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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

Software:

CSparse
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References:

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