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

Summary  We consider the numerical treatment of second kind integral equations on the real line of the form (abbreviatedφ =ψ +K _z φ) in whichκ εL ₁(ℝ),z εL _∞ (ℝ), andψ εBC(ℝ), the space of bounded continuous functions on ℝ, are assumed known andφ εBC(ℝ) is to be determined. We first derive sharp error estimates for the finite section approximation (reducing the range of integration to [−A, A]) via bounds on (I − K _z )⁻¹ as an operator on spaces of weighted continuous functions. Numerical solution by a simple discrete collocation method on a uniform grid on ℝ is then analysed: in the case whenz is compactly supported this leads to a coefficient matrix which allows a rapid matrix-vector multiply via the FFT. To utilise this possibility we propose a modified two-grid iteration, a feature of which is that the coarse grid matrix is approximated by a banded matrix, and analyse convergence and computational cost. In cases wherez is not compactly supported a combined finite section and two-grid algorithm can be applied and we extend the analysis to this case. As an application we consider acoustic scattering in the half-plane with a Robin or impedance boundary condition which we formulate as a boundary integral equation of the class studied. Our final result is that ifz (related to the boundary impedance in the application) takes values in an appropriate compact subsetQ of the complex plane, then the difference betweenφ(s) and its finite section approximation computed numerically using the iterative scheme proposed is ≤C ₁[khlog(1/kh)+(1−θ)^−1/2(kA)^−1/2] in the interval [−θA, θA] (θ<1), forkh sufficiently small, wherek is the wavenumber andh the grid spacing. Moreover this numerical approximation can be computed in ≤C ₂ N logN operations, whereN = 2A/h is the number of degrees of freedom. The values of the constantsC ₁ andC ₂ depend only on the setQ and not on the wavenumberk or the support ofz. This work was supported by the UK Engineering and Physical Sciences Research Council and by the Radio Communications Research Unit, Rutherford Appleton Laboratory.