Large-scale simulation of mantle convection based on a new matrix-free approach

In this paper, we report on a two-scale approach for efficient matrix-free finite element simulations. It is an extended version of our previous conference publication [1]. The proposed method is based on surrogate element matrices constructed by low order polynomial approximations. It is applied to a Stokes-type PDE system with variable viscosity as is a key component in mantle convection models. We set the ground for a rigorous performance analysis inspired by the concept of parallel textbook multigrid efficiency and study the weak scaling behavior on SuperMUC, a peta-scale supercomputer system. For a complex geodynamical model, we achieve, on up to 47 250 compute cores, a parallel efficiency of 93% for application of the discrete operator and 83% for a complete Uzawa V-cycle including the coarse grid solve. Our largest simulation uses a trillion (O(10 ¹² )) degrees of freedom for a global mesh resolution of 1.5 km. Applicability of our new approach for geodynamical problems is demonstrated by investigating dynamic topography for classical benchmark settings as well as for high-resolution models with lateral viscosity variations.