A robust matrix-free approach for large-scale non-isothermal high-contrast viscosity Stokes flow on blended domains with applications to geophysics

TL;DR

This paper introduces a scalable, matrix-free iterative solver for large-scale non-isothermal Stokes flow with high viscosity contrast, applied to geophysical mantle convection, using advanced discretization and time-stepping methods.

Contribution

It develops a robust, matrix-free solver with specialized preconditioners for high contrast viscosity Stokes problems on blended domains, enabling efficient large-scale geophysical simulations.

Findings

Achieved scalable performance on high-resolution Earth convection models

Demonstrated robustness for high viscosity contrast scenarios

Implemented a hybrid hierarchical grid for parallel computations

Abstract

We consider a compressible Stokes problem in the quasi-stationary case coupled with a time dependent advection-diffusion equation with special emphasis on high viscosity contrast geophysical mantle convection applications. In space, we use a P2-P1 Taylor--Hood element which is generated by a blending approach to account for the non-planar domain boundary without compromising the stencil data structure of uniformly refined elements. In time, we apply an operator splitting approach for the temperature equation combining the BDF2 method for diffusion and a particle method for advection, resulting in an overall second order scheme. Within each time step, a stationary Stokes problem with a high viscosity contrast has to be solved for which we propose a matrix-free, robust and scalable iterative solver based on Uzawa type block preconditioners, polynomial Chebyshev smoothers and a BFBT type Schur complement approximation. Our implementation is using a hybrid hierarchical grid approach allowing for massively parallel, high resolution Earth convection simulations.