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CivilComp Proceedings
ISSN 17593433 CCP: 107
PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON PARALLEL, DISTRIBUTED, GRID AND CLOUD COMPUTING FOR ENGINEERING Edited by: P. Iványi and B.H.V. Topping
Paper 14
Scalable Parallel Multigrid Preconditioning for High Fidelity Finite Element and Finite Difference Simulations P.K. Jimack^{1}, M.A. Walkley^{1} and J. Zhang^{2}
^{1}School of Computing, University of Leeds, United Kingdom
P.K. Jimack, M.A. Walkley, J. Zhang, "Scalable Parallel Multigrid Preconditioning for High Fidelity Finite Element and Finite Difference Simulations", in P. Iványi, B.H.V. Topping, (Editors), "Proceedings of the Fourth International Conference on Parallel, Distributed, Grid and Cloud Computing for Engineering", CivilComp Press, Stirlingshire, UK, Paper 14, 2015. doi:10.4203/ccp.107.14
Keywords: multigrid, weak scalability, preconditioning, finite elements, finite differences, elliptic problems.
Summary
A major challenge in undertaking high resolution numerical simulations for
engineering problems comes from the growth in the computational work that
occurs as the underlying finite difference or the finite element meshes are
refined in order to improve accuracy. For most solution algorithms this growth
in work is superlinear with the number of degrees of freedom. In such cases
it is impossible for a parallel implementation with p processors to solve a
problem with p x N degrees of freedom in the same time as it can solve
the problem with N degrees of freedom on a single processor (socalled
weak scalability). Because multigrid algorithms have the property that
they may solve a discrete problem in O(N) operations they provide a natural
approach for seeking weakly scalable parallel solvers. Unfortunately,
developing a highly efficient parallel implementation is a challenging task,
that can prevent perfect weak scalability. In this paper we explain why this is
so, and suggest ways in which these difficulties may be overcome. In particular
we demonstrate that, if multigrid is used as a preconditioner for a different
iterative scheme (based upon Krylov subspace methods for example), the coarse
grid part of the multigrid problem does not need to be solved exactly (unlike
for pure multigrid) in order to obtain an O(N) algorithm. Since this is the
part of multigrid that is most challenging to implement in parallel we are able
to show the effectiveness of this approach for both finite difference and
finite element discretizations of selected applications in both two and three
dimensions.
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