BPX-like preconditioned conjugate gradient solvers for Poisson problem and their CUDA implementations

Abstract

In this paper, we firstly introduce two BPX-like preconditioners B1J and B3J, and present an equivalent but more robust BPX-like preconditioner B2J for the solution of the linear finite element discretization of Poisson problem. Secondly, we implement these preconditioners and their preconditioned conjugate gradient (PCG) solvers Bpl -CG(l = 1, 2, 3) under Compute Unified Device Architecture (CUDA), where we exploit the hierarchical and the overall storage structure, take advantage of the multicolored Gauss–Seidel smoother. Finally, comparisons are made among these PCG solvers and the state-of-the-art SA-AMG preconditioned CG solver (SACG) in CUSP library. Numerical results demonstrate that the iteration numbers of Bp2-CG holds the weakest dependence on the grid size, while Bp3-CG is the most efficient solver. Furthermore, Bp3-CG possesses considerable advantages over SACG in computational capability and efficiency. In particular, Bp3-CG runs 3.67 times faster than SA-CG when solving a problem with about one-million unknowns.