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Multigrid algorithms for \(\varvec{hp}\)-version interior penalty discontinuous Galerkin methods on polygonal and polyhedral meshes

  • P. F. Antonietti
  • P. Houston
  • X. Hu
  • M. Sarti
  • M. Verani
Article

Abstract

In this paper we analyze the convergence properties of two-level and W-cycle multigrid solvers for the numerical solution of the linear system of equations arising from hp-version symmetric interior penalty discontinuous Galerkin discretizations of second-order elliptic partial differential equations on polygonal/polyhedral meshes. We prove that the two-level method converges uniformly with respect to the granularity of the grid and the polynomial approximation degree p, provided that the number of smoothing steps, which depends on p, is chosen sufficiently large. An analogous result is obtained for the W-cycle multigrid algorithm, which is proved to be uniformly convergent with respect to the mesh size, the polynomial approximation degree, and the number of levels, provided the number of smoothing steps is chosen sufficiently large. Numerical experiments are presented which underpin the theoretical predictions; moreover, the proposed multilevel solvers are shown to be convergent in practice, even when some of the theoretical assumptions are not fully satisfied.

Keywords

hp-discontinuous Galerkin methods Polygonal/polyhedral grids Two-level and multigrid algorithms 

Mathematics Subject Classification

65N30 65N55 65N22 

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Copyright information

© Springer-Verlag Italia 2017

Authors and Affiliations

  • P. F. Antonietti
    • 1
  • P. Houston
    • 2
  • X. Hu
    • 3
  • M. Sarti
    • 1
  • M. Verani
    • 1
  1. 1.MOX-Laboratory for Modeling and Scientific Computing, Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly
  2. 2.School of Mathematical SciencesUniversity of Nottingham, University ParkNottinghamUK
  3. 3.Department of MathematicsTufts UniversityMedfordUSA

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