (with $k$ denoting the polynomial degree) and provide a Ritz-projection-based analysis together with geometric-perturbation estimates to justify convergence; numerical experiments in 2D and 3D verify the predicted rates. The methodology and results advance high-order, geometry-aware simulations of transport-diffusion in evolving domains, with practical implications for applications in fluid-structure interaction, material science, and biology where moving interfaces are essential.

Abstract

The aim of this paper is to develop a numerical scheme to approximate evolving interface problems for parabolic equations based on the abstract evolving finite element framework proposed in (C M Elliott, T Ranner, IMA J Num Anal, 41:3, 2021, doi:10.1093/imanum/draa062). An appropriate weak formulation of the problem is derived for the use of evolving finite elements designed to accommodate for a moving interface. Optimal order error bounds are proved for arbitrary order evolving isoparametric finite elements. The paper concludes with numerical results for a model problem verifying orders of convergence.

Figures (6)

• Figure 2.1: An example configuration of the domain.
• Figure 3.1: Showing the difference between a non viable initial mesh and an adequate one for a circle enclosed in a square. The one on the left breaking condition \ref{['M5']} whereas the one on the right following condition \ref{['M5']}.
• Figure 3.2: The intersection of two interface elements (teal and blue respectively) of different domains. The shared interface facet is then pushed by the map $\mathbf{\Psi}^h$ to become a piece of $\Gamma(0)$. The map $\widetilde{I}^h \mathbf{\Psi}^h$ maps the original mesh to an isoparametric mesh approximating the interface.
• Figure 3.3: Example of the temporal deformation of an interior element in three space dimensions.
• Figure 3.4: Schematic of the setup used. $\Lambda^h(t;x)$ might be needed, depending on the problem, to define the discrete data. However once the discrete problem is known, only the knowledge of $\Omega^h_i(0)$, $\Gamma^h(0)$ and $\mathbf{\Phi}^h(t;\cdot)$ are needed to calculate the discrete solution $U^h(t;\cdot)$. $\mathbf{\Phi^l}$ is only needed in the analysis of theoretical error estimates.

Theorems & Definitions (61)

• Remark 2.1
• Lemma 2.2: diogo, Thm. 2.4
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Theorem 2.6: The Transport Theorem
• Lemma 2.7: Characterisation of Material Derivative
• Remark 2.8
• Remark 2.9
• Remark 2.10