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Journal of Scientific Computing

, Volume 81, Issue 3, pp 2188–2212 | Cite as

Superconvergent Interpolatory HDG Methods for Reaction Diffusion Equations I: An HDG\(_{k}\) Method

  • Gang Chen
  • Bernardo Cockburn
  • John Singler
  • Yangwen ZhangEmail author
Article
  • 47 Downloads

Abstract

In our earlier work (Cockburn et al. in J Sci Comput 79(3):1777–1800, 2019), we approximated solutions of a general class of scalar parabolic semilinear PDEs by an interpolatory hybridizable discontinuous Galerkin (interpolatory HDG) method. This method reduces the computational cost compared to standard HDG since the HDG matrices are assembled once before the time integration. Interpolatory HDG also achieves optimal convergence rates; however, we did not observe superconvergence after an element-by-element postprocessing. In this work, we revisit the Interpolatory HDG method for reaction diffusion problems, and use the postprocessed approximate solution to evaluate the nonlinear term. We prove this simple change restores the superconvergence and keeps the computational advantages of the Interpolatory HDG method. We present numerical results to illustrate the convergence theory and the performance of the method.

Keywords

Interpolatory hybridizable discontinuous Galerkin method Superconvergence Nonlinear reaction diffusion Error analysis 

Notes

Acknowledgements

G. Chen is supported by National natural science Foundation of China (NSFC) under Grant Number 11801063 and China Postdoctoral Science Foundation under Grant Number 2018M633339. The research of Y. Zhang is partially supported by the US National Science Foundation (NSF) under Grant Number DMS-1619904.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.College of MathematicsSichun UniversityChengduChina
  2. 2.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  3. 3.Department of Mathematics and StatisticsMissouri University of Science and TechnologyRollaUSA
  4. 4.Department of Mathematical SciencesUniversity of DelawareNewarkUSA

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