A semi-analytic accuracy benchmark for Stokes flow in 3-D spherical mantle convection codes

One of the main challenges in numerical mantle circulation simulations is the determination of accurate solutions to the Stokes equation in combination with an additional constraint arising from the continuity equation. Here we derive a semi-analytic solution to the Stokes equation in a spherical shell using scalar and vector spherical harmonics. For the simple case of an incompressible flow with uniform viscosity we show that a direct relation exists between the flow velocity and the driving force through a fourth-order ordinary differential equation. The latter can be exploited to derive the forcing term from a prescribed velocity field. This feature lends itself to the design of a self-contained benchmark set-up that can be performed without relying on the numerical output from other codes. We demonstrate the applicability of the benchmark by verifying the convergence behaviour of the Stokes solver in the prototype of a new mantle convection modelling framework for high performance computing based on Hierarchical Hybrid Grids (HHG).