A substructuring preconditioner with vertex-related interface solvers for elliptic-type equations in three dimensions
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A substructuring preconditioner with vertex-related interface solvers for elliptic-type equations in three dimensions Qiya Hu1,2 · Shaoliang Hu1,2 Received: 26 December 2017 / Accepted: 7 November 2018 / Published online: 7 January 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract In this paper, we propose a variant of the substructuring preconditioner for solving three-dimensional elliptic-type equations with strongly discontinuous coefficients. In the proposed preconditioner, we use the simplest coarse solver associated with the finite element space induced by the coarse partition and construct inexact interface solvers based on overlapping domain decomposition with small overlaps. This new preconditioner has an important merit: its construction and efficiency do not depend on the concrete form of the considered elliptic-type equations. We apply the proposed preconditioner to solve the linear elasticity problems and Maxwell’s equations in three dimensions. Numerical results show that the convergence rate of PCG method with the preconditioner is nearly optimal, and also robust with respect to the (possibly large) jumps of the coefficients in the considered equations. Keywords Domain decomposition · Substructuring preconditioner · Linear elasticity problems · Maxwell’s equations · PCG iteration · Convergence rate Mathematics Subject Classification (2010) 65N30 · 65N55
Communicated by: Jan Hesthaven This work was funded by Natural Science Foundation of China G11571352. Qiya Hu
[email protected] Shaoliang Hu
[email protected] 1
LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
2
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
1130
Q. Hu, S. Hu
1 Introduction There are many works to study domain decomposition methods (DDMs) for solving the systems generated by finite element discretization of elliptic-type partial differential equations ([1–6, 8–25, 28–44, 48–53, 55] and the references therein). It is known that, for three-dimensional problems, non-overlapping domain decomposition methods (DDMs) are more difficult to construct and implement than overlapping DDMs although the non-overlapping DDMs have some advantages over the overlapping DDMs in the treatment of jump coefficients. In fact, the construction of non-overlapping DDMs heavily depends on the considered models in three dimensions. For example, non-overlapping DDMs for positive definite Maxwell’s equations are essentially different from the usual elliptic equation (comparing [14, 31, 33, 50]). These drawbacks restrict applications of the non-overlapping DDMs. A key ingredient in the construction of non-overlapping domain decomposition methods is the choice of a suitable coarse subspace. There are two main ways to construct coarse subspaces in the existing works: (i) use some degrees of freedom on the joint-set (BPS method, FETI-DP method, BDDC method) and (ii) use the local kernel spaces of the consid
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