Devising superconvergent HDG methods for partial differential equations

Abstract

The DG methods are ideally suited for numerically solving hyperbolic problems. However this is not the case for diffusion problems,even though they are ideally suited for hp-adaptivity. Indeed, when compared with the classical continuous Galerkin methods on the same mesh, they have many more global degrees of freedom and they are not easy to implement. When compared with the mixed methods, they do not provide optimally convergent approximations to the flux and do not display superconvergence properties of the scalar variable. As a response to these disadvantages, the HDG methods were introduced in [6]. Therein, it was shown that HDG methods can be implemented as efficiently as the mixed methods. Later in [7] it was proven that the HDG methods do share with mixed methods their superior convergence properties while retaining the advantages typical of the DG methods. Inspired by these results, in this Thesis we are trying to explore HDG methods in a wider circumstance.