TY - JOUR
T1 - A Geometric Space-Time Multigrid Algorithm for the Heat Equation
AU - Tobias Weinzierl & Tobias Köppl
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 1
SP - 110
EP - 130
PY - 2012
DA - 2012/05
SN - 5
DO - http://doi.org/10.4208/nmtma.m12si07
UR - https://global-sci.org/intro/article_detail/nmtma/5931.html
KW - Adaptive Cartesian grids, geometric multiscale methods, heat equation, octree, spacetree, space-time discretisation.
AB - <p style="text-align: justify;">We study the time-dependent heat equation on its space-time domain that
is discretised by a $k$-spacetree. $k$-spacetrees are a generalisation of
the octree concept and are a discretisation paradigm yielding a multiscale
representation of dynamically adaptive Cartesian grids with low memory
footprint.
The paper presents a full approximation storage geometric multigrid
implementation for this setting that combines the smoothing properties of
multigrid for the equation&#39;s elliptic operator with a multiscale solution propagation
in time.
While the runtime and memory overhead for tackling the all-in-one space-time
problem is bounded, the holistic approach promises to exhibit a better
parallel scalability than classical time stepping, adaptive dynamic refinement
in space and time fall naturally into
place, as well as the treatment of periodic boundary conditions of steady cycle
systems, on-time computational steering is eased as the algorithm delivers
guesses for the solution&#39;s long-term behaviour immediately, and, finally,
backward problems arising from the adjoint equation benefit from the the
solution being available for any point in space and time.</p>