## Abstract

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 language | English (US) |
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Pages (from-to) | 477-499 |

Number of pages | 23 |

Journal | Communications on Applied Mathematics and Computation |

Volume | 4 |

Issue number | 2 |

DOIs | |

State | Published - 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.

## Keywords

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