“An introduction to the Hybrid High-Order method with application to nonlinear problems”
Thursday, May 13th, 2021 at 16:00 - Zoom - Daniele Di Pietro (Université de Montpellier)
Abstract
Originally introduced in [1,2] (see also [3]), Hybrid High-Order (HHO) methods provide a framework for the discretisation of models based on Partial Differential Equations (PDEs) with features that set them apart from traditional ones. The construction hinges on discrete unknowns that are broken polynomials on the mesh and on its skeleton, from which two key ingredients are devised:Local reconstructions obtained by solving small, embarrassingly parallel prob- lems inside each element, and typically conceived so that their composition with the natural interpolator of sufficiently smooth functions yields a physics- and problem-dependent projector on local polynomial spaces; Stabilisation terms that penalise residuals designed at the element level so as to ensure stability while preserving the approximation properties of the reconstruction. These ingredients are combined to formulate local contributions, which are then assembled as in standard Finite Element Methods. From this construction, several appealing features ensue: the support of polytopal meshes and arbitrary approximation orders in any space dimension, an enhanced compliance with the physics, and a reduced computational cost thanks to the compact stencil along with the possibility of locally eliminating a large portion of the unknowns. In this presentation we discuss the numerical approximation of nonlinear problems of Leray-Lions type through HHO methods [4–5].
1] D. A. Di Pietro, A. Ern, and S. Lemaire, An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators, Comput. Meth. Appl. Math., 2014, 14(4):461–472. Open access (editor’s choice). DOI: 10.1515/cmam-2014-0018
[2] D. A. Di Pietro and A. Ern, A hybrid high-order locking-free method for linear elasticity on general meshes, Comput. Meth. Appl. Mech. Engrg., 2015, 283:1–21. DOI: 10.1016/j.cma.2014.09.009
[3] D. A. Di Pietro and J. Droniou, The Hybrid High-Order Method for Polytopal Meshes. Design, Analysis, and Applications Number 19 in Modeling, Simulation and Applications, Springer International Publishing, 2020. DOI: 10.1007/978-3-030-37203-3
[4] D. A. Di Pietro and J. Droniou, A Hybrid High-Order method for Leray–Lions elliptic equations on general meshes, Math. Comp., 2017, 86(307):2159–2191. DOI: 10.1090/mcom/3180
[5] D. A. Di Pietro and J. Droniou, Wsp-approximation properties of elliptic projectors on polynomial spaces, with application to the error analysis of a Hybrid High-Order discretisation of Leray–Lions problems, Math. Models Methods Appl. Sci., 2017, 27(5):879–908. DOI: 10.1142/S0218202517500191
[6] D. A. Di Pietro, J. Droniou, and A. Harnist, Improved error estimates for Hybrid High-Order discretizations of Leray–Lions problems, Calcolo, 2021, 58(19). DOI: 10.1007/s10092-021-00410-z
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