Superconvergent discontinuous Galerkin methods for elliptic problems

This thesis is devoted to devising and analyzing new discontinuous Galerkin (DG) methods with optimal convergence properties for elliptic and hyperbolic problems. It has three parts.;In the first part, we investigate if by reducing the stabilization (or dissipation) of the DG method we can enhance its accuracy. This approach is motivated by the fact that this is what actually happens in the one-dimensional case for the so-called the minimal dissipation local discontinuous Galerkin (MD-LDG) method. Thus, we carry out the first error analysis of the MD-LDG method for multidimensional convection-diffusion problems. We find that the orders of convergence of the approximations for the potential and the flux using polynomials of degree k are (k + 1) and k, respectively. Thus, the naive elimination of dissipativity effects does not lead to an improvement of the order of convergence of the flux, so we try different approaches.;In the second part of the thesis, we identify and study an LDG-hybridizable Galerkin method for second-order elliptic problems in several space dimensions with remarkable convergence properties. We prove that, if the method uses polynomials of degree k ≥ 0 for both the potential and the flux, the order of convergence in L2 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 devising of a new element-by-element postprocessing of the approximate solution which provides an approximation of the potential converging with order k + 2 in L2.;In the third part of the thesis, using the above mentioned projection operators we obtain a new optimal convergence result of the original DG method for the transport-reaction equation in multidimensional space, provided the meshes are suitably chosen.