Deflated preconditioned conjugate gradient solvers for the Pressure–Poisson equation

A deflated preconditioned conjugate gradient technique has been developed for the solution of the Pressure–Poisson equation within an incompressible flow solver. The deflation is done using a region-based decomposition of the unknowns, making it extremely simple to implement. The procedure has shown a considerable reduction in the number of iterations. For grids with large graph-depth the savings exceed an order of magnitude. Furthermore, the technique has shown a remarkable insensitivity to the number of groups/regions chosen, and to the way the groups are formed

Deflated preconditioned conjugate gradient solvers for the pressure‐Poisson equation: Extensions and improvements

Extensions and improvements to a deflated preconditioned conjugate gradient technique for the solution of the pressure‐Poisson equation within an incompressible flow solver are described. In particular, the use of the technique for embedded grids, for cases where volume of fluid or level set schemes are required and its implementation on parallel machines are considered. Several examples are included that demonstrate a considerable reduction in the number of iterations and a remarkable insensitivity to the number of groups/ regions chosen and/or to the way the groups are formed

Deflated preconditioned conjugate gradient solvers for linear elasticity

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

Enhanced balancing Neumann-Neumann preconditioning in computational fluid and solid mechanics

In this work, we propose an enhanced implementation of balancing Neumann-Neumann (BNN) preconditioning together with a detailed numerical comparison against the balancing domain decomposition by constraints (BDDC) preconditioner. As model problems, we consider the Poisson and linear elasticity problems. On one hand, we propose a novel way to deal with singular matrices and pseudo-inverses appearing in local solvers. It is based on a kernel identication strategy that allows us to eciently compute the action of the pseudo-inverse via local indenite solvers. We further show how, identifying a minimum set of degrees of freedom to be xed, an equivalent denite system can be solved instead, even in the elastic case. On the other hand, we propose a simple modication of the preconditioned conjugate gradient (PCG) algorithm that reduces the number of Dirichlet solvers to only one per iteration, leading to similar computational cost as additive methods. After these improvements of the BNN PCG algorithm, we compare its performance against that of the BDDC preconditioners on a pair of large-scale distributed-memory platforms. The enhanced BNN method is a competitive preconditioner for three-dimensional Poisson and elasticity problems, and outperforms the BDDC method in many cases

Enhanced balancing Neumann-Neumann preconditioning in computational fluid and solid mechanics

Manuscript submitted for publication in International Journal for Numerical Methods in Engineering. Under review.Preprin

On parallel pre-conditioners for pressure Poisson equation in LES of complex geometry flows

China and India Award of Royal Academy of Engineering, UK to Drs Eldad Avital and Krishna M. Singh. Grant Numbers: Grant No. SEMF1A4R, Grant No. EP/L000261/