A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems

We identify and study an LDG-hybridizable Galerkin method, which is not an LDG method, for second-order elliptic problems in several space dimensions with remarkable convergence properties. Unlike all other known discontinuous Galerkin methods using polynomials of degree $k\ge0$ for both the potential as well as the flux, the order of convergence in $L^2$ of both unknowns is $k+1$. Moreover, both the approximate potential as well as its numerical trace superconverge in $L^2$-like norms, to suitably chosen projections of the potential, with order $k+2$. This allows the application of element-by-element postprocessing of the approximate solution which provides an approximation of the potential converging with order $k+2$ in $L^2$. The method can be thought to be in between the hybridized version of the Raviart-Thomas and that of the Brezzi-Douglas-Marini mixed methods.