A note on improvement by iteration for the approximate solutions of second kind Fredholm integral equations with Green's kernels

1. Problem Statement

The paper addresses the numerical approximation of the linear Fredholm integral equation of the second kind:
$$x - Kx = f$$
where $f$ is a given function in $X = C[0, 1]$, and $K$ is a compact linear integral operator defined by:
$$(Kx)(s) = \int_0^1 \kappa(s, t)x(t) dt$$
Specific Challenge: The kernel $\kappa(s, t)$ is of Green's function type. This means the kernel is continuous but possesses a discontinuity in its derivatives along the diagonal $s=t$. Specifically, $\kappa$ is defined piecewise on $\Omega_1 = \{0 \le t \le s \le 1\}$ and $\Omega_2 = \{0 \le s \le t \le 1\}$, with $\kappa_1, \kappa_2$ being sufficiently smooth within their respective domains but potentially non-smooth across the diagonal.

Context: Previous literature established that for smooth kernels, using iterated collocation or modified collocation methods with even-degree piecewise polynomials (specifically at Gauss points) improves the order of convergence (superconvergence). However, for Green's function kernels, it was known that standard iteration does not improve convergence if the collocation points are not Gauss points. This paper investigates whether iteration can improve convergence for Green's kernels when using non-Gauss collocation points (specifically, equidistant points) on a uniform partition.

2. Methodology

2.1 Approximation Space and Projection

• Partition: A uniform partition of $[0, 1]$ is defined with step size $h = 1/n$.
• Space ($X_n$): The space of piecewise polynomials of degree $\le 2r$ associated with the partition.
• Collocation Points: In each subinterval $[t_{j-1}, t_j]$, $2r+1$ distinct points are chosen. The paper specifically considers equidistant points (not Gauss points):
  $$\tau^k_j = t_{j-1} + \frac{kh}{2r}, \quad k = 0, 1, \dots, 2r$$
• Projection ($P_n$): An interpolatory projection operator $P_n: X \to X_n$ is defined such that $P_n x$ interpolates $x$ at the chosen collocation points.

2.2 Numerical Schemes

The paper analyzes three specific approximation schemes:

1. Standard Collocation ($\phi^C_n$): Solves $\phi^C_n - P_n K \phi^C_n = P_n f$.
2. Iterated Collocation ($\phi^S_n$): Defined by one step of iteration: $\phi^S_n = K \phi^C_n + f$.
3. Modified Collocation ($\phi^M_n$): Based on Kulkarni's method, solving $\phi^M_n - K^M_n \phi^M_n = f$, where $K^M_n = P_n K + K P_n - P_n K P_n$.
4. Iterated Modified Collocation ($\tilde{\phi}^M_n$): Defined by $\tilde{\phi}^M_n = K \phi^M_n + f$.

2.3 Analytical Tools

The authors employ divided differences to handle the non-smoothness of the Green's kernel.

• They define divided differences of the operator $Kx$ at the collocation points and an arbitrary point $s$.
• Key Lemmas: Two crucial lemmas (Lemma 2.1 and 2.2) are proven to bound the divided differences of $Kx$ involving the kernel's singularity. These bounds show that despite the kernel's lack of smoothness, the divided differences scale predictably with $h$ (e.g., $O(h^{-2r})$).
• Error Estimation: The proofs rely on Taylor series expansions and the properties of the interpolatory polynomial remainder term $\Psi_j(t)$.

3. Key Contributions and Results

The paper's primary contribution is demonstrating that iteration improves the order of convergence for Green's function kernels even when using non-Gauss (equidistant) collocation points, provided the degree of the piecewise polynomials is even ($2r$).

3.1 Convergence Orders for Standard and Iterated Collocation

Theorem 4 establishes the following error bounds (assuming $\phi \in C^{2r+2}$):

• Standard Collocation: $\|\phi - \phi^C_n\|_\infty = O(h^{2r+1})$
• Iterated Collocation: $\|\phi - \phi^S_n\|_\infty = O(h^{2r+2})$

Significance: This is a one-order improvement over the standard method. The paper proves that for Green's kernels, the iterated solution converges faster than the standard collocation solution, even without Gauss points.

3.2 Convergence Orders for Modified and Iterated Modified Collocation

Theorem 5 establishes the error bounds for the modified schemes:

• Modified Collocation ($\phi^M_n$):
  • If $r \ge 1$: $O(h^{2r+2})$
  • If $r = 0$: $O(h^3)$
• Iterated Modified Collocation ($\tilde{\phi}^M_n$):
  • If $r \ge 1$: $O(h^{2r+3})$
  • If $r = 0$: $O(h^4)$

Significance: The iterated modified collocation method achieves a superconvergence of order $2r+3$, which is significantly higher than the standard collocation order of $2r+1$.

4. Technical Insights and Mechanism

The paper resolves a gap in the literature where it was previously thought that iteration does not improve convergence for Green's kernels with non-Gauss points. The mechanism for this improvement relies on:

1. Cancellation of Errors: The specific structure of the divided differences of the Green's kernel allows the error terms in the iteration step to cancel out to a higher order.
2. Lemma 2.1 & 2.2: These lemmas prove that the divided differences of $Kx$ behave in a way that, when integrated against the polynomial basis functions, yields higher-order smallness ($O(h^{2r+2})$ or better) compared to the standard interpolation error.
3. Proposition 1 & 3: These propositions rigorously bound terms like $\|K(I-P_n)x\|$ and $\|K(I-P_n)K(I-P_n)x\|$, showing that the operator $K$ "smooths" the interpolation error sufficiently to boost the convergence rate of the iterated solutions.

5. Conclusion and Significance

• Main Finding: The paper successfully demonstrates that for Fredholm integral equations with Green's function kernels, using iterated collocation and iterated modified collocation methods with equidistant collocation points on piecewise polynomials of even degree ($2r$) yields improved convergence rates.
• Comparison to Literature:
  • Previous results suggested that without Gauss points, iteration offered no benefit for Green's kernels.
  • This paper refutes that for the specific case of even-degree piecewise polynomials and equidistant points, showing that iteration does provide a gain of 1 order for standard iteration and 2 orders for modified iteration.
• Practical Implication: Numerical analysts can achieve higher accuracy for Green's function problems without the computational complexity of finding Gauss points, simply by applying an iteration step to the standard collocation solution.

The paper concludes with a numerical example (though brief) illustrating that while standard superconvergence conditions (ratio of errors tending to 0) might fail for simple piecewise constant cases with specific points, the theoretical framework developed for higher-degree polynomials holds true, validating the improved orders of convergence derived in Theorems 4 and 5.