On Discontinuous and Continuous Approximations to Second-Kind Volterra Integral Equations

Collocation and Galerkin methods in the discontinuous and globally con-tinuous piecewise polynomial spaces, in short, denoted as DC, CC, DG and CG methods respectively, are employed to solve second-kind Volterra integral equations (VIEs). It is proved that the quadrature DG and CG (QDG and QCG) methods ob-tained from the DG and CG methods by approximating the inner products by suitable numerical quadrature formulas, are equivalent to the DC and CC methods, respec-tively. In addition, the fully discretised DG and CG (FDG and FCG) methods are equivalent to the corresponding fully discretised DC and CC (FDC and FCC) meth-ods. The convergence theories are established for DG and CG methods, and their semi-discretised (QDG and QCG) and fully discretized (FDG and FCG) versions. In particular, it is proved that the CG method for second-kind VIEs possesses a similar convergence to the DG method for first-kind VIEs. Numerical examples illustrate the theoretical results.