A review of Hybrid High-Order methods: formulations, computational aspects, comparison with other methods

Abstract : Hybrid High-Order (HHO) methods are formulated in terms of discrete unknowns attached to mesh faces and cells (hence, the term hybrid), and these unknowns are polynomials of arbitrary order k>=0 (hence, the term high-order). HHO methods are devised from local reconstruction operators and a local stabilization term. The discrete problem is assembled cellwise, and cell-based unknowns can be eliminated locally by static condensation. HHO methods support general meshes, are locally conservative, and allow for a robust treatment of physical parameters in various situations, e.g., heterogeneous/anisotropic diffusion, quasi-incompressible linear elasticity, and advection-dominated transport. This paper reviews HHO methods for a variable-diffusion model problem with nonhomogeneous, mixed Dirichlet–Neumann boundary conditions, including both primal and mixed formulations. Links with other discretization methods from the literature are discussed.
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Daniele Di Pietro, Alexandre Ern, Simon Lemaire. A review of Hybrid High-Order methods: formulations, computational aspects, comparison with other methods. Barrenechea G.; Brezzi F.; Cangiani A.; Georgoulis E. Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, 114, Springer, pp.205-236, 2016, Lecture Notes in Computational Science and Engineering, 978-3-319-41638-0. ⟨10.1007/978-3-319-41640-3_7⟩. ⟨hal-01163569v4⟩

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