The physics of the Earth's mantle are as fascinating as computationally challenging. While being a fundamental geophysical process, enormous knowledge gaps about Earth's mantle convection remain. The reasons: Realistic mantle models pose computational challenges due to highly nonlinear rheologies, severe heterogeneities, anisotropies, and wide ranges of spatial scales.

Presented are advances in nonlinear implicit solvers for Earth's instantaneous mantle flow governed by nonlinear instantaneous Stokes PDEs:

[1] Heterogeneity-robust Schur complement preconditioning (weighted BFBT),
[2] Hybrid spectral--geometric--algebraic multigrid (HMG), and
[3] Nonlinear preconditioning of an inexact Newton--Krylov method (primal-dual Newton).

These methods operate on aggressively adapted meshes and mixed continuous--discontinuous discretizations with high-order accuracy. Our goals are the realistic representation of mantle physics, maximizing accuracy and minimizing runtime, and achieving optimal algorithmic performance. While targeting mantle flow problems, the mentioned advances can be applied to many problems at the intersection of computational science and engineering with high-performance and large-scale computing.