A posteriori error estimates for a modified Morley FEM

1. Problem Statement

The paper addresses the numerical approximation of two specific fourth-order partial differential equations (PDEs) using the Morley Finite Element Method (FEM), a non-conforming method. The standard Morley FEM enforces continuity only at vertices and normal derivatives at edge midpoints, meaning the discrete functions do not belong to the Sobolev space $H^1(\Omega)$, let alone $H^2(\Omega)$.

The authors focus on two problems where the Morley FEM is applied in a modified manner (introducing a nodal interpolation operator in lower-order terms):

1. Singularly Perturbed Biharmonic Equation:
   $$\varepsilon^2 \Delta^2 u - \Delta u = f \quad \text{in } \Omega$$
   where $0 < \varepsilon \ll 1$. This problem exhibits boundary layers when $\varepsilon$ is small, leading to reduced convergence rates for standard uniform meshes.
2. Nonlinear von Kármán Equations:
   $$\Delta^2 \psi_1 = [\psi_1, \psi_2] + f, \quad \Delta^2 \psi_2 = -\frac{1}{2}[\psi_1, \psi_1]$$
   where $[\cdot, \cdot]$ is the von Kármán bracket (Monge-Ampère type nonlinearity). The non-conformity of the Morley element makes the treatment of the nonlinear term challenging, particularly regarding the integration by parts required for error analysis.

2. Methodology

The core methodology involves deriving residual-based a posteriori error estimators that are both reliable (upper bound) and efficient (lower bound) for the modified Morley FEM.

Key Technical Components:

• Modified Discretization:
  • For the singularly perturbed problem, the gradient term $\nabla u \cdot \nabla v$ is discretized using the nodal interpolation operator $I_h$ applied to the Morley functions ($\nabla I_h u_h \cdot \nabla I_h v_h$). This ensures the term is computable since $I_h u_h$ is continuous.
  • For the von Kármán equations, the nonlinear bracket term is treated similarly, incorporating $I_h$ to handle the non-conforming nature of the test and trial functions.
• Error Estimator Construction:
  The global error estimator $\eta$ is defined as the square root of the sum of local contributions $\eta(T)$, which include:
  • Non-conformity terms: Jumps in the gradient across edges ($\mu_{NC}$) and the difference between the discrete solution and its nodal interpolant ($\mu_{Ih}$).
  • Data oscillation: Terms involving the source function $f$ and mesh size $h$.
  • Residuals: Jumps in the normal derivatives of the Hessian and jumps in the normal derivatives of the interpolated gradient.
• Analytical Tools:
  • Enriching Operators: The authors utilize an enriching operator $J: M_0(\mathcal{T}) \to H^2_0(\Omega)$ to map non-conforming functions to conforming ones, allowing the use of standard duality arguments.
  • Smoothing Operators: A specialized smoothing operator $\hat{\Pi}_C$ is constructed to handle the non-conformity error in the $\varepsilon$-dependent norm, ensuring the bounds are independent of the ratio $h/\varepsilon$.
  • Bubble Functions: Standard bubble function techniques (volume and edge bubbles) are employed to prove the efficiency (lower bound) of the estimators.

3. Key Contributions

1. Reliability and Efficiency for Singular Perturbation:
   The paper proves that the derived estimator $\eta$ satisfies:
   $$|||u - u_h|||_{\varepsilon, pw} + \|\nabla(u - I_h u_h)\|_{L^2} \lesssim \eta \lesssim |||u - u_h|||_{\varepsilon, pw} + \|\nabla(u - I_h u_h)\|_{L^2} + \text{osc}$$
   Crucially, the efficiency proof holds without assuming high regularity of the exact solution, overcoming limitations of previous works (e.g., [ZW08]) where efficiency relied on smoothness. The analysis explicitly handles the boundary layer regime ($\varepsilon \ll h$).

2. Extension to Nonlinear von Kármán Equations:
   The authors adapt the techniques from the linear singularly perturbed case to the nonlinear von Kármán system. They successfully derive a reliable and efficient estimator for the modified Morley FEM applied to these equations, resolving issues related to the symmetry and integration of the nonlinear bracket term.

3. Adaptive Algorithm Validation:
   The paper investigates an adaptive mesh refinement algorithm driven by these estimators. It demonstrates that the estimator effectively captures both boundary layers (in the singularly perturbed case) and corner singularities (in the von Kármán case on non-convex domains).

4. Results

The theoretical findings are validated through four numerical experiments:

• Singularly Perturbed Biharmonic (Example 5.1 & 5.2):

  • Boundary Layers: For small $\varepsilon$ ($10^{-2}$), uniform meshes fail to capture the boundary layer, resulting in suboptimal convergence. The adaptive algorithm, driven by the new estimator, refines the mesh near the boundary, recovering optimal convergence rates.
  • Singularities: On an L-shaped domain, the estimator correctly identifies the re-entrant corner singularity and the boundary layer, guiding refinement to both features.
  • Pre-asymptotic Behavior: The study analyzes the transition where the mesh size $h_T$ becomes smaller than $\varepsilon$, showing the estimator remains robust.

• von Kármán Equations (Example 5.3 & 5.4):

  • Convex Domain: On a square domain with a smooth solution, both uniform and adaptive meshes achieve optimal convergence rates ($O(N^{-0.5})$).
  • Non-Convex Domain: On a domain with a cusp (re-entrant corner), uniform meshes suffer from suboptimal convergence due to the singularity. The adaptive algorithm successfully resolves the singularity, restoring optimal convergence rates.
  • Component Analysis: The breakdown of the estimator shows that the non-conformity terms ($\mu_{NC}, \mu_{Ih}$) and the nonlinear jump terms ($\eta_1, \eta_2$) are all significant and correctly bounded by the error.

5. Significance

• Robustness: The derived estimators are robust with respect to the singular perturbation parameter $\varepsilon$, making them suitable for problems with boundary layers where standard estimators often fail or require smoothness assumptions.
• Non-conforming Nonlinearities: The work provides a rigorous framework for handling nonlinear terms in non-conforming FEMs, specifically addressing the difficulties arising from the lack of $H^1$ continuity in the Morley element.
• Practical Applicability: The numerical experiments confirm that the theoretical estimators are effective in practice, driving adaptive mesh refinement to resolve complex features like boundary layers and geometric singularities efficiently.
• Theoretical Advancement: By modifying the smoothing operator and utilizing specific interpolation properties, the authors extend the applicability of a posteriori error analysis to a broader class of non-conforming methods and singularly perturbed problems than previously established.

In summary, this paper establishes a rigorous theoretical foundation and practical validation for using the modified Morley FEM with adaptive mesh refinement for challenging fourth-order problems involving singular perturbations and nonlinearities.