Nyström method

Remarks

Clause 1 is the Fredholm alternative for the compact integral operator $(\mathcal K u)(x) = \int_a^b K(x, y) u(y)\, dy$ on $C[a, b]$. Write $(\mathcal K_n u)(x) = \sum_j w_j^{(n)} K(x, x_j^{(n)}) u(x_j^{(n)})$; the values $u_n(x_i^{(n)})$ equal $u_i^{(n)}$, and $u_n$ solves $u_n - \mathcal K_n u_n = f$. By the uniform boundedness principle the sums $\sum_j \abs{w_j^{(n)}}$ are bounded, so $\mathcal K_n \to \mathcal K$ pointwise and the family $(\mathcal K_n)$ is collectively compact. Anselone's theory of collectively compact operators then shows that $I - \mathcal K_n$ is invertible for large $n$ with uniformly bounded inverses, and $u_n - u = (I - \mathcal K_n)^{-1}(\mathcal K_n - \mathcal K) u$ gives clause 3. So the error is of the order of the quadrature error on the smooth functions $y \mapsto K(x, y) u(y)$: with Gauss–Legendre nodes and smooth $K$ and $f$ it decays very fast. The formula for $u_n$ is the Nyström interpolation; it extends the nodal values to all of $[a, b]$ with the accuracy of the nodal values. Without unique solvability of the homogeneous equation the method has no limit to converge to: for $K = 1$ on $[0, 1]$ and $f = 0$ every constant solves the equation.

Going deeper

For a student

The Nyström method is the simplest and often the most accurate way to solve second-kind Fredholm equations with smooth kernels. Its error analysis goes through the operator $\mathcal K_n$, which is not norm-convergent to $\mathcal K$ but is collectively compact, which is enough. It contrasts with projection methods such as collocation and Galerkin methods, which approximate $u$ in a finite-dimensional subspace. For weakly singular kernels one uses product integration rules, where the singular factor is integrated exactly. Equations of the first kind are ill-posed and need regularization, so this theorem does not cover them.