A fitted space-time finite element method for an advection-diffusion problem with moving interfaces

TL;DR

We address a parabolic advection-diffusion problem with a moving interface across which the diffusion coefficient jumps. The authors develop a space-time interface-fitted finite element method using a continuous Galerkin formulation on fully unstructured space-time meshes, and prove well-posedness via the Banach-Nečas-Babuška theorem. They derive a priori error estimates in a discrete energy norm under a globally low but locally high regularity assumption by employing Stein extension operators to handle interface-induced low global regularity. Numerical experiments in 2D and 3D space-time domains validate the theoretical rates and demonstrate robustness to interface motion and coefficient discontinuities, underscoring the method’s practical utility for moving-interface problems.

Abstract

This paper presents a space-time interface-fitted finite element method for solving a parabolic advection-diffusion problem with a nonstationary interface. The jumping diffusion coefficient gives rise to the discontinuity of the solution gradient across the interface. We use the Banach-Necas-Babuska theorem to show the well-posedness of the continuous variational problem. A fully discrete finite-element based scheme is analyzed using the Galerkin method and unstructured interface-fitted meshes. An optimal error estimate is established in a discrete energy norm under a globally low but locally high regularity condition. Some numerical results corroborate our theoretical results.

Figures (8)

• Figure 1: The domain $\Omega$ consisting of two subdomains $\Omega_1(t)$ and $\Omega_2(t)$ moving with a velocity $\mathbf{v}$, separated by the interface $\Gamma(t)$.
• Figure 2: Illustration of a space-time domain $Q_T\subset \mathbb{R}^2$ and its space-time interface-fitted decomposition: The gray elements are in $\overline{Q_1}$, the blue elements are in $\overline{Q_2}$, and the green ones intersect both $Q_1$ and $Q_2$.
• Figure 3: Illustration of the discrepancy region $S_h = S_h^1\cup S_h^2$ with $d=1$. The red curve represents the space-time interface $\Gamma^\ast$, the black diagonal represents the discrete interface $\Gamma_h^\ast$.
• Figure 4: The space-time domain $Q_T$ with an interface evolving at the velocity $\mathrm{v} = 0.1$ in the first example (left), and $\mathrm{v} = 0.1 \pi \cos(2\pi t)$ in the second example (right).
• Figure 5: The numerical solutions of Example 1 (left) and Example 2 (right).

Theorems & Definitions (22)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.1
• Lemma 3.1
• proof
• Theorem 3.1
• Lemma 3.2
• Lemma 3.3