A natural nonconforming FEM for the Bingham flow problem is quasi-optimal

This paper introduces a novel three-field formulation for the Bingham flow problem and its two-dimensional version named after Mosolov together with low-order discretizations: a nonconforming for the classical formulation and a mixed finite element method for the three-field model. The two discretizations are equivalent and quasi-optimal in the sense that the H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1$$\end{document} error of the primal variable is bounded by the error of the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} best-approximation of the stress variable. This improves the predicted convergence rate by a log factor of the maximal mesh-size in comparison to the first-order conforming finite element method in a model scenario. Despite that numerical experiments lead to comparable results, the nonconforming scheme is proven to be quasi-optimal while this is not guaranteed for the conforming one.