Superconvergent HDG methods on isoparametric elements for second-order elliptic problems (Q2903043)

The authors propose a projection-based a priori error analysis of a class of mixed and hybridizable discontinuous Galerkin (HDG) methods for the model problem \( \mathbf{q}+\nabla u=0,\nabla \cdot \mathbf{q}=f\in L^{2}\left( \Omega \right) \) in \(\Omega \subset\mathbb{R}^{n},u=g\in H^{1/2}\left( \partial \Omega \right) \) on \(\partial \Omega \). They consider elements which are related to the reference elements by nonlinear mappings and local spaces on the reference elements satisfying suitable conditions. Imposing some regularity and compatibility conditions for the mapping used to define the mesh and global spaces, they construct an operator mapping the exact solution into an approximation in the finite element space which is convergent for both unknowns and superconvergent for the scalar variable. The error analysis is almost identical to that obtained in the case of affine mappings by introducing two new spaces of traces and associated projections.