TY - JOUR

T1 - A Geometric Space-Time Multigrid Algorithm for the Heat Equation

AU - Köppl, null

N1 - KAUST Repository Item: Exported on 2020-11-18
Acknowledged KAUST grant number(s): UKc0020
Acknowledgements: This publication is partially based on work supported by Award No. UKc0020, made by the King Abdullah University of Science and Technology (KAUST).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2020/11/16

Y1 - 2020/11/16

N2 - 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. © 2012 Global-Science Press.

AB - 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. © 2012 Global-Science Press.

UR - http://hdl.handle.net/10754/665995

UR - http://global-sci.org/intro/article_detail/nmtma/5931.html

UR - http://www.scopus.com/inward/record.url?scp=84856411276&partnerID=8YFLogxK

U2 - 10.4208/nmtma.m12si07

DO - 10.4208/nmtma.m12si07

M3 - Article

SN - 1004-8979

VL - 5

SP - 110

EP - 130

JO - Numerical Mathematics: Theory, Methods and Applications

JF - Numerical Mathematics: Theory, Methods and Applications

IS - 1

ER -