A condition for positive-definiteness of the Schur complement matrix of the magnetostatic problem

抄録

Until now, authors have considered the Balancing Domain Decomposition DIAGonal scaling (BDD-DIAG) preconditioner as a preconditioner of non-overlapping domain decomposition analysis of 3-dimensional magnetostatic problems taking the magnetic vector potential as an unknown function. One interesting fact is that the direct solver without pivoting can be used in many cases for solving the coarse problem in BDD-DIAG. In this paper, they show a sufficient condition for positive-definiteness of the Schur complement matrix of the original magnetostatic problem in which the coefficient matrix of the finite element approximation is a well-known singular one. Authors only considered a well-known perturbed problem previously. But, in this paper, they consider the original magnetostatic problem. Furthermore, with a numerical example, they notice that the BDD-DIAG preconditioner includes the DIAGonal scaling (diag) preconditioner as its special case.