Deflated preconditioned conjugate gradient solvers for linear elasticity

Abstract

Extensions of deflation techniques previously developed for the Poisson equation to static elasticity are presented. Compared to the (scalar) Poisson equation (J. Comput. Phys. 2008; 227 (24):10196–10208; Int. J. Numer. Meth. Engng 2010; DOI: 10.1002/nme.2932; Int. J. Numer. Meth. Biomed. Engng 2010; 26 (1):73–85), the elasticity equations represent a system of equations, giving rise to more complex low‐frequency modes (Multigrid . Elsevier: Amsterdam, 2000). In particular, the straightforward extension from the scalar case does not provide generally satisfactory convergence. However, a simple modification allows to recover the remarkable acceleration in convergence and CPU time reached in the scalar case. Numerous examples and timings are provided in a serial and a parallel context and show the dramatic improvements of up to two orders of magnitude in CPU time for grids with moderate graph depths compared to the non‐deflated version. Furthermore, a monotonic decrease of iterations with increasing subdomains, as well as a remarkable acceleration for very few subdomains are also observed if all the rigid body modes are included.

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