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An Immersed Finite Element Method for Anisotropic Elliptic Interface Problems with Nonhomogeneous Jump Conditions

Haifeng Ji, Zhilin Li

TL;DR

This work presents an immersed finite element method (IFE) for anisotropic elliptic interface problems with nonhomogeneous jump conditions on unfitted meshes. It preserves the standard nonconforming degrees of freedom while modifying the local space to encode jump conditions, introduces a correction function to handle nonhomogeneous data, and derives explicit IFE basis and correction functions; the analysis proves existence/uniqueness of the IFE space and optimal $H^1$- and $L^2$-error estimates. The resulting stiffness matrix is shown to have a condition number comparable to that of a standard nonconforming FEM, enabling a robust preconditioner based on a Gauss–Seidel smoother with interface correction. Numerical experiments in 2D and 3D confirm optimal convergence and robustness against high diffusion-contrast, demonstrating the method’s effectiveness for unfitted mesh interface problems. The approach advances unfitted mesh techniques for complex anisotropic interfaces with practical solvers and provable guarantees.

Abstract

A new finite element method (FEM) using meshes that do not necessarily align with the interface is developed for two- and three-dimensional anisotropic elliptic interface problems with nonhomogeneous jump conditions. The degrees of freedom of the proposed method are the same as those of traditional nonconforming FEMs, while the function space is modified to account for the jump conditions of the solution. The modified function space on an interface element is shown to exist uniquely, independent of the element's shape and the manner in which the interface intersects it. Optimal error estimates for the method, along with the usual bound on the condition number of the stiffness matrix, are proven, with the error constant independent of the interface's location relative to the mesh. To solve the resulting linear system, a preconditioner is proposed in which a Gauss-Seidel smoother with the interface correction is employed to ensure robustness against large jumps in the diffusion matrix. Numerical experiments are provided to demonstrate the optimal convergence of the proposed method and the efficiency of the preconditioner.

An Immersed Finite Element Method for Anisotropic Elliptic Interface Problems with Nonhomogeneous Jump Conditions

TL;DR

This work presents an immersed finite element method (IFE) for anisotropic elliptic interface problems with nonhomogeneous jump conditions on unfitted meshes. It preserves the standard nonconforming degrees of freedom while modifying the local space to encode jump conditions, introduces a correction function to handle nonhomogeneous data, and derives explicit IFE basis and correction functions; the analysis proves existence/uniqueness of the IFE space and optimal - and -error estimates. The resulting stiffness matrix is shown to have a condition number comparable to that of a standard nonconforming FEM, enabling a robust preconditioner based on a Gauss–Seidel smoother with interface correction. Numerical experiments in 2D and 3D confirm optimal convergence and robustness against high diffusion-contrast, demonstrating the method’s effectiveness for unfitted mesh interface problems. The approach advances unfitted mesh techniques for complex anisotropic interfaces with practical solvers and provable guarantees.

Abstract

A new finite element method (FEM) using meshes that do not necessarily align with the interface is developed for two- and three-dimensional anisotropic elliptic interface problems with nonhomogeneous jump conditions. The degrees of freedom of the proposed method are the same as those of traditional nonconforming FEMs, while the function space is modified to account for the jump conditions of the solution. The modified function space on an interface element is shown to exist uniquely, independent of the element's shape and the manner in which the interface intersects it. Optimal error estimates for the method, along with the usual bound on the condition number of the stiffness matrix, are proven, with the error constant independent of the interface's location relative to the mesh. To solve the resulting linear system, a preconditioner is proposed in which a Gauss-Seidel smoother with the interface correction is employed to ensure robustness against large jumps in the diffusion matrix. Numerical experiments are provided to demonstrate the optimal convergence of the proposed method and the efficiency of the preconditioner.
Paper Structure (20 sections, 18 theorems, 149 equations, 4 tables, 2 algorithms)

This paper contains 20 sections, 18 theorems, 149 equations, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finite element approximation
  4. Local approximations
  5. Correction functions
  6. IFE spaces and interpolation operators
  7. The IFE method
  8. Explicit IFE basis and correction functions
  9. Proof of Lemma \ref{['lem_unique']}
  10. Explicit IFE basis functions
  11. Explicit correction functions
  12. Error estimates for the IFE method
  13. Coercivity and continuity
  14. Norm-equivalence for IFE functions
  15. Consistency
  16. ...and 5 more sections

Key Result

Lemma 3.1

For each interface element $T$ (a triangle for $N=2$ or a tetrahedron for $N=3$) without any angle restrictions, the pair $(\phi^+,\phi^-)\in\mathbb{P}_1(\mathbb{R}^N)\times\mathbb{P}_1(\mathbb{R}^N)$ is uniquely determined by $\mathcal{J}_{i,T}$ for all $i=0, \cdots, N$ and $\mathcal{M}_{F,T}$ for

Theorems & Definitions (34)

  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Remark 3.4
  • Remark 3.5
  • Lemma 3.6
  • proof
  • proof
  • Lemma 3.7
  • proof
  • ...and 24 more