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A Geometric Space-Time Multigrid Algorithm for the Heat Equation

Published online by Cambridge University Press:  28 May 2015

Tobias Weinzierl  and
Tobias Köppl
Tobias Weinzierl*
Affiliation:
Institut für Informatik, Technische Universität München, Boltzmannstr. 3, 85748 Garching, Germany
Tobias Köppl*
Affiliation:
Institut für Mathematik, Technische Universität München, Boltzmannstr. 3, 85748 Garching, Germany
*
Corresponding author.Email address:weinzier@in.tum.de
Corresponding author.Email address:koeppl@ma.tum.de
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Abstract

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’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’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.

Keywords

65M5065M5565N5065N5565M22Adaptive Cartesian gridsgeometric multiscale methodsheat equationoctreespace-treespace-time discretisation
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 5 , Issue 1 , February 2012 , pp. 110 - 130
Copyright
Copyright © Global Science Press Limited 2012

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