MultiGrid Preconditioners for Mixed Finite Element Methods of the Vector Laplacian

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Abstract

Due to the indefiniteness and poor spectral properties, the discretized linear algebraic system of the vector Laplacian by mixed finite element methods is hard to solve. A block diagonal preconditioner has been developed and shown to be an effective preconditioner by Arnold et al. (Acta Numer 15:1–155, 2006). The purpose of this paper is to propose alternative and effective block diagonal and approximate block factorization preconditioners for solving these saddle point systems. A variable V-cycle multigrid method with the standard point-wise Gauss–Seidel smoother is proved to be a good preconditioner for the discrete vector Laplacian operator. The major benefit of our approach is that the point-wise Gauss–Seidel smoother is more algebraic and can be easily implemented as a black-box smoother. This multigrid solver will be further used to build preconditioners for the saddle point systems of the vector Laplacian. Furthermore it is shown that Maxwell’s equations with the divergent free constraint can be decoupled into one vector Laplacian and one scalar Laplacian equation.

Keywords

Saddle point system Multigrid methods Mixed finite elements Vector Laplacian Maxwell equations 

Mathematics Subject Classification

65N55 65F10 65N22 65N30 

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at IrvineIrvineUSA
  2. 2.Beijing Institute for Scientific and Engineering ComputingBeijing University of TechnologyBeijingChina
  3. 3.School of Mathematical SciencesUniversity of Electronic Science and Technology of ChinaChengduChina
  4. 4.School of Mathematical and Computational SciencesXiangtan UniversityXiangtanChina

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