For investigation of fluid dynamics in the Earth's outer core and mantle, we have been developing thermal-hydraulic subsystem of GeoFEM, which gives a parallel finite element method (FEM) platform. This subsystem is a three dimensional and time-dependent simulation code for an incompressible fluid with the Boussinesq approximation. This subsystem is tested on an analysis of lid-driven convection in a cubic cavity. To investigate focus on the Earth's core dynamics in this study, we carry out a thermal convection analysis in a rotating spherical shell which is modeled on the Earth's outer core. The results of the simulation are compared to a simulation using the spectral method. In both cases, the results of the convection pattern show that three pairs of convection columns are formed and that these columns propagate to westward.