The Frenet immersed finite element method for elliptic interface problems: An error analysis

TL;DR

The paper establishes an error analysis for the Frenet immersed finite element method applied to elliptic interface problems. By proving a key trace inequality for Frenet IFE functions and embedding the method within a symmetric interior penalty DG framework, it demonstrates coercivity of the discrete bilinear form and derives optimal convergence rates in both $L^2$ and energy norms, with explicit dependence on the contrast parameters $\beta^+$ and $\beta^-$. The analysis leverages a Frenet transformation that maps curved interface elements to a reference domain, enabling exact satisfaction of interface jump conditions and robust geometric control. These results provide theoretical justification for the Frenet IFE approach and its ability to deliver high-order accuracy on unfitted meshes without interface penalties. Overall, the work solidifies the method’s reliability for simulations in heterogeneous media with moving or complex interfaces.

Abstract

This article presents an error analysis of the recently introduced Frenet immersed finite element (IFE) method. The Frenet IFE space employed in this method is constructed to be locally conforming to the function space of the associated weak form for the interface problem. This article further establishes a critical trace inequality for the Frenet IFE functions. These features enable us to prove that the Frenet IFE method converges optimally under mesh refinement in both $L^2$ and energy norms.

Figures (6)

• Figure 1: An illustration of an interface element $K$ and the fictitious element $K_F$ (right), and the corresponding Frenet interface element $\hat{K}$ and the Frenet fictitious element/rectangle $\hat{K}_F$.
• Figure 2: Element $K(n)$, $n=2,3,4,5$ (left) with different sizes. The sets $\hat{K}(n)$ (in red) and $\check{K}(n)$ (in blue) for $n=2,3,4,5$ (right).
• Figure 3: An illustration of a reference Frenet element $\hat{K}$ (red), a Frenet parallelogram $\check{K}$ (blue) and a fictitious Frenet parallelogram $\check{\check{K}}$ (black).
• Figure 4: An illustration of the parametrizations $\hat{\gamma}$ and $\check{\gamma}$.
• Figure 5: An illustration of the curved triangles described in the proof of \ref{['lem:stable_extensions_poly']}

Theorems & Definitions (41)

• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6