Two discretisations of the time-dependent Bingham problem

Abstract

AbstractThis paper introduces two methods for the fully discrete time-dependent Bingham problem in a three-dimensional domain and for the flow in a pipe also named after Mosolov. The first time discretisation is a generalised midpoint rule and the second time discretisation is a discontinuous Galerkin scheme. The space discretisation in both cases employs the non-conforming first-order finite elements of Crouzeix and Raviart. The a priori error analyses for both schemes yield certain convergence rates in time and optimal convergence rates in space. It guarantees convergence of the fully-discrete scheme with a discontinuous Galerkin time-discretisation for consistent initial conditions and a source term $$f\in H^1(0,T;L^2(\Omega ))$$ f ∈ H 1 ( 0 , T ; L 2 ( Ω ) ) .