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 the relative named after Mosolov and 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 error of the primal variable is bounded by the error of the L2 bestapproximation 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. AMS subject classifications 65N30, 76M10 key words Bingham flow problem, Mosolov’s problem, nonconforming finite element methods, three-field formulation, mixed variational inequalities ∗Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, D10099 Berlin, Germany; Department of Computational Science and Engineering, Yonsei University, Seoul, Korea †Department of Mathematics and Applied Mathematics, University of Cape Town, 7701 Rondebosch, South Africa ‡Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, D10099 Berlin, Germany