Analysis on an HDG Method for the p-Laplacian Equations

In Cockburn and Shen (SIAM J Sci Comput 38(1):A545–A566, 2016) the authors propose the first hybridizable discontinuous Galerkin method (HDG) for the p-Laplacian equation. Several iterative algorithms are developed and tested. The main purpose of this paper is to provide rigorous error estimates for the proposed HDG method. We first develop the error estimates based on general polyhedral/polygonal triangulations, under standard regularity assumption of the solution, the convergence analysis is presented for 1<p<2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<2$$\end{document} and p>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>2$$\end{document}. Nevertheless, when p approaches to the limits (p→1+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \rightarrow 1^+$$\end{document} or p→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \rightarrow \infty $$\end{document}), the convergence rate shows degeneration for both cases. Finally, this degeneration can be recovered if we use simplicial triangulation of the domain with sufficient large stabilization parameter for the method.