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Accurate simulation of global mantle dynamics, and in particular of
plate motion requires the resolution of features on very different
length and time scales: While faulted plate margins can need a mesh
resolution down to ~1km, a significantly coarser mesh (up to several
100km) is sufficient away from these local features, for instance in
the lower mantle. Simulations on adaptively refined meshes (AMR) can
reduce the number of unknowns drastically: While a uniform mesh with
1km resolution would lead to $~10^12$ unknowns, adapted meshes with
the same resolution around plate boundaries can get by with only
$10^8$-$10^9$ unknowns, rendering such high-resolution simulations
possible on large parallel computers.  However, the efficient handling
of locally refined meshes with hundreds of millions of unknowns in
parallel is still very challenging.

We are developing the parallel adaptive finite element mantle
convection code Rhea to efficiently solve global mantle flow problems
on parallel supercomputers. Rhea uses adaptive meshes based on
forest-of-octrees combined with efficient iterative solvers for the
viscous flow problem.

We present a study from global flow simulations using a realistic
nonlinear rheology and a local mesh resolution that is sufficiently
high to resolve large variations in the viscosity. For these models we
carefully incorporated the details of the plate boundaries at a fine
scale, and use a thermal model of the seismicity-defined slabs which
grades into the more diffuse buoyancy resolved with tomography. The
rheology law combines Newtonian and non-Newtonian flow along with
yielding under high stress. Using these global models we address the
change of plate motions and, by comparing with data hope to enhance
our understanding of the dynamics in the Earth's mantle.