Superconvergent Interpolatory HDG Methods for Reaction Diffusion Equations II: HHO-Inspired Methods

Gang Chen, Bernardo Cockburn, John R. Singler, Yangwen Zhang

Research output: Contribution to journalArticlepeer-review


In Chen et al. (J. Sci. Comput. 81(3): 2188–2212, 2019), we considered a superconvergent hybridizable discontinuous Galerkin (HDG) method, defined on simplicial meshes, for scalar reaction-diffusion equations and showed how to define an interpolatory version which maintained its convergence properties. The interpolatory approach uses a locally postprocessed approximate solution to evaluate the nonlinear term, and assembles all HDG matrices once before the time integration leading to a reduction in computational cost. The resulting method displays a superconvergent rate for the solution for polynomial degree k⩾ 1. In this work, we take advantage of the link found between the HDG and the hybrid high-order (HHO) methods, in Cockburn et al. (ESAIM Math. Model. Numer. Anal. 50(3): 635–650, 2016) and extend this idea to the new, HHO-inspired HDG methods, defined on meshes made of general polyhedral elements, uncovered therein. For meshes made of shape-regular polyhedral elements affine-equivalent to a finite number of reference elements, we prove that the resulting interpolatory HDG methods converge at the same rate as for the linear elliptic problems. Hence, we obtain superconvergent methods for k⩾ 0 by some methods. We thus maintain the superconvergence properties of the original methods. We present numerical results to illustrate the convergence theory.

Original languageEnglish (US)
Pages (from-to)477-499
Number of pages23
JournalCommunications on Applied Mathematics and Computation
Issue number2
StatePublished - Jun 2022

Bibliographical note

Funding Information:
J. Singler and Y. Zhang thank the IMA for funding research visits, during which some of this work was completed.

Publisher Copyright:
© 2021, Shanghai University.


  • Hybrid high-order methods
  • Hybridizable discontinuous Galerkin methods
  • Interpolatory method
  • Superconvergence


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