Foundations of Computational Mathematics

, Volume 12, Issue 5, pp 671–718

Convergence Rates of AFEM with H−1 Data

  • Albert Cohen
  • Ronald DeVore
  • Ricardo H. Nochetto
Article

DOI: 10.1007/s10208-012-9120-1

Cite this article as:
Cohen, A., DeVore, R. & Nochetto, R.H. Found Comput Math (2012) 12: 671. doi:10.1007/s10208-012-9120-1

Abstract

This paper studies adaptive finite element methods (AFEMs), based on piecewise linear elements and newest vertex bisection, for solving second order elliptic equations with piecewise constant coefficients on a polygonal domain Ω⊂ℝ2. The main contribution is to build algorithms that hold for a general right-hand side fH−1(Ω). Prior work assumes almost exclusively that fL2(Ω). New data indicators based on local H−1 norms are introduced, and then the AFEMs are based on a standard bulk chasing strategy (or Dörfler marking) combined with a procedure that adapts the mesh to reduce these new indicators. An analysis of our AFEM is given which establishes a contraction property and optimal convergence rates Ns with 0<s≤1/2. In contrast to previous work, it is shown that it is not necessary to assume a compatible decay s<1/2 of the data estimator, but rather that this is automatically guaranteed by the approximability assumptions on the solution by adaptive meshes, without further assumptions on f; the borderline case s=1/2 yields an additional factor logN. Computable surrogates for the data indicators are introduced and shown to also yield optimal convergence rates Ns with s≤1/2.

Keywords

Adaptivity H−1 data Contraction Rates Cardinality Approximation classes 

Mathematics Subject Classification

65N30 65N15 41A25 65N50 65Y20 

Copyright information

© SFoCM 2012

Authors and Affiliations

  • Albert Cohen
    • 1
    • 2
  • Ronald DeVore
    • 3
  • Ricardo H. Nochetto
    • 4
  1. 1.UMR 7598, Laboratoire Jacques-Louis LionsUPMC Univ Paris 06ParisFrance
  2. 2.UMR 7598, Laboratoire Jacques-Louis LionsCNRSParisFrance
  3. 3.Department of MathematicsTexas A&M UniversityCollege StationUSA
  4. 4.Department of Mathematics and Institute for Physical Science and TechnologyUniversity of MarylandCollege ParkUSA

Personalised recommendations