Summary
We study the numerical solution of Volterra integral equations of the first kind by collocation methods using piecewise polynomials of degreep. We show that when superconvergence occurs, the error in the collocation solution at the node points alternates in sign, independent from the kernel or the exact solution. Evaluation at certain points of interpolating polynomials of degreep+2 annihilates, the leading term in the asymptotic expansion of the error, thus yielding convergence of orderp+3. The derivation of the asymptotic expansion of the error for the general equation is reduced to the special case of numerical differentiation by the use of an asymptotic continuity result for the projectors associated with the collocation method under compact perturbations. We present some numerical results on the improved order of convergence forp=2 and 3.
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Research supported by the United States Army under Contract No. DAAG 29-83-K-0109
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Eggermont, P.P.B. Improving the accuracy of collocation solutions of Volterra integral equations of the first kind by local interpolation. Numer. Math. 48, 263–279 (1986). https://doi.org/10.1007/BF01389475
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DOI: https://doi.org/10.1007/BF01389475