A conservative Mixed Finite Element Method for a Regularised Nonlinear Long-Wave Model

In this paper, we develop and analyze a mixed finite element method for a nonlinear, higher-order model describing nonlinear wave phenomena and exhibiting important conservation properties. A central goal of our approach is to ensure that these properties are preserved at the discrete level while avoiding the challenges typically encountered when constructing finite element subspaces of H2(Omega)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>2(\Omega )$$\end{document} as would be required in a standard continuous Galerkin framework. At the continuous level, we establish well-posedness and characterize the solution through energy laws and mass conservation. For the semi-discrete formulation, we derive error estimates in various B & ocirc;chner spaces. Furthermore, we establish that the implicit fully discrete scheme is well-posed, converges with optimal order and consistent with both mass conservation and an entropy dissipation law. Finally, we confirm the theoretical findings and conservation properties on a set of benchmark problems.