A collocation method for weakly singular integral equations with super-algebraic convergence rate

We consider biperiodic weakly singular integral equations of the second kind such as they arise in boundary integral equation methods. These are solved numerically using a collocation scheme based on trigonometric polynomials. The singularity is removed by a local change to polar coordinates. The resulting operators have smooth kernels and are discretized using the tensor product composite trapezodial rule. We prove stability and convergence of the scheme achieving algebraic convergence of any order under appropriate regularity assumptions. The method can be applied to typical boundary value problems such as potential and scattering problems both for bounded obstacles and for periodic surfaces. We present numerical results demonstrating that the expected convergence rates can be observed in practice.