A hybridizable discontinuous Galerkin method for fractional diffusion problems

Bernardo Cockburn, Kassem Mustapha

Research output: Contribution to journalArticlepeer-review

39 Scopus citations

Abstract

We study the use of the hybridizable discontinuous Galerkin (HDG) method for numerically solving fractional diffusion equations of order (Formula presented.) with (Formula presented.). For exact time-marching, we derive optimal algebraic error estimates assuming that the exact solution is sufficiently regular. Thus, if for each time (Formula presented.)∈[0,T] the approximations are taken to be piecewise polynomials of degree (Formula presented.) on the spatial domain (Formula presented.), the approximations to u in the (Formula presented.)-norm and to ∇u in the (Formula presented.)-norm are proven to converge with the rate (Formula presented.), where h is the maximum diameter of the elements of the mesh. Moreover, for k≥1 and quasi-uniform meshes, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for u converging with a rate of (Formula presented.).

Original languageEnglish (US)
Pages (from-to)293-314
Number of pages22
JournalNumerische Mathematik
Volume130
Issue number2
DOIs
StatePublished - Jun 1 2015

Bibliographical note

Funding Information:
The valuable comments of the editor and the referees improved the paper. The support of the Science Technology Unit at KFUPM through King Abdulaziz City for Science and Technology (KACST) under National Science, Technology and Innovation Plan (NSTIP) project No. 13-MAT1847-04 is gratefully acknowledged.

Publisher Copyright:
© 2014, Springer-Verlag Berlin Heidelberg.

Keywords

  • 26A33
  • 35L65
  • 65M12
  • 65M15
  • 65M60
  • 65N30

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