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A Hybrid High-Order method for Leray-Lions elliptic equations on general meshes


Authors: Daniele A. Di Pietro and Jérôme Droniou
Journal: Math. Comp. 86 (2017), 2159-2191
MSC (2010): Primary 65N08, 65N30, 65N12
DOI: https://doi.org/10.1090/mcom/3180
Published electronically: December 21, 2016
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Abstract: In this work, we develop and analyze a Hybrid High-Order (HHO) method for steady nonlinear Leray-Lions problems. The proposed method has several assets, including the support for arbitrary approximation orders and general polytopal meshes. This is achieved by combining two key ingredients devised at the local level: a gradient reconstruction and a high-order stabilization term that generalizes the one originally introduced in the linear case. The convergence analysis is carried out using a compactness technique. Extending this technique to HHO methods has prompted us to develop a set of discrete functional analysis tools whose interest goes beyond the specific problem and method addressed in this work: (direct and) reverse Lebesgue and Sobolev embeddings for local polynomial spaces, $ L^{p}$-stability and $ W^{s,p}$-approximation properties for $ L^{2}$-projectors on such spaces, and Sobolev embeddings for hybrid polynomial spaces. Numerical tests are presented to validate the theoretical results for the original method and variants thereof.


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Additional Information

Daniele A. Di Pietro
Affiliation: University of Montpellier, Institut Montpéllierain Alexander Grothendieck, 34095 Montpellier, France
Email: daniele.di-pietro@umontpellier.fr

Jérôme Droniou
Affiliation: School of Mathematical Sciences, Monash University, Clayton, Victoria 3800, Australia
Email: jerome.droniou@monash.edu

DOI: https://doi.org/10.1090/mcom/3180
Keywords: Hybrid High-Order methods, nonlinear elliptic equations, $p$-Laplacian, discrete functional analysis, convergence analysis, $W^{s, p}$-approximation properties of $L^2$-projection on polynomials
Received by editor(s): August 8, 2015
Received by editor(s) in revised form: March 23, 2016
Published electronically: December 21, 2016
Additional Notes: This work was partially supported by ANR project HHOMM (ANR-15-CE40-0005).
Article copyright: © Copyright 2016 American Mathematical Society