Convergence acceleration method of large-scale parallel iterative solvers for heterogeneous properties

In large-scale scienti c computing, linear sparse solver is one of the most time-consuming process. In GeoFEM, various types of preconditioned iterative method is implemented on massively parallel computers. It has been well-known that ILU(0) factorization is very e ective preconditioning method for iterative solver. But it's also well-known that this method requires global data dependency and this is not the optimal way on parallel computers where locality is of utmost importance. In this paper, "Localized" ILU(0) preconditioning method has been implemented to various type of iterative solvers. This method provides data locality on each processor and good parallelization e ect. Developed system performance has been also evaluated on workstation cluster with MPI. Linear Solvers In GeoFEM In GeoFEM, preconditioned iterative method is implemented on massively parallel computers in order to solve large scale problems with more than 10 DOFs. GeoFEM solves both of symmetric and un-symmetric matrices. Therefore CG (Conjugate Gradient) for symmetric matrices and BiCGSTAB (Bi-Conjugate Gradient Stabilized) and GMRES (Generalized Minimal Residual) methods are implemented. GMRES is especially suitable for nonlinear problems. Message passing type programming model is adopted and the program is written in Fortran 90 with MPI [1]. Whole region is partitioned by node-based manner. Local operation is considered to be very important in order to handle large data easily and to attain good parallelization e ect. Local data structure with communication table described in [2] is implemented. Thus data handling and matrix assemble operations are fully localized and global operation occurs only in the solver subsystem (Fig.1). Actually, no global information except partition to partition connectivity is required. In iterative solvers, preconditioning is very important for convergence. In this paper strong and stable preconditioning method for parallel computing is developed and discussed.