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

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.

Department(s)

Mathematics and Statistics

Comments

National Science Foundation, Grant DMS-1217122

Keywords and Phrases

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

International Standard Serial Number (ISSN)

2661-8893; 2096-6385

Document Type

Article - Journal

Document Version

Citation

File Type

text

Language(s)

English

Rights

© 2023 Springer, All rights reserved.

Publication Date

01 Jun 2022

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