A performance evaluation of a new flexible preconditioning method on a parallel finite element structure analysis program, FrontISTR

Variable preconditioning methods for Krylov-type linear equation solvers have become popular thanks to their faster convergence speed compared to conventional methods, such as, the Incomplete Lower-Upper factorization and point Jacobi methods. Recently, Kushida and Okuda have introduced a variable preconditioning method, which updates the preconditioning matrix using the Broyden-Fletcher-Goldfarb-Shanno scheme, and applied it to the generalized minimum residual recursive scheme (GMRESR), which is a variant of the well-known GMRES method. Although their method indicated a superior performance to the conventional methods, the problems employed in their study were academic, and its performance on practical problems is of interest. In this study, we evaluate the feasibility of their variable preconditioning for practical problems. FrontISTR, which is a well established open-source parallel finite element analysis program and well used in the industrial field, is employed as the framework to implement the above-mentioned Kushida and Okuda's preconditioning method in GMRESR (Self-Updating Preconditioning GMRESR; SUP-GMRESR). As results, (1) SUP-GMRESR indicated approximately a three-fold faster convergence than GMRES, which is one of the default linear equation solvers implemented on FrontISTR, using a 600 million degrees of freedom problem, and (2) SUP-GMRESR converged even when GMRES suffered from a stagnation.