Uniform-in-time superconvergence of HDG methods for the heat equation

Abstract

We prove that the superconvergence properties of the hybridizable discontinuous Galerkin method for second-order elliptic problems do hold uniformly in time for the semidiscretization by the same method of the heat equation provided the solution is smooth enough. Thus, if the approximations are piecewise polynomials of degree $k$, the approximation to the gradient converges with the rate $h^{k+1}$ for $k\ge 0$ and the $L^2$-projection of the error into a space of lower polynomial degree superconverges with the rate $\sqrt {\log (T/h^2)} h^{k+2}$ for $k\ge 1$ uniformly in time. As a consequence, an element-by-element postprocessing converges with the rate $\sqrt {\log (T/h^2)} h^{k+2}$ for $k\ge 1$ also uniformly in time. Similar results are proven for the Raviart-Thomas and the Brezzi-Douglas-Marini mixed methods.