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Convergence of the multigrid reduction in time algorithm for the linear elasticity equations
Author(s) -
Hessenthaler A.,
Nordsletten D.,
Röhrle O.,
Schroder J. B.,
Falgout R. D.
Publication year - 2018
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2155
Subject(s) - multigrid method , speedup , elasticity (physics) , bottleneck , concurrency , computer science , algorithm , convergence (economics) , reduction (mathematics) , linear elasticity , relaxation (psychology) , mathematical optimization , running time , mathematics , parallel computing , finite element method , partial differential equation , distributed computing , psychology , mathematical analysis , social psychology , materials science , geometry , physics , economics , composite material , thermodynamics , embedded system , economic growth
Summary This paper presents some recent advances for parallel‐in‐time methods applied to linear elasticity. With recent computer architecture changes leading to stagnant clock speeds, but ever increasing numbers of cores, future speedups will be available through increased concurrency. Thus, sequential algorithms, such as time stepping, will suffer a bottleneck. This paper explores multigrid reduction in time (MGRIT) for an important application area, linear elasticity. Previously, efforts at parallel‐in‐time for elasticity have experienced difficulties, for example, the beating phenomenon. As a result, practical parallel‐in‐time algorithms for this application area currently do not exist. This paper proposes some solutions made possible by MGRIT (e.g., slow temporal coarsening and FCF‐relaxation) and, more importantly, a different formulation of the problem that is more amenable to parallel‐in‐time methods. Using a recently developed convergence theory for MGRIT and Parareal, we show that the changed formulation of the problem avoids the instability issues and allows the reduction of the error using two temporal grids. We then extend our approach to the multilevel case, where we demonstrate how slow temporal coarsening improves convergence. The paper ends with supporting numerical results showing a practical algorithm enjoying speedup benefits over the sequential algorithm.