Journal of Scientific Computing

, Volume 79, Issue 2, pp 1161–1181 | Cite as

Optimal Quasi-diagonal Preconditioners for Pseudodifferential Operators of Order Minus Two

  • Thomas FührerEmail author
  • Norbert Heuer


We present quasi-diagonal preconditioners for piecewise polynomial discretizations of pseudodifferential operators of order minus two in any space dimension. Here, quasi-diagonal means diagonal up to a sparse transformation. Considering shape regular simplicial meshes and arbitrary fixed polynomial degrees, we prove, for dimensions larger than one, that our preconditioners are asymptotically optimal. Numerical experiments in two, three and four dimensions confirm our results. For each dimension, we report on condition numbers for piecewise constant and piecewise linear polynomials.


Pseudodifferential operator of negative order Diagonal scaling Additive Schwarz method Preconditioner Negative order Sobolev spaces 

Mathematics Subject Classification

65F35 65N30 



This work was supported by CONICYT through FONDECYT Projects 11170050 and 1150056


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Authors and Affiliations

  1. 1.Facultad de MatemáticasPontificia Universidad Católica de ChileSantiagoChile

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