Superconvergent HDG methods for linear, stationary, third-order equations in one-space dimension

We design and analyze the first hybridizable discontinuous Galerkin methods for stationary, third-order linear equations in one-space dimension. The methods are defined as discrete versions of characterizations of the exact solution in terms of local problems and transmission conditions. They provide approximations to the exact solution uu and its derivatives q:=u′q:=u’ and p:=up:=u which are piecewise polynomials of degree kuk_u, kqk_q and kpk_p, respectively. We consider the methods for which the difference between these polynomial degrees is at most two. We prove that all these methods have superconvergence properties which allows us to prove that their numerical traces converge at the nodes of the partition with order at least 2k+12\,k+1, where kk is the minimum of ku,kqk_u,k_q, and kpk_p. This allows us to use an element-by-element post-processing to obtain new approximations for u,qu, q and pp converging with order at least 2k+12k+1 uniformly. Numerical results validating our error estimates are displayed.