A Numerical Experimental Study of Inverse Preconditioning for the Parallel Iterative Solution to 3D Finite Element Flow Equations

Integration of the subsurface flow equation by Finite Elements (FE) in space and Finite Differences (FD) in time requires the repeated solution to sparse symmetric positive definite systems of linear equations. Iterative techniques based on Preconditioned Conjugate Gradients (PCG) are one of the most attractive tool to solve the problem on sequential computers. A present challenge is to make PCG attractive in a parallel computing environment as well. To this aim a key factor is the development of an efficient parallel preconditioner. FSAI (Factorized Sparse Approximate Inverse) and enlarged FSAI relying on the approximate inverse of the coefficient matrix appears to be a most promising parallel preconditioner. In the present paper PCG using FSAI, diagonal and pARMS (parallel Algebraic Recursive Multilevel Solvers) preconditioners is implemented on the IBM SP4/512 and CLX/768 supercomputers with up to 32 processors to solve underground flow problems of a large size. The results show that FSAI may allow for a parallel relative efficiency E_p* larger than 50% on the largest problems with p=32 processors. Moreover FSAI turns out to be significantly less expensive and more robust than pARMS. Finally it is shown that E_p* for p in the upper range may be much improved if PCG-FSAI is implemented on CLX.