A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane
The numerical treatment of second kind integral equations on the real line \[ \varphi(s)=\psi(s)+\int_{-\infty}^{+\infty}k(s-t)z(t)\varphi(t) dt,\quad s\in \mathbb{R}, \] is considered where \(k\in L_1(\mathbb{R}),z\in L_{\infty}(\mathbb{R})\)\ and \(\psi\in BC(\mathbb{R})\) (bounded continuous functions on \(\mathbb{R}\)). \(\varphi\) is the unknown function. A sharp error estimate for the finite section approximation is derived. The numerical solution by a simple discrete collocation method on a uniform grid on \(\mathbb{R}\) is analysed. The case when \(z\)\ is compactly supported leads to a coefficient matrix allowing a rapid matrix vector multiplication by the fast Fourier transform and a modified two-grid iteration is used. If \(z\)\ is not compactly supported a finite section and a two-grid algorithm are used. As an application an acoustic scattering problem is presented. An error estimate of the numerical solution is given.
convolution type integral equation of the second kind
scientific article; zbMATH DE number 1837415