Superconvergent Interpolatory HDG Methods for Reaction Diffusion Equations II: HHO-Inspired Methods
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.
G. Chen et al., "Superconvergent Interpolatory HDG Methods for Reaction Diffusion Equations II: HHO-Inspired Methods," Communications on Applied Mathematics and Computation, vol. 4, no. 2, pp. 477 - 499, Springer, Jun 2022.
The definitive version is available at https://doi.org/10.1007/s42967-021-00128-3
Mathematics and Statistics
Keywords and Phrases
Hybrid high-order methods; Hybridizable discontinuous Galerkin methods; Interpolatory method; Superconvergence
International Standard Serial Number (ISSN)
Article - Journal
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01 Jun 2022
National Science Foundation, Grant DMS-1217122