Superconvergence of interpolated collocation solutions for weakly singular Volterra integral equations of the second kind

Abstract

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 $$u_h$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:math> , two different interpolation postprocessing approximations of higher accuracy: $$I_{2h}^{2m-1}u_h$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>I</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>h</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>m</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> </mml:math> based on the collocation points and $$I_{2h}^{m}u_h$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>I</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>h</mml:mi> </mml:mrow> <mml:mi>m</mml:mi> </mml:msubsup> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> </mml:math> 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.