André Harnist: Thursday 1 April at 11:00 am
Abstract:
We consider Hybrid High-Order (HHO) approximations of Leray-Lions problems set in W^(1,p) with p in (1,2]. For this class of problems, negative powers of the gradient of the solution can appear in the flux. Depending on the expression of the latter, this can lead to a degeneracy of the problem when the gradient of the solution vanishes or becomes large.
The goal of this presentation is to derive novel error estimates depending on the degeneracy of the problem inspired by [1,2,3]. Specifically, we show that, for the globally non-degenerate case, the energy-norm of the error has a convergence rate of (k+1), with k denoting the degree of the HHO approximation. In the globally degenerate case, on the other hand, the energy-norm of the error converges with a rate of (k+1)(p-1), coherently with the estimate originally proved in [4].
We additionally introduce, for each mesh element, a dimensionless number that captures the local degeneracy of the model and identifies the contribution of the element to the global error: from the fully degenerate regime, corresponding to a contribution in (k+1)(p-1), to the non-degenerate regime, corresponding to a contribution in (k+1), through all intermediate regimes. These regime-dependent error estimates are illustrated by a complete panel of numerical experiments.
[1] M. Botti, D. Castanon Quiroz, D. A. Di Pietro and A. Harnist, A Hybrid High-Order method for creeping flows of non-Newtonian fluids, Submitted. 2020. URL: https://hal.archives-ouvertes.fr/hal-02519233.
[2] D. A. Di Pietro and J. Droniou, The Hybrid High-Order Method for Polytopal Meshes. Modelling, Simulation and Application 19. Springer International Publishing, 2020. ISBN: 978-3-030-37202-6 (Hardcover) 978-3-030-37203-3 (eBook). DOI: 10.1007/978-3-030-37203-3.
[3] D. A. Di Pietro and J. Droniou, A Hybrid High-Order method for Leray–Lions elliptic equations on general meshes, Math. Comp., volume 86, 2017, number 307, pages 2159–2191, DOI:10.1090/mcom/318
[4] D. A. Di Pietro and J. Droniou, W s,p -approximation properties of elliptic projectors on poly-nomial spaces, with application to the error analysis of a hybrid high-order discretisation of Leray-Lions problems, Math. Models Methods Appl. Sci., DOI: 10.1142/S0218202517500191.
