In this thesis, the Galerkin and iterated Galerkin methods are investigated for solving linear second-kind Volterra integral equations with weakly singular kernels of mixed type. In these methods, the approximate solution is sought in a space of piecewise polynomials. Due to the simultaneous presence of logarithmic and algebraic singularities in the kernel, the exact solution typically exhibits reduced smoothness near the origin; therefore, a suitably graded mesh is employed to achieve optimal convergence rates. Moreover, the grading exponent of the mesh is chosen to match the strength of the singularity. It is shown that the standard Galerkin method attains a convergence rate of order \(\mathcal{O}(n^{-m})\), while the iterated Galerkin method achieves a superconvergent rate of order \(\mathcal{O}(n^{-2m})\), where \(n\) denotes the number of subintervals and \(m\) the polynomial degree. The theoretical analysis is corroborated by numerical experiments, and the proposed methods are compared with collocation and iterated collocation methods, confirming the predicted convergence behavior.
