Projection method for integral equations

Remarks

Two choices of $P_n$ are standard. In the collocation method, $X_n \subseteq C[a, b]$ and $P_n$ is interpolation at points $x_1, \ldots, x_N$, so the equation is required to hold at these points. In the Galerkin method, $X$ is a Hilbert space and $P_n$ is the orthogonal projection onto $X_n$. So the residual is required to be orthogonal to $X_n$; see the Galerkin method.

Example: $X = L^2(0, 1)$, $(K\varphi)(x) = \int_0^1 x t \, \varphi(t) \, dt$, $f(x) = x$, and $X_1$ spanned by $u_1(x) = x$ with $P_1$ orthogonal. With $\varphi_1 = c x$, one has $K\varphi_1 = \frac{c}{3} x$, so the equation is $c - \frac{c}{3} = 1$ and $c = \frac32$. Since $K$ maps into $X_1$ and $f \in X_1$, this is the exact solution: $\frac32 x - \frac{x}{3} \cdot \frac32 = x$.

Non-example: the Nyström method replaces the integral in $K$ by a quadrature rule and solves for the values of the unknown at the nodes. It approximates the operator $K$ instead of projecting the unknown onto a fixed subspace, so it is not a projection method in this sense. Its analysis uses collectively compact operator approximation.

The convergence of projection methods for second-kind equations is described by the convergence theorem for second-kind equations.