A condition for positive-definiteness of the coarse matrix in BDD-DIAG of a perturbed magnetostatic problem

Abstract

Until now, we 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 can be used in many cases for solving the coarse problem in BDD-DIAG. In this paper, we show a sufficient condition for positive-definiteness of the coarse matrix in BDD-DIAG of a perturbed maganetostatic problem. Though our numerical results are got in many cases of the original magnetostatic problem, we mainly consider the well-known perturbed problem in this paper. A conjecture is also given for the original magnetostatic problem.