Projection-based approximations for eigenvalue problems of Fredholm integral operators with Green's kernels

1. Problem Statement

The paper addresses the numerical solution of the eigenvalue problem for a compact linear integral operator $K$ defined on the space $X = L^2[0, 1]$ or $C[0, 1]$:
$$K\phi = \lambda\phi$$
where the kernel $\kappa(s, t)$ is of Green's function type. Unlike smooth kernels, Green's kernels are continuous on the square $\Omega = [0, 1] \times [0, 1]$ but possess limited differentiability along the diagonal $s=t$. Specifically, the kernel is defined piecewise as $\kappa_1(s, t)$ for $t \leq s$ and $\kappa_2(s, t)$ for $s \leq t$, where $\kappa_1, \kappa_2 \in C^\alpha(\Omega_i)$ with $\alpha > r$.

The goal is to approximate the eigenvalues $\lambda$ and eigenfunctions $\phi$ using finite-rank operators $K_n$ derived from projection methods, while establishing rigorous convergence rates for these approximations.

2. Methodology

The authors employ a unified framework analyzing both classical projection methods and modified projection methods (including iterated variants).

2.1 Discretization Setup

• Partition: A uniform partition of $[0, 1]$ is used with step size $h = 1/n$.
• Approximating Space ($X_n$): The space of piecewise polynomials of even degree $\leq 2r$ on each subinterval. Continuity at partition points is not required.
• Projections: Two types of projections mapping into $X_n$ are analyzed:
  1. Orthogonal Projection ($P_n$): The $L^2$-orthogonal projection onto $X_n$.
  2. Interpolatory Projection ($Q_n$): Defined by interpolation at $2r+1$ equidistant points within each subinterval. Notably, the authors do not rely on Gauss points (zeros of Legendre polynomials), which simplifies implementation compared to prior works.

2.2 Approximation Schemes

The paper compares three main approaches:

1. Classical Projection (Galerkin/Collocation): Solves $\pi_n K \phi_n = \lambda_n \phi_n$, where $\pi_n$ is either $P_n$ or $Q_n$.
2. Modified Projection: Solves $K_n^M \phi_n^M = \lambda_n^M \phi_n^M$, where the operator is defined as:
   $$K_n^M = \pi_n K + K \pi_n - \pi_n K \pi_n$$
3. Iterated Methods:
   • Iterated Classical: $\phi_n^S = \frac{1}{\lambda_n} K \phi_n$.
   • Iterated Modified: $\tilde{\phi}_n^M = \frac{1}{\lambda_n^M} K \phi_n^M$.

3. Key Contributions

The primary contributions of this work are:

• Extension to Green's Kernels: Previous superconvergence results largely focused on smooth kernels. This paper extends the analysis to Green's function-type kernels, which are non-smooth across the diagonal.
• Relaxation of Collocation Nodes: The authors demonstrate that superconvergence can be achieved using equidistant collocation points rather than requiring the computationally more complex Gauss points.
• Convergence Rate Analysis: The paper establishes explicit error bounds in the supremum norm for eigenvalues and spectral subspaces for both orthogonal and interpolatory projections.
• Superconvergence via Iteration: It proves that iterating the projection method significantly enhances the convergence order of eigenfunctions, even for non-smooth kernels.

4. Main Results (Convergence Rates)

Let $h$ be the mesh size and $r$ be the degree parameter. The convergence orders for the error in eigenvalues ($|\lambda - \lambda_n|$) and eigenvectors ($\|\phi - \phi_n\|_\infty$) are summarized below:

A. Orthogonal Projection (Galerkin Method)

• Standard Galerkin:
  • Eigenvalue: $O(h^{2r+3})$ (for $r \geq 1$), $O(h^2)$ (for $r=0$).
  • Eigenvector: $O(h^{2r+1})$ (for $r \geq 1$), $O(h)$ (for $r=0$).
• Iterated Galerkin:
  • Eigenvector: $O(h^{2r+3})$ (for $r \geq 1$), $O(h^2)$ (for $r=0$).
• Modified Galerkin:
  • Eigenvalue: $O(h^{2r+5})$ (for $r \geq 1$), $O(h^4)$ (for $r=0$).
  • Eigenvector: $O(h^{2r+3})$ (for $r \geq 1$), $O(h^3)$ (for $r=0$).
• Iterated Modified Galerkin:
  • Eigenvector: $O(h^{2r+5})$ (for $r \geq 1$), $O(h^4)$ (for $r=0$).

B. Interpolatory Projection (Collocation Method)

• Standard Collocation:
  • Eigenvalue: $O(h^{2r+2})$.
  • Eigenvector: $O(h^{2r+1})$.
• Iterated Collocation:
  • Eigenvector: $O(h^{2r+2})$.
• Modified Collocation:
  • Eigenvalue: $O(h^{2r+3})$ (for $r \geq 1$), $O(h^4)$ (for $r=0$).
  • Eigenvector: $O(h^{2r+2})$ (for $r \geq 1$), $O(h^3)$ (for $r=0$).
• Iterated Modified Collocation:
  • Eigenvector: $O(h^{2r+3})$ (for $r \geq 1$), $O(h^4)$ (for $r=0$).

Key Observation: The modified projection method consistently yields higher convergence orders for eigenvalues compared to the classical method. Furthermore, the iterated modified method achieves the highest accuracy for eigenvectors (superconvergence).

5. Numerical Verification

The theoretical results were validated using a specific Green's kernel example: $\kappa(s, t) = \min(s, t) - st$ (related to the operator with eigenvalues $1/\pi^2$).

• Piecewise Constant Case ($r=0$):
  • Standard Galerkin/Collocation achieved $O(h^2)$ for eigenvalues and $O(h)$ for eigenvectors.
  • Modified methods achieved $O(h^4)$ for eigenvalues.
  • Iterated modified methods achieved $O(h^4)$ for eigenvectors.
• The numerical experiments confirmed the theoretical convergence rates, showing that the modified and iterated methods significantly outperform classical approaches for the same mesh size.

6. Significance and Conclusion

This paper provides a robust theoretical and numerical framework for solving eigenvalue problems involving Green's kernels, which are common in boundary value problems but difficult to handle due to their lack of smoothness.

• Efficiency: The modified projection method offers a superior accuracy-to-cost ratio. While it increases the dimension of the resulting matrix eigenvalue problem by a factor of roughly 2 (due to the term $K\pi_n K$), the gain in convergence order (e.g., from $O(h^2)$ to $O(h^4)$) means fewer grid points are needed to achieve a desired accuracy.
• Flexibility: By proving that equidistant points suffice for superconvergence, the authors remove the dependency on Gauss points, making the implementation simpler and more flexible for various applications.
• Robustness: The results hold even when the kernel is not fully smooth, extending the applicability of high-order projection methods to a broader class of integral operators.

The authors conclude that the modified projection approach is an efficient and reliable tool for eigenvalue computations involving Green's kernels, with future work suggested for singular kernels and asymptotic expansions.