# Superconvergent interpolatory HDG methods for reaction diffusion equations I: An HDG$_{k}$ method

## TL;DR

This paper enhances an interpolatory HDG method for reaction diffusion equations by enabling superconvergence through a modified postprocessing step, maintaining computational efficiency and improving accuracy.

## Contribution

It introduces a simple modification to the interpolatory HDG method that restores superconvergence for reaction diffusion problems, combining efficiency with higher accuracy.

## Key findings

- Restores superconvergence with a modified postprocessing step.
- Maintains computational efficiency of the original interpolatory HDG method.
- Numerical results confirm theoretical convergence and performance improvements.

## Abstract

In our earlier work [8], we approximated solutions of a general class of scalar parabolic semilinear PDEs by an interpolatory hybridizable discontinuous Galerkin (Interpolatory HDG) method. This method reduces the computational cost compared to standard HDG since the HDG matrices are assembled once before the time integration. Interpolatory HDG also achieves optimal convergence rates; however, we did not observe superconvergence after an element-by-element postprocessing. In this work, we revisit the Interpolatory HDG method for reaction diffusion problems, and use the postprocessed approximate solution to evaluate the nonlinear term. We prove this simple change restores the superconvergence and keeps the computational advantages of the Interpolatory HDG method. We present numerical results to illustrate the convergence theory and the performance of the method.