| In this thesis, we focus on the discontinuous Galerkin (DG) methods for the functional integro-differential equations and on the cascading multigrid (CMG) methods for the parabolic PDEs, Volterra integro-differential equations (VIDEs) and Fredholm equations.;Two new cascading multilevel algorithms are analyzed to the semi-linear parabolic PDEs and extended to the partial Volterra integro-differential equations (PVIDEs) and the parabolic PDEs with delays. More distinctly the cascading multigrid method could very well solve the Fredholm equations without dealing with the full stiffness matrix directly. Therefore we can save much more computing time. Most importantly, we contribute to the multigrid arts by developing an abstract cascading multigrid method in Besov spaces and a discontinuous Galerkin cascading multigrid method. We extend these methods to evolutionary equations and PVIDEs. Finally, we discuss briefly the future works on (partial) VIDEs with blow-up solutions and artificial boundary methods for PVIDEs on unbounded domains.;We give both a priori and a posteriori error estimates of the DG method for linear, semilinear and nonstandard VIDEs. Furthermore the superconvergence of the mesh-dependent Galerkin method for VIDEs is also considered. The fully discretized DG method for VIDEs is also analyzed. Numerical examples are provided to compare the DG method with the continuous Galerkin (CG) method and the continuous collocation (CC) method. We study the primary discontinuities of several classes of VIDEs with time dependent delays, which include the functional VIDEs of Hale's type, delay VIDEs with weakly singular kernels and delay VIDEs of neutral type (with weakly singular kernels). According to the regularity information established, we construct an adaptive DG method for functional VIDEs of Hale's type. |
