This paper studies a finite element discretization of the regularized Bingham equations that describe viscoplastic flow. Convergence analysis is provided, as we prove optimal convergence with respect to the spatial mesh width but depending inversely on the regularization parameter $\varepsilon$, and also suboptimal (by one order) convergence that is independent of the regularization parameter. An efficient nonlinear solver for the discrete model is then proposed and analyzed. The solver is based on Anderson acceleration (AA) applied to a Picard iteration, and we prove accelerated convergence of the method by applying AA theory (recently developed by authors) to the iteration, after showing sufficient smoothness properties of the associated fixed point operator. Numerical tests of spatial convergence are provided, as are results of the model for 2D and 3D driven cavity simulations. For each numerical test, the proposed nonlinear solver is also tested and shown to be very effective and robust with respect to the regularization parameter as it goes to zero.

翻译：本文对描述粘贴体流的正规化宾汉姆方程式的有限分解元素进行了研究。 提供了趋同分析, 因为我们证明在空间网格宽度方面最佳趋同, 但反过来又取决于正规化参数$\varepsilon$, 以及独立于正规化参数的亚最佳(按一个顺序)趋同。 然后为离散模型提议并分析一个高效的非线性求解器。 解答器以Anderson加速( AAA)为基础, 应用到皮卡尔迭代法, 在显示相关固定点操作员的足够平稳性之后, 我们通过对迭代法应用 AA 理论( 作者最近开发的理论) 证明该方法加快了趋同速度。 提供了空间趋同的数值测试, 正如2D 和 3D 驱动的轨迹模拟模型的结果一样。 对于每一项数字测试, 拟议的非线性求解解解器也经过测试, 并显示对零点的正规化参数非常有效和有力。