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：针对带有一阶吸收边界条件的亥姆霍兹方程的混合间断Galerkin（HDG）离散格式，我们提出了一种基于残差最小化的可靠且高效的后验误差估计子。该残差最小化过程在HDG格式提供的标量解逼近的局部超收敛后处理方案上执行。由此，除了获得标量解的超收敛逼近外，我们还得到了一个Riesz意义上的残差代表元，该代表元进一步被用于推导后验估计子。通过两个特设数值实验，我们验证了理论发现及后验误差估计子的表现。