In this work mantle convection simulation with Terra is investigated from a numerical point of view, theoretical analysis as well as practical tests are performed. The stability criteria for the numerical formulation of the physical model will be made clear. For the incompressible case and the Terra specific treatment of the anelastic approximation, two inf-sup stable grid modifications are presented, which are both compatible with hanging nodes. For the Q1hQ12h element pair a simple numeric test is introduced to prove the stability for any given grid. For the Q1h Pdisc 12h element pair and 1-regular refinements with hangig nodes an existing general proof can be adopted. The influence of the slip boundary condition is found to be destabilizing. For the incompressible case a cure can be adopted from the literature. The necessary conditions for the expansion of the stability results to the anelastic approximation will be pointed out. A numerical framework is developed in order to measure the effect of different numerical approaches to improve the handling of strongly varying viscosity. The framework is applied to investigate how block smoothers with different block sizes, combination of different block smoothers, different prolongation schemes and semi coarsening influence the multigrid performance. A regression-test framework for Terra will be briefly introduced.