Challenges for Mantle Convection Simulations at the Exa-Scale: Numerics, Algorithmics and Software

Upcoming supercomputers provide theoretical compute performance that is hard to imagine, but even harder to exploit at its full potential. The potential of extreme-scale computing forces us to reconsider classical approaches for finite-element-based scientific computing. In this contribution, we provide some ideas in that direction using the problem of modeling the dynamics of the Earth’s mantle as an example. Resolving the mantle with a global resolution of 1 km results in more than 10^12 mesh cells. Thus, an associated convection model will require solvers capable of handling correspondingly large systems resulting from the discretization of the underlying Stokes problem. Multigrid algorithms are asymptotically optimal so that they can deliver the scalability needed for systems of such size. Here a monolithic multigrid method is employed based on computationally inexpensive Uzawa-type smoothers. Furthermore, advanced techniques must be employed to reduce memory consumption, data transport, and parallel communication overhead. This can be achieved by matrix-free methods that employ polynomial surrogates to compute approximations for the matrix coefficients on the fly. The combination of these techniques leads to highly efficient methods as can be shown by a suitable performance analysis that considers both algorithmic efficiency and also the specific parallel efficiency of a given implementation.

@incollection{id2898,
  author = {Mohr, Marcus and R\"ude, Ulrich and Wohlmuth, Barbara and Bunge, Hans-Peter},
  booktitle = {Impact of Scientific Computing on Science and Society},
  doi = {10.1007/978-3-031-29082-4\_4},
  editor = {Pekka Neittaanm\"aki and Marja-Leena Rantalainen},
  language = {en},
  pages = {75-92},
  publisher = {Springer International Publishing},
  series = {Computational Methods in Applied Sciences},
  title = {Challenges for Mantle Convection Simulations at the Exa-Scale: Numerics, Algorithmics and Software},
  volume = {58},
  year = {2023},
}

%O Book Section
%A Mohr, Marcus
%A Rüde, Ulrich
%A Wohlmuth, Barbara
%A Bunge, Hans-Peter
%B Impact of Scientific Computing on Science and Society
%R 10.1007/978-3-031-29082-4_4
%E Pekka Neittaanmäki
%E Marja-Leena Rantalainen
%G en
%P 75-92
%I Springer International Publishing
%S Computational Methods in Applied Sciences
%T Challenges for Mantle Convection Simulations at the Exa-Scale: Numerics, Algorithmics and Software
%V 58
%D 2023