Minimum-residual a posteriori error estimates for a hybridizable discontinuous Galerkin discretization of the Helmholtz equation

We propose a reliable and efficient a posteriori error estimator for a hybridizable discontinuous Galerkin (HDG) discretization of the Helmholtz equation, with a first-order absorbing boundary condition, based on residual minimization. Such a residual minimization is performed on a local and superconvergent postprocessing scheme of the approximation of the scalar solution provided by the HDG scheme. As a result, in addition to the super convergent approximation for the scalar solution, a residual representative in the Riesz sense, which is further employed to derive the a posteriori estimators. We illustrate our theoretical findings and the behavior of the a posteriori error estimator through two ad-hoc numerical experiments.