Minimum-Residual a Posteriori Error Estimates for Hdg Discretizations of the Helmholtz Equation

We propose and analyze two a posteriori error indicators for hybridizable discontinuous Galerkin (HDG) discretizations of the Helmholtz equation. These indicators are built to minimize the residual associated with a local superconvergent postprocessing scheme for the primal variable, measured in a dual norm of an enlarged discrete test space. The residual minimization is reformulated into equivalent local saddle-point problems, yielding a superconvergent postprocessed approximation of the primal variable in the asymptotic regime for sufficiently regular exact solutions and a built-in residual representation with minimal computational effort. Both error indicators are based on frequency-dependent postprocessing schemes and verify reliability and efficiency estimates for a frequency-weighted H1-error for the scalar unknown and the L2-error for the flux. We illustrate our theoretical findings through ad-hoc numerical experiments. © 2025 Elsevier B.V.