Superconvergent hdg methods on isoparametric elements for second-order elliptic problems

摘要

We propose a projection-based a priori error analysis of a wide class of mixed and hybridizable discontinuous Galerkin methods for diffusion problems for which the mappings relating the elements to the reference elements are nonlinear. We show that if the local spaces on the reference elements satisfy suitable conditions, and if the mappings used to define the mesh and global spaces satisfy simple regularity and compatibility conditions, the methods provide optimally convergent approximations for both unknowns as well as superconvergent approximations for the scalar variable. A crucial feature of the analysis of the methods is the use of two new spaces of traces and two associated, suitably defined projections thanks to which the error analysis then becomes almost identical to that obtained by the authors in [Math. Comp., 81 (2012), pp. 1327-1353] where the case in which the mappings are affine is considered.