A hybrid spectral element method for Volterra integral equations with weakly singular kernel

In this paper, we propose a hybird generalized Jacobi function and Legendre spectral element method for the linear and nonlinear Volterra integral equations (VIEs) with weakly singular kernels. It is shown that the suggested method is easy to implement and possesses the high-order accuracy. We also establish the existence and uniqueness of the numerical solution, and characterize the $hp$-error analysis of the suggested method under $L^2(I)$-norm. Numerical results confirm the theoretical analysis.