Superconvergent interpolatory HDG methods for reaction diffusion equations I: An HDG$_{k}$ method
For researchers using HDG methods for parabolic PDEs, this work provides a simple fix to recover superconvergence without increasing computational cost.
The authors modified an interpolatory HDG method for reaction-diffusion equations by using a postprocessed solution to evaluate the nonlinear term, restoring superconvergence while maintaining computational efficiency. Numerical results confirm the theoretical convergence rates.
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.