This 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 \(f\in H^1(0,T;L^2(\Omega ))\).

本文介绍了两种方法来解决三维域中完全离散的瞬态 Bingham 问题和管道中的流动，该管道也以 Mosolov 命名。第一次离散化是广义中点规则，第二次离散化是不连续的伽辽金格式。两种情况下的空间离散化都采用了 Crouzeix 和 Raviart 的非一致性一阶有限元。两种方案的先验误差分析在时间上产生一定的收敛率，在空间上产生最佳收敛率。对于一致的初始条件和源项\(f\in H^1(0,T;L^2(\Omega ))\) ，它保证了具有不连续伽辽金时间离散化的全离散方案的收敛。