Numerische Mathematik

, Volume 135, Issue 4, pp 1073–1119 | Cite as

Convergence and optimality of \({\mathbf {hp}}\)-AFEM

  • Claudio Canuto
  • Ricardo H. Nochetto
  • Rob Stevenson
  • Marco Verani


We design and analyze an adaptive hp-finite element method (\({\mathbf {hp}}\)-AFEM) in dimensions \(n=1,2\). The algorithm consists of iterating two routines: \({\mathbf {hp}}\)-NEARBEST finds a near-best hp-approximation of the current discrete solution and data to a desired accuracy, and REDUCE improves the discrete solution to a finer but comparable accuracy. The former hinges on a recent algorithm by Binev for adaptive hp-approximation, and acts as a coarsening step. We prove convergence and instance optimality.

Mathematics Subject Classification

65N30 65N12 65N50 



C. Canuto and M. Verani are partially supported by GNCS-INdAM and the Italian research Grant Prin 2012 2012HBLYE4 “Metodologie innovative nella modellistica differenziale numerica”. R. H. Nochetto is partially supported by NSF Grants DMS-1109325 and DMS-1411808. We would like to thank the referees for their insightful comments and suggestions.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Claudio Canuto
    • 1
  • Ricardo H. Nochetto
    • 2
  • Rob Stevenson
    • 3
  • Marco Verani
    • 4
  1. 1.Dipartimento di Scienze MatematichePolitecnico di TorinoTurinItaly
  2. 2.Department of Mathematics and Institute for Physical Science and TechnologyUniversity of MarylandCollege ParkUSA
  3. 3.Korteweg-de Vries Institute for MathematicsUniversity of AmsterdamAmsterdamThe Netherlands
  4. 4.MOX, Dipartimento di MatematicaPolitecnico di MilanoMilanItaly

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