A collocation method for weakly singular integral equations with super-algebraic convergence rate
The paper explores collocation schemes based on trigonometric polynomials for the numerical solution of biperiodic weakly singular integral equations of the second kind. The approach uses a co-ordinate transformation to remove the singularity and a discretisation based on the product trapezoidal scheme. Stability and convergence results are provided under regularity assumptions and the results are compared to actual practical examples where numerical examples are used to demonstrate that the high convergence rates predicted by the theory can be achieved in realistic problems. As the authors point out in their introduction, the method has to compete with simpler strategies that achieve \(4\)th-order convergence by neglecting the singularity in a clever way. The examples and the authors' conclusions demonstrate that their approach competes effectively.
biperiodic weakly singular integral equations of the second kind
scientific article; zbMATH DE number 6666240
scientific article