Abstract
In this paper, we discuss the superconvergence of the “interpolated” collocation solutions for weakly singular Volterra integral equations of the second kind. Based on the collocation solution u_h, two different interpolation postprocessing approximations of higher accuracy: I_{2h}^{2m-1}u_h based on the collocation points and I_{2h}^{m}u_h based on the least square scheme are constructed, whose convergence order are the same as that of the iterated collocation solution. Such interpolation postprocessing methods are much simpler in computation. We further apply this interpolation postprocessing technique to hybrid collocation solutions and similar results are obtained. Numerical experiments are shown to demonstrate the efficiency of the interpolation postprocessing methods.
Highlights
In this paper, we consider the following weakly singular Volterra integral equation (VIE) of the second kind:t u(t) = g(t) + (t − s)−α K (t, s)u(s)ds, t ∈ I := [0, T ], (1)Communicated by Hui Liang
The second kind Volterra integral equations with weakly singular kernels typically have solutions whose derivatives are unbounded at the left endpoint of the interval of integration
We apply the interpolation postprocessing technique to the collocation solution under graded mesh and the hybrid collocation solution under “looser” graded mesh to get the same superconvergence as the iterated methods
Summary
The second kind Volterra integral equations with weakly singular kernels typically have solutions whose derivatives are unbounded at the left endpoint of the interval of integration Due to this singular behavior, the optimal global and local (super-) convergence results of collocation solutions in piecewise polynomial spaces on uniform meshes will no longer be valid (Brunner 1983). We apply the interpolation postprocessing technique to the collocation solution under graded mesh and the hybrid collocation solution under “looser” graded mesh to get the same superconvergence as the iterated methods. This interpolation postprocessing is simpler in computation than the iterated postprocessing method.
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