An Enriched Immersed Finite Element Method for 3D Interface Problems

TL;DR

This work introduces a three-dimensional enriched immersed finite element method for interface problems with non-homogeneous jump conditions. By decomposing the interface element space into a homogeneous part and a jump-enrichment part determined directly from the jump data, the global discretization remains isomorphic to the standard FE space on the same mesh, yielding conditioning that scales as $\mathcal{O}(h^{-2})$ independently of interface location and enabling efficient multigrid solvers. The authors develop data-aware interpolation and quasi-interpolation operators to handle non-homogeneous data, provide rigorous error and conditioning analyses, and validate the approach with extensive 3D numerical experiments, including moving and real-world interface geometries. The results show optimal convergence rates in multiple norms and robust solver performance across high-contrast coefficient jumps, making the method practically impactful for complex 3D interface simulations. Overall, the enriched IFE framework combines homogenization-inspired enrichment with a fixed, well-conditioned linear system, offering a scalable tool for 3D unfitted-interface problems with non-homogeneous jumps.

Abstract

We introduce an enriched immersed finite element method for addressing interface problems characterized by general non-homogeneous jump conditions. Unlike many existing unfitted mesh methods, our approach incorporates a homogenization concept. The IFE trial function set is composed of two components: the standard homogeneous IFE space and additional enrichment IFE functions. These enrichment functions are directly determined by the jump data, without adding extra degrees of freedom to the system. Meanwhile, the homogeneous IFE space is isomorphic to the standard finite element space on the same mesh. This isomorphism remains stable regardless of interface location relative to the mesh, ensuring optimal $\mathcal{O}(h^2)$ conditioning that is independent of the interface location and facilitates an immediate development of a multigrid fast solver; namely the iteration numbers are independent of not only the mesh size but also the relative interface location. Theoretical analysis and extensive numerical experiments are carried out in the efforts to demonstrate these features.

Figures (13)

• Figure 1: A three-dimensional domain with interface.
• Figure 2: Interface elements: 3 cutting points (left) and 4 cutting points (right).
• Figure 3: Partition a cuboid into 5 tetrahedra.
• Figure 4: Comparison of FE and IFE basis functions: the homogeneous IFE shape functions $\phi_{j,T}$ are shown in second row, while non-homogeneous IFE shape functions $\xi_{j,T}$ are shown in the third row.
• Figure 5: Convergence of IFE solutions: Example 1 spherical interface (left) and Example 2 orthocircle interface (right).

Theorems & Definitions (49)

• Lemma 2.1
• proof
• Lemma 2.2: Theorem 2.2,2020GuoLin2
• Remark 3.1
• Remark 3.2
• Theorem 3.1: Bounds of IFE shape functions
• proof
• Remark 3.3
• Lemma 3.1
• proof