Numerical Methods for Mantle Convection

Over the past 40 years, numerical methods used to model mantle convection have matured significantly and now occupy a distinctive niche in computational fluid dynamics. The central issue for the partial differential equations governing mantle dynamics concerns the rheology governing the flow of mantle rocks under relevant temperature and pressure conditions. The momentum equation is the most computationally intensive portion of this coupled system because of its strongly nonlinear, elliptical character. We review the various discretization methods, ranging from finite differences to finite elements, which have been employed. We present in detail finite-element implementations of the Uzawa algorithm for the momentum equation and of the streamline Petrov-Galerkin algorithm for the energy equation. We review the application of modern techniques such as the Schur complement solvers and least-squares commutator preconditioners in the solution of the elliptical equation. We discuss the computational libraries available for massively parallel simulation and their application to this system of equations and give a preview of approaches suitable for exascale computing, which is a major challenge for the decade ahead.