Superconvergent Interpolatory HDG methods for reaction diffusion equations II: HHO-inspired methods

TL;DR

This paper extends superconvergent interpolatory HDG methods to HHO-inspired schemes on polyhedral meshes, achieving high convergence rates for reaction diffusion equations with reduced computational costs.

Contribution

It introduces HHO-inspired HDG methods on general polyhedral meshes, maintaining superconvergence and computational efficiency for reaction diffusion problems.

Findings

Methods achieve superconvergence for polynomial degree k≥0.

Interpolatory approach reduces computational cost.

Numerical results confirm theoretical convergence rates.

Abstract

In J. Sci. Comput., 81: 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 intergration leading to a reduction in computational cost. The resulting method displays a superconvergent rate for the solution for polynomial degree $k\ge 1$. In this work, we take advantage of the link found between the HDG and the hybrid high-order (HHO) methods, in ESAIM Math. Model. Numer. Anal., 50: 635-650, 2016 and extend this idea to the new, HHO-inspired HDG methods, defined on meshes made of general polyhedral elements, uncovered therein}. 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 $k\ge 0$ by some methods. We present numerical results to illustrate the convergence theory.