Abstract. 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 ≥ 0 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.

Abstract. We identify discontinuous Galerkin methods for second-order elliptic problems in several space dimensions having superconvergence properties similar to those of the Raviart-Thomas and the Brezzi-Douglas-Marini mixed methods. These methods use polynomials of degree k ≥ 0 for both the potential as well as the flux. We show that the approximate flux converges in L 2 with the optimal order of k + 1, and that the approximate potential and its numerical trace superconverge, in L 2 -like norms, to suitably chosen projections of the potential, with order k + 2. We also apply element-by-element postprocessing of the approximate solution to obtain new approximations of the flux and the potential. The new approximate flux is proven to have normal components continuous across inter-element boundaries, to converge in L 2 with order k + 1, and to have a divergence converging in L 2 also with order k + 1. The new approximate potential is proven to converge with order k + 2 in L 2 . Numerical experiments validating these theoretical results are presented.

In this article, we propose a novel discontinuous Galerkin method for convectiondiffusion-reaction problems, characterized by three main properties. The first is that the method is hybridizable; this renders it efficiently implementable and competitive with the main existing methods for these problems. The second is that, when the method uses polynomial approximations of the same degree for both the total flux and the scalar variable, optimal convergence properties are obtained for both variables; this is in sharp contrast with all other discontinuous methods for this problem. The third is that the method exhibits superconvergence properties of the approximation to the scalar variable; this allows us to postprocess the approximation in an element-by-element fashion to obtain another approximation to the scalar variable which converges faster than the original one. In this paper, we focus on the efficient implementation of the method and on the validation of its computational performance. With this aim, extensive numerical tests are devoted to explore the convergence properties of the novel scheme, to compare it with other methods in the diffusiondominated regime, and to display its stability and accuracy in the convection-dominated case.

We present a family of conforming finite elements for the Stokes problem on general triangular meshes in two dimensions. The lowest order case consists of enriched piecewise linear polynomials for the velocity and piecewise constant polynomials for the pressure. We show that the elements satisfy the inf-sup condition and converges with order k for both the velocity and pressure. Moreover, the pressure space is exactly the divergence of the corresponding space for the velocity. Therefore the discretely divergence-free functions are divergence-free pointwise. We also show how the proposed elements are related to a class of C 1 elements through the use of a discrete de Rham complex.