Domain Decomposition Methods for Magnetostatic Problems by Using Higher Order Polytopal Elements Satisfying Discrete de Rham Sequences

Abstract

A Balancing Domain Decomposition (BDD) method is considered as the preconditioner of an iterative Domain Decomposition Method (DDM) for perturbed magnetostatic problems, where the magnetic vector potential is regarded as an unknown function approximated by the Ned´ elec curl conforming finite element. To reduce the number of the Degrees´ Of Freedom (DOF) of coarse spaces in BDD methods, Polynomial Element Methods (PEMs) are introduced. Owing to the introduction of PEMs and the result of BDD method originally proposed by Mandel, the condition number of coefficient matrices derived from the iterative DDM is evaluated. Therefore, the number of iterations of the iterative DDM can be kept even when the number of the subdomains becomes larger. Moreover, the approximate coarse space by PEMs can admit that each subdomain becomes more general polyhedron. The results in case of higher order PEMs are also shown.