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In this paper, we firstly introduce two BPX-like preconditioners \(B_J^1\) and \(B_J^3\), and present an equivalent but more robust BPX-like preconditioner \(B_J^2\) for the solution of the linear finite element discretization of Poisson problem. Secondly, we implement these preconditioners and their preconditioned conjugate gradient (PCG) solvers \(B_l^p\)-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 (SA-CG) in CUSP library. Numerical results demonstrate that the iteration numbers of \(B_2^p\)-CG holds the weakest dependence on the grid size, while \(B_3^p\)-CG is the most efficient solver. Furthermore, \(B_3^p\)-CG possesses considerable advantages over SA-CG in computational capability and efficiency. In particular, \(B_3^p\)-CG runs 3.67 times faster than SA-CG when solving a problem with about one-million unknowns.

Incomplete LU (ILU) methods are widely used in petroleum reservoir simulation and many other applications. However high complexity often makes them the hotspot in the whole simulation due to high complexity when problem size is large. ILU’s inherent serial nature also makes them difficult to take full advantage of computing power of multi-core and many-core devices. In this paper, a greedy graph coloring method is applied to the ILU(k) factorization and triangular solution phases. This method increases degree of parallelism and improves load balance. A block-wise storage format is employed in our ILU implementation in order to take advantage of hierarchical memory structures. Moreover, a dual intensive parallel model is proposed to further improve the performance of ILU(k) on GPUs. We test the performance of the proposed parallel ILU(k) with a set of Jacobian systems arising from petroleum reservoir simulation. Numerical results suggest that the proposed parallel ILU(k) method is effective and robust on multi-core and many-core architectures. On an Intel Xeon E5 multi-core CPU, the speedup compared with the serial execution time is \(5.6\times \) and \(5.4\times \) for factorization and triangular solution, respectively; on an Nvidia K40c GPU card, the speedup can reach \(8.6\times \) and \(12.7\times \) for factorization and triangular solution, respectively.

In this paper, we propose a preconditioner for solving large-scale linear systems arising from fully implicit petroleum reservoir simulation. First, we take several analytical and physical considerations into account, and choose appropriate auxiliary spaces (problems) to design a preconditioner in the framework of auxiliary space preconditioning method. Secondly, we present an efficient implementation of the proposed preconditioner in modern multicore computer environment. Finally, we test the efficiency and robustness of the proposed methods with a large benchmark problem.