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Optimal $L^2$ error estimation for the unfitted interface finite element method based on the non-symmetric Nitsche's methods

Gang Chen, Chaoran Liu, Yangwen Zhang

TL;DR

The paper develops a robust theoretical framework to achieve optimal $L^2$ error estimates for unfitted interface finite element methods based on the non-symmetric Nitsche formulation. By constructing a tailored dual problem that preserves adjoint-consistency and performing detailed regularity analysis for both interface and dual problems, it overcomes the non-symmetry barrier in Aubin–Nitsche-type arguments. The work delivers coefficient-robust stability, a penalty-free variant, and precise $L^2$-norm error bounds, thereby strengthening the theoretical foundations and practical applicability of unfitted FEM for interface problems. This advances accurate simulation of multiphase and heterogeneous media where interfaces and material contrasts are complex or evolving.

Abstract

This paper establishes optimal error estimates in the $L^2$ for the non-symmetric Nitsche method in an unfitted interface finite element setting. Extending our earlier work, we give a complete analysis for the Poisson interface model and, by formulating a tailored dual problem that restores adjoint consistency, derive the desired bounds.

Optimal $L^2$ error estimation for the unfitted interface finite element method based on the non-symmetric Nitsche's methods

TL;DR

The paper develops a robust theoretical framework to achieve optimal error estimates for unfitted interface finite element methods based on the non-symmetric Nitsche formulation. By constructing a tailored dual problem that preserves adjoint-consistency and performing detailed regularity analysis for both interface and dual problems, it overcomes the non-symmetry barrier in Aubin–Nitsche-type arguments. The work delivers coefficient-robust stability, a penalty-free variant, and precise -norm error bounds, thereby strengthening the theoretical foundations and practical applicability of unfitted FEM for interface problems. This advances accurate simulation of multiphase and heterogeneous media where interfaces and material contrasts are complex or evolving.

Abstract

This paper establishes optimal error estimates in the for the non-symmetric Nitsche method in an unfitted interface finite element setting. Extending our earlier work, we give a complete analysis for the Poisson interface model and, by formulating a tailored dual problem that restores adjoint consistency, derive the desired bounds.
Paper Structure (18 sections, 31 theorems, 251 equations)

This paper contains 18 sections, 31 theorems, 251 equations.

Table of Contents

  1. Introduction.
  2. Notations and Basic results
  3. Notations
  4. Basic results
  5. Finite element method
  6. Some novel regularity results
  7. The basic regularity results
  8. The regularity for interface problems
  9. Regularity for Robin interface condition
  10. Regularity for the dual problem
  11. Error estimations
  12. The Scott-Zhang interpolation
  13. Energy Error Estimation
  14. $L^2$ error estiamtion
  15. Error estimations for penalty-free
  16. ...and 3 more sections

Key Result

Lemma 1

For any $T\in\mathcal{T}_h$ and $v_h\in \mathcal{P}_k(T)$, there holds where $\mathcal{P}_k(T)$ is the polynomial space on the element $T$.

Theorems & Definitions (59)

  • Lemma 1: The trace inequality
  • Lemma 2: The inverse inequality
  • Lemma 3
  • Lemma 4
  • proof
  • Remark 1
  • Lemma 5: Existence of a unique solution
  • Lemma 6: Orthogonality
  • Theorem 1
  • Theorem 2
  • ...and 49 more