Adaptive collocation methods for Volterra integral and integro-differential equations

In this paper, the a posteriori error estimates of collocation methods are derived for Volterra integral and integro-differential equations with smooth and weakly singular kernels. The derived error indicators are constructed in an iterative way and used to adaptively refine the meshes in the ports of large gradients. This approach is much more efficient than the collocation methods on the fixed meshes. Numerical computations confirm the analysis via several examples.