スペクトル選点法における最適パラメータ前処理

抄録

A general preconditioning is presented in this paper for iterative solution of a problem approximated by spectral collocation methods. In this preconditioning, a parameter is introduced in the mass matrix of the finite element preconditioning so that the finite difference preconditioning and the finite element preconditioning are the special cases of this preconditioning corresponding to the specific parameters respectively. The eigenvalues of this preconditioned matrix are calculated and the condition number is investigated. The results reveal that there exsits an optimal parameter corresponding to a minimun condition number which is still less than that of the finite element precondition. The efficiency of this optimal preconditioning is demonstrated by the Richardson iteration. Around 73% - 82% iterations of the finite element preconditioning still can be reduced by this optimal parameter preconditioning.