Numerische Mathematik

, Volume 132, Issue 1, pp 85–130 | Cite as

Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and applications

  • Ricardo H. Nochetto
  • Enrique Otárola
  • Abner J. SalgadoEmail author


We develop a constructive piecewise polynomial approximation theory in weighted Sobolev spaces with Muckenhoupt weights for any polynomial degree. The main ingredients to derive optimal error estimates for an averaged Taylor polynomial are a suitable weighted Poincaré inequality, a cancellation property and a simple induction argument. We also construct a quasi-interpolation operator, built on local averages over stars, which is well defined for functions in \(L^1\). We derive optimal error estimates for any polynomial degree on simplicial shape regular meshes. On rectangular meshes, these estimates are valid under the condition that neighboring elements have comparable size, which yields optimal anisotropic error estimates over \(n\)-rectangular domains. The interpolation theory extends to cases when the error and function regularity require different weights. We conclude with three applications: nonuniform elliptic boundary value problems, elliptic problems with singular sources, and fractional powers of elliptic operators.

Mathematics Subject Classification

35J70 35J75 65D05 65N30 65N12 



We dedicate this paper to R.G. Durán, whose work at the intersection of real and numerical analysis has been inspirational to us.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Ricardo H. Nochetto
    • 1
  • Enrique Otárola
    • 2
    • 3
  • Abner J. Salgado
    • 4
    Email author
  1. 1.Department of Mathematics, Institute for Physical Science and TechnologyUniversity of MarylandCollege ParkUSA
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA
  3. 3.Department of Mathematical SciencesGeorge Mason UniversityFairfaxUSA
  4. 4.Department of MathematicsUniversity of TennesseeKnoxvilleUSA

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