Optimization of the deflated Conjugate Gradient algorithm for the solving of elliptic equations on massively parallel machines

• Authors: Mathias Malandain;Nicolas Maheu;Vincent Moureau
• Venue: Journal of Computational Physics
• Year: 2013

The discretization of Partial Differential Equations often leads to the need of solving large symmetric linear systems. In the case of the Navier-Stokes equations for incompressible flows, solving the elliptic pressure Poisson equation can represent the most important part of the computational time required for the massively parallel simulation of the flow. The need for efficiency that this issue induces is completed with a need for stability, in particular when dealing with unstructured meshes. Here, a stable and efficient variant of the Deflated Preconditioned Conjugate Gradient (DPCG) solver is first presented. This two-level method uses an arbitrary coarse grid to reduce the computational cost of the solving. However, in the massively parallel implementation of this technique for very large linear systems, the coarse grids generated can count up to millions of cells, which makes direct solvings on the coarse level impossible. The solving on the coarse grid, performed with a Preconditioned Conjugate Gradient (PCG) solver for this reason, may involve a large number of communications, which reduces dramatically the performances on massively parallel machines. To this effect, two methods developed in order to reduce the number of iterations on the coarse level are introduced, that is the creation of improved initial guesses and the adaptation of the convergence criterion. The design of these methods make them easy to implement in any already existing DPCG solver. The structural requirements for an efficient massively parallel unstructured solver and the implementation of this solver are described. The novel DPCG method is assessed for applications involving turbulence, heat transfers and two-phase flows, with grids up to 17.8billion elements. Numerical results show a two- to 12-fold reduction of the number of iterations on the coarse level, which implies a reduction of the computational time of the Poisson solver up to 71% and a global reduction of the proportion of communication times up to 53%. As a result, the weak scaling of the LES solver is shown to be clearly improved for massively parallel uses.