Nitsche-based FEM for the Laplace eigenvalue problem: spectral approximation and a posteriori error analysis

Technical Summary: Nitsche-Based FEM for the Laplace Eigenvalue Problem

Problem Statement
This paper addresses the numerical approximation of the Laplace eigenvalue problem with mixed boundary conditions, specifically focusing on the weak imposition of essential (Dirichlet) boundary conditions via the Nitsche method. While continuous finite element methods and Discontinuous Galerkin (DG) methods are well-established for such problems, the authors note a gap in the rigorous spectral approximation theory for eigenvalue problems where Dirichlet conditions are enforced weakly using Nitsche's technique. Previous studies have largely focused on source problems or specific variants without a comprehensive spectral convergence analysis for the Laplace operator with mixed boundaries.

Methodology
The authors formulate the problem using a primal variational approach. They introduce a finite element discretization where the Dirichlet boundary condition is not enforced strongly but is incorporated into the bilinear form through three distinct Nitsche variants:

1. Symmetric ($\varepsilon = 1$): The standard symmetric formulation.
2. Incomplete ($\varepsilon = 0$): A formulation lacking symmetry.
3. Skew-symmetric ($\varepsilon = -1$): A formulation proven to be elliptic for any positive stabilization parameter $\alpha$.

The analysis is conducted within the framework of compact operator theory. The authors define discrete solution operators and utilize the Babuška-Osborn abstract spectral approximation theory to study the convergence of the discrete spectrum to the continuous spectrum. Key steps include proving the norm convergence of the discrete solution operator to the continuous one and establishing error estimates for eigenvalues and eigenfunctions.

Additionally, the paper develops a residual-based a posteriori error estimator specifically for the symmetric Nitsche formulation. This estimator comprises element residuals, jumps of the normal derivative across interior facets, and boundary residuals arising from the weak imposition of the Dirichlet condition. The reliability of this estimator is proven under a saturation assumption, while local efficiency is established using standard bubble-function arguments.

Key Contributions and Results

• Spectral Convergence Theory: The paper provides the first rigorous proof of convergence for the Laplace eigenproblem with mixed boundary conditions where the Dirichlet part is imposed weakly via Nitsche's method.
• Error Estimates:
  • For the symmetric variant, the authors prove optimal order error estimates for eigenfunctions and a double order of convergence for eigenvalues.
  • For the non-symmetric variants (incomplete and skew-symmetric), the analysis yields suboptimal convergence rates for eigenvalues.
• Stabilization Parameter Analysis: The study highlights the critical role of the Nitsche stabilization parameter ($\alpha$). The authors demonstrate that if $\alpha$ is not sufficiently large, the method may produce spurious eigenvalues (non-physical modes) or complex eigenvalues (in non-symmetric cases). Conversely, for sufficiently large $\alpha$, the method achieves accuracy comparable to standard strong imposition methods.
• A Posteriori Analysis: A reliable and efficient residual-based error estimator is derived for the symmetric case, providing the theoretical foundation for adaptive mesh refinement.
• Numerical Validation: Extensive numerical experiments in 2D and 3D validate the theoretical findings.
  • 2D Tests: Experiments on a rectangle and an L-shaped domain confirm the predicted convergence rates and illustrate the impact of $\alpha$ on spectral cleanliness. The L-shaped domain tests demonstrate the method's ability to handle geometric singularities.
  • 3D Tests: Experiments on a unit cube and a non-convex Fichera-type domain (with both fully Dirichlet and mixed boundary conditions) show that the adaptive strategy effectively concentrates degrees of freedom near singularities (corners and edges), recovering optimal convergence rates even in configurations where regularity is reduced.

Significance and Claims
The authors claim that this work fills a theoretical gap by establishing a rigorous spectral approximation theory for Nitsche-based finite element methods applied to eigenvalue problems. They emphasize that while the Nitsche method offers flexibility in handling boundary conditions and interfaces, its application to spectral problems requires careful tuning of the stabilization parameter to avoid spurious modes.

The paper positions itself as a foundational step ("stepping stone") for analyzing Nitsche methods in more complex eigenvalue problems, such as the Stokes eigenvalue problem with slip boundary conditions. The authors modestly note that their a posteriori analysis is strictly proven for the symmetric formulation with full Dirichlet conditions, though numerical experiments suggest the adaptive strategy remains robust in mixed boundary and non-convex 3D scenarios beyond the current theoretical scope. The work concludes that the skew-symmetric variant appears most robust regarding spectral computation (avoiding complex eigenvalues) despite its suboptimal convergence rates, while the symmetric variant offers optimal rates provided the stabilization parameter is chosen correctly.