### Applicant

Prof. Dr. Ulrich Rüde

Chair for Computer Science 10 – System simulation

Friedrich-Alexander-Universität Erlangen-Nürnberg

### Project Overview

Convective mass transport in the Earth’s mantle is a key component of the Earth system, as it provides the forces that shape the surface of our planet through processes such as plate tectonics or dynamic topography, which in turn lead to mountain building and sea level changes. On geologic time scales (10^6a), the rocks that compose the mantle can be treated as a viscous fluid and the process as a fluid dynamics problem. The governing equations describing its behavior include a central component a generalized Stokes problem.

A quantitative understanding of mantle convection requires accurate numerical simulation. The sheer size of the discrete problem, however, poses a challenge to standard approaches. A discretization of the Earth’s mantle with about 1 km resolution results in linear systems with approximately a trillion (10^12) unknowns. Only matrix-free solvers with optimal computational complexity can handle such problems on current state-of-the-art supercomputers.

Not only are solvers required that can cope with the size of the discrete problem, but also numerical methods that can handle the complexity of the underlying differential equations. The focus of this project is to address the difficulties that arise from the drastically varying viscosity, which can change by orders of magnitudes on short length scales in some scenarios. Special preconditioners must be chosen and developed, that are efficient and scalable for extreme-scale applications and simultaneously able to handle non-smooth coefficients.

The goal of this project is the implementation and performance evaluation of multigrid solvers and preconditioners for Stokes problems with strongly varying or even jumping coefficients. In particular, a promising candidate method is the so-called weighted BFBT preconditioner, which is specifically adapted to the requirement of robustness in the presence of drastically varying viscosity. The primary goal of this project is to implement and evaluate the performance of this method in the context of the HYTEG software framework.

In particular, alongside the matrix-free implementation in HYTEG, we will focus on the performance analysis of the solver. For this we are going to follow two orthogonal approaches. Firstly, a model based on the notion of textbook multigrid efficiency (TME) will be developed and applied to the solver for some benchmark problems. TME sets the ambitious goal to not only achieve h-independent convergence, but moreover suggests a limit for the constants of the expected asymptotically linear complexity of a multigrid solver. Secondly, the analysis is augmented by evaluation of the computational performance of relevant compute kernels via appropriate performance models, such as the roofline or ECM model. Finally, having analysed the serial and parallel performance, we aim to perform numerical studies in scenarios with real-world geophysical data on the supercomputers at RRZE.