Adaptive Multilevel Methods with Local Smoothing for $H^1$- and $H^{\mathrm{curl}}$-Conforming High Order Finite Element Methods

Abstract

A multilevel method on adaptive meshes with hanging nodes is presented, and the additional matrices appearing in the implementation are derived. Smoothers of overlapping Schwarz type are discussed; smoothing is restricted to the interior of the subdomains refined to the current level; thus it has optimal computational complexity. When applied to conforming finite element discretizations of elliptic problems and Maxwell equations, the method's convergence rates are very close to those for the nonadaptive version. Furthermore, the smoothers remain efficient for high order finite elements. We discuss the implementation in a general finite element code using the example of the deal.II library.

MSC codes

  1. 65N55
  2. 65N22
  3. 65N30
  4. 65F08

Keywords

  1. multigrid methods
  2. adaptive mesh refinement
  3. finite element method
  4. hanging nodes
  5. Maxwell equations

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Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: 2095 - 2114
ISSN (online): 1095-7197

History

Submitted: 30 November 2009
Accepted: 2 June 2011
Published online: 25 August 2011

MSC codes

  1. 65N55
  2. 65N22
  3. 65N30
  4. 65F08

Keywords

  1. multigrid methods
  2. adaptive mesh refinement
  3. finite element method
  4. hanging nodes
  5. Maxwell equations

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