Continuous/Discontinuous Galerkin Difference Discretizations of High-Order Differential Operators

We develop continuous/discontinuous discretizations for high-order differential operators using the Galerkin Difference approach. Grid dispersion analyses are performed that indicate a nodal superconvergence in the ℓ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^2$$\end{document} norm. A treatment of the boundary conditions is described that ultimately leads to moderate growth in the spectral radius of the operators with polynomial degree, and in general the norms of the Galerkin Difference differentiation operators are significantly smaller than those arising from standard elements. Lastly, we observe that with the use of the Galerkin Difference space, the standard penalty terms required for discretizing high-order operators are not needed. Numerical results confirm the conclusions of the analyses performed.