A projective hybridizable discontinuous Galerkin mixed method for second-order diffusion problems

Abstract

Abstract In this paper, we present a hybridizable discontinuous Galerkin (HDG) mixed method for second-order diffusion problems using a projective stabilization function and broken Raviart–Thomas functions to approximate the dual variable. The proposed HDG mixed method is inspired by the primal HDG scheme with reduced stabilization suggested by Lehrenfeld and Schoberl in 2010, and the standard hybridized version of the Raviart–Thomas (H-RT) method. Indeed, we use the broken Raviart–Thomas space of degree k ≥ 0 for the flux, a piecewise polynomial of degree k + 1 for the potential, and a piecewise polynomial of degree k for its numerical trace. This unconventional polynomial combination is made possible by the projective Lehrenfeld–Schoberl (LS) stabilization function. Its introduction and the use of Raviart-Thomas spaces will have beneficial effects: no postprocessing is required to improve the accuracy of the potential uh, and a straightforward flux reconstruction is sufficient to obtain a H(div)–conforming flux variable. The convergence and accuracy of our method are investigated through numerical experiments in two-dimensional space by using h and p refinement strategies. An optimal convergence order ( k + 1 ) for the H(div)-conforming flux and superconvergence ( k + 2 ) for the potential is observed. Comparative tests with the classical H-RT and the well-known hybridizable local discontinuous Galerkin (H-LDG) mixed methods are also performed and exposed in terms of CPU time.