Superconvergent Interpolatory HDG Methods for Reaction Difusion 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,defned on simplicial meshes,for scalar reaction-difusion equations and showed how to defne 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,defned on meshes made of general polyhedral elements,uncovered therein.For meshes made of shape-regular polyhedral elements afne-equivalent to a fnite 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.

A New Ensemble HDG Method for Parameterized Convection Diffusion PDEs

A new second order time stepping ensemble hybridizable discontinuous Galerkin method for parameterized convection diffusion PDEs with various initial and boundary conditions,body forces,and time depending coefficients is developed.For ensemble solutions in L_(∞)(0,T;L^(2)(Ω)),a superconvergent rate with respect to the freedom degree of the globally coupled unknowns for all the polynomials of degree k≥0 is established.The results of numerical experiments are consistent with the theoretical findings.