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.

翻译：我们提出了一种可靠和高效的后验误差估计器，适用于带一阶吸收边界条件的混合矛盾Galerkin（HDG）离散化Helmholtz方程。此类残差最小化是在一个本地和超收敛的HDG方案提供的标量解的后处理方案上进行的。因此，除了标量解的超收敛逼近之外，还提出了一个在Riesz意义上的残差代表，并进一步用于推导后验估计量。我们通过两个特定的数值实验说明了我们的理论发现和后验误差估计器的行为。