Schwarz Preconditioning for High Order Edge Element Discretizations of the Time-Harmonic Maxwell’s Equations

Abstract

We focus on high order edge element approximations of waveguide problems. For the associated linear systems, we analyze the impact of two Schwarz preconditioners, the Optimized Additive Schwarz (OAS) and the Optimized Restricted Additive Schwarz (ORAS), on the convergence of the iterative solver.

Parallel preconditioners for high‐order discretizations arising from full system modeling for brain microwave imaging

“…The preconditioner without the partition of unity matrices D s , , which is called optimized additive Schwarz, would be symmetric for symmetric problems, but in practice, it gives a slower convergence with respect to , as shown for instance in another study …”

“…for all edges e of the tetrahedron T , and for all multi-indices k of weight k. Note that these high order elements still yield a conformal discretization of H(curl, Ω): indeed, they are products between the degree 1 Nédélec elements w e , which are curl-conforming, and the continuous functions λ k . However, some of these high order generators (r > 1) are linearly dependent: the selection of a linearly independent subset to constitute an actual basis is described in [7], which provides further details about the implementation of these finite elements. Moreover, the duality property, which is practical for the implementation, is not satisfied for high order generators, but it can be easily restored as explained in [8].…”

“…which is called Optimized Additive Schwarz (OAS), would be symmetric for symmetric problems, but in practice it gives a slower convergence with respect to M −1 ORAS , as shown for instance in [7]. These domain decomposition preconditioners are implemented in the library HPDDM [13], an open source high-performance unified framework for domain decomposition methods.…”