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

Abstract : In this paper, we present a projective hybridizable Raviart-Thomas mixed method (H-RT p) for second-order diffusion problems. The proposed method is inspired by the hybridizable discontinuous Galerkin (HDG) formalism, as it introduces a residual flux terms in the (hy-bridized version of) Raviart-Thomas mixed method. Specifically, we add a projective-type stabilization function in the definition of the normal trace of the flux on the mesh skeleton. Indeed, we use 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 in the general framework of HDG methods thanks to the projective-based stabilization function introduced. The convergence and accuracy of the H-RT p method are investigated through numerical experiments in two-dimensional space by using hand p-refinement strategies. An optimal convergence order (k + 1) for the H(div)-conforming flux variable (obtained after a straightforward reconstruction) and superconvergence (k + 2) for the potential (without any postprocessing) is observed. Comparative tests with the classical H-RT method and the well-known hybridizable local discontinuous Galerkin (H-LDG) method are also performed and exposed in terms of CPU time and mesh refinement.
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Loïc Dijoux, Vincent Fontaine, Thierry A. Mara. A projective hybridizable discontinuous Galerkin mixed method for second-order diffusion problems. Applied Mathematical Modelling, Elsevier, 2019, 75 (1), pp.663-677. ⟨10.1016/j.apm.2019.05.054⟩. ⟨hal-02289119⟩



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