A substructuring preconditioner with vertex-related interface solvers for elliptic-type equations in three dimensions

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.

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Correspondence to Qiya Hu or Shaoliang Hu.

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This work was funded by Natural Science Foundation of China G11571352.

Communicated by: Jan Hesthaven

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Hu, Q., Hu, S. A substructuring preconditioner with vertex-related interface solvers for elliptic-type equations in three dimensions. Adv Comput Math 45, 1129–1161 (2019). https://doi.org/10.1007/s10444-018-9648-y

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Keywords

  • Domain decomposition
  • Substructuring preconditioner
  • Linear elasticity problems
  • Maxwell’s equations
  • PCG iteration
  • Convergence rate

Mathematics Subject Classification (2010)

  • 65N30
  • 65N55