## A projective hybridizable discontinuous Galerkin mixed method for second-order diffusion problems.(English)Zbl 1481.65223

Summary: 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 C. Lehrenfeld and J. Schöberl [“High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows”, Comput. Methods Appl. Mech. Eng. 307, 339–361 (2016; doi:10.1016/j.cma.2016.04.025)], and the standard hybridized version of the Raviart-Thomas (H-RT) method. Indeed, we use the broken Raviart-Thomas space of degree $$k \geq 0$$ 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-Schöberl (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 $$u_h$$, and a straightforward flux reconstruction is sufficient to obtain a $$H(\operatorname{div})$$-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 $$H(\operatorname{div})$$-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.

### MSC:

 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations

CSparse
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