  In the Exa-Dune project we have developed, implemented and optimised
numerical algorithms and software for the scalable solution of partial
differential equations (PDEs) on future exascale systems exhibiting a
heterogeneous massively parallel architecture. In order to cope with the
increased probability of hardware failures, one aim of the project was to add
flexible, application-oriented resilience capabilities into the framework.
Continuous improvement of the underlying hardware-oriented numerical methods
have included GPU-based sparse approximate inverses, matrix-free
sum-factorisation for high-order discontinuous Galerkin discretisations as well
as partially matrix-free preconditioners. On top of that, additional
scalability is facilitated by exploiting massive coarse grained parallelism
offered by multiscale and uncertainty quantification methods where we have
focused on the adaptive choice of the coarse/fine scale and the overlap region
as well as the combination of local reduced basis multiscale methods and the
multilevel Monte-Carlo algorithm. Finally, some of the concepts are applied in
a land-surface model including subsurface flow and surface runoff.

Mathematical Software;Numerical Analysis;Numerical Analysis