Superconvergent Interpolatory HDG Methods for Reaction Diffusion Equations I: An HDG K Method
In our earlier work (Cockburn et al. in J Sci Comput 79(3):1777—1800, 2019), 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.
G. Chen et al., "Superconvergent Interpolatory HDG Methods for Reaction Diffusion Equations I: An HDG K Method," Journal of Scientific Computing, vol. 81, no. 3, pp. 2188-2212, Springer, Dec 2019.
The definitive version is available at https://doi.org/10.1007/s10915-019-01081-3
Mathematics and Statistics
Center for High Performance Computing Research
Keywords and Phrases
Error Analysis; Interpolatory Hybridizable Discontinuous Galerkin Method; Nonlinear Reaction Diffusion; Superconvergence
International Standard Serial Number (ISSN)
Article - Journal
© 2019 Springer, All rights reserved.
01 Dec 2019