  We present a new hybrid discrete exterior calculus (DEC) and finite
difference (FD) method to simulate fully three-dimensional Boussinesq
convection in spherical shells subject to internal heating and basal heating,
relevant in the planetary and stellar phenomenon. We employ DEC to compute the
surface spherical flows, taking advantage of its unique features of structure
preservation (e.g., conservation of secondary quantities like kinetic energy)
and coordinate system independence, while we discretize the radial direction
using FD method. The grid employed for this novel method is free of problems
like the coordinate singularity, grid non-convergence near the poles, and the
overlap regions. We have developed a parallel in-house code using the PETSc
framework to verify the hybrid DEC-FD formulation and demonstrate convergence.
We have performed a series of numerical tests which include quantification of
the critical Rayleigh numbers for spherical shells characterized by aspect
ratios ranging from 0.2 to 0.8, simulation of robust convective patterns in
addition to stationary giant spiral roll covering all the spherical surface in
moderately thin shells near the weakly nonlinear regime, and the quantification
of Nusselt and Reynolds numbers for basally heated spherical shells. The method
exhibits slightly better than second order error convergence with the mesh
size.
