Resumen de Performance of Krylov subspace method with SOR preconditioner supported by Eisenstat’s technique for linear system derived from time-periodic FEM

Makoto Kumagai, Shimpei Kakita, Yoshifumi Okamoto

• Purpose – This paper aims to present the affinity of BiCGStab and BiCGStab2 with successive over-relaxation (SOR) preconditioner supported by Eisenstat’s technique for a linear system derived from the time-periodic finite element method (TP-FEM). To solve the time domain electromagnetic field problem with long transient state, TP-FEM is very useful from the perspective of rapidly achieving a steady state. Because TP-FEM solves all of the state variables at once, the linear system derived from TP-FEM becomes the large scale and nonsymmetric, whereas the detailed performance of some preconditioned Krylov subspace method is not reported.

  Design/methodology/approach – In this paper, BiCGStab and BiCGStab2 are used as the linear solver for a large-sparse nonsymmetric linear system derived from TP-FEM. In addition, incomplete LU (ILU) factorization is applied as a preconditioner to compare SOR supported by Eisenstat’s technique. As examples, the pot-type reactor and three-phase transformer is analyzed.

  Findings – In the problem of the pot-type reactor, when SOR preconditioner supported by Eisenstat’s technique is applied to BiCGStab and BiCGStab2, the elapsed time can be reduced dramatically. However, in the problem of the three-phase transformer, the iterative process of the linear solvers with SOR preconditioner is not terminated, whereas the iterative process of linear solvers with ILU preconditioner is terminated. The preconditioner that can be supported by Eisenstat’s technique is not necessarily appropriate for the problem to derive from TP-FEM.

  Originality/value – In this paper, the affinity of preconditioned linear solver supported by Eisenstat’s technique for the nonsymmetric linear system derived from TP-FEM is demonstrated.