The auto-convolution Volterra integral and integro-differential equations arise in many applications, for example, in the identification of memory kernels in the theory of viscoelasticity and in the computation of certain special functions. The convergence analysis of piecewise polynomial collocation solutions for these two kinds of equations is now largely well understood. However, the convergence analysis on Galerkin methods is still not clear. In this talk, we will show that the quadrature Galerkin method obtained from the Galerkin method by approximating the inner products by suitable numerical quadrature formulas, is equivalent to the continuous piecewise polynomial collocation method. In addition, the convergence and superconvergence of the numerical solution based on Galerkin methods are investigated.