Robust a posteriori error estimates for HDG method for convection-diffusion equations (Q2794713)

This paper is concerned with establishing a posteriori error estimates of approximate solutions, obtained by a Galerkin finite element method for the convection-diffusion equation, presented in the form NEWLINE\[NEWLINE\begin{aligned} \varepsilon^{-1} q + \nabla u = 0 \,\, &\text{ on} \quad \Omega \subset \mathbb R^{d},\\ \text{div} q + \beta \nabla u + c u = f \,\, &\text{ on} \quad \Omega \subset \mathbb R^{d},\\ u = g\,\, &\text{ on} \quad \partial \Omega. \end{aligned}NEWLINE\]NEWLINE The mesh is constructed by a regular simplicial triangulation \(T_{h}\) of \( \Omega\) and the finite element spaces by polynomials interpolation of the order \( p\geq 1\) for either \( T \in T_{h}\).NEWLINENEWLINEThe a posteriori error estimates are presented as a sum of the corresponding error on the interior of \(T\), on the interior edges of \( \partial T\), and on the boundary \(\partial T \subset \partial\Omega\). The numerical results are illustrated for three examples (with the known exact solution) for different degrees of polynomials \(p=1,2,3\).