This dissertation is composed of three studies addressing different but related problems on dynamic processes occurring in a Plume-fed Asthenosphere system, as well as techniques for improving numerical models of mantle convection. The first paper, 2D Numerical Experiments on a Plume-fed Asthenosphere: Necessary Preconditions and Implications for Geoid and Dynamic Topography, performs a suite of 2D finite element-based experiments that explore what conditions are needed so that mantle flow includes a plume-fed asthenosphere (PFA) as a key part of its flow pattern. We find that a plume flux ~1.2 times big as the slab flux is needed for a persistent PFA. The numerical experiments also demonstrate that, instead of generating dynamic topography on the sea floor, flow-induced dynamic relief due to sub-asthenospheric density anomalies will preferentially form at the base of a buoyant asthenosphere, which is a promising mechanism to explain why Earth's ±100m Geoid variations are associated with much less than ~2km of dynamic topography at Earth's surface. The second paper, A Quasi-Cspline Interpolation Algorithm for Data on Unstructured Triangular and Tetrahedral Meshes, develops a quasi-cubic Hermite spline interpolation algorithm for 2D and 3D scattered data, fitting both nodal values and slopes to the edges of triangular or tetrahedral cubic serendipity elements. This explicit recipe for 2D and 3D interpolation has been tested in vectorized and parallelized Matlab code, and has been used in both 2D and 3D large numerical simulations using unstructured triangular and tetrahedral meshes. The third paper, Plumeasthenosphere-lithosphere Interactions Within a Mantle with a Plume-fed Asthenosphere: Implications for Hawaii- and Iceland-type Plume Dynamics, studies the effects of on- and off-axis deep-mantle plumes with thermally controlled density and viscosity variations, assuming that thermal expansion controls density and that viscosity is governed by a temperature-dependent Arrhenius-type relation. The code we use is a parallel Matlab-based 3-D Finite Element code that we have developed, which utilizes unstructured tetrahedral meshes, and which can handle large and abrupt (6 orders of magnitude) viscosity contrasts (Hasenclever, PhD Dissertation 2010). In this paper, We show the results of: 1) the necessary conditions (plume flux, density contrast, viscosity contrast) for the existence of a PFA system with an on/off-axis plume; 2) resulting 3-D flow patterns in the asthenosphere, and the dynamic topography that is associated with them; 3) the decoupling effect of a buoyant and less viscous asthenosphere layer to the underlying mantle, and how this helps lead to relatively fixed hot spots.