This paper is devoted to the numerical solution of the non-isothermal instationary Bingham flow with temperature dependent parameters by semismooth Newton methods. We discuss the main theoretical aspects regarding this problem. Mainly, we focus on existence of solutions and a multiplier formulation which leads us to a coupled system of PDEs involving a Navier-Stokes type equation and a parabolic energy PDE. Further, we propose a Huber regularization for this coupled system of partial differential equations, and we briefly discuss the well posedness of these regularized problems. A detailed finite element discretization, based on the so called (cross-grid $\mathbb{P}_1$) - $\mathbb{Q}_0$ elements, is proposed for the space variable, involving weighted stiffness and mass matrices. After discretization in space, a second order BDF method is used as a time advancing technique, leading, in each time iteration, to a nonsmooth system of equations, which is suitable to be solved by a semismooth Newton algorithm. Therefore, we propose and discuss the main properties of a SSN algorithm, including the convergence properties. The paper finishes with two computational experiment that exhibit the main properties of the numerical approach.
翻译：本文专门用半斯穆特 牛顿方法讨论这个问题的主要理论方面。 我们主要侧重于存在解决方案和乘数配制,导致我们建立包含纳维尔-斯托克斯式方程式和抛光能量PDE的PDE组合系统。 此外,我们提议对这一部分差异方程式的组合系统建立一个枢纽规范,我们简要讨论这些常规化问题的构成情况。基于所谓的(跨格 $\mathbb{P ⁇ 1$) - $\mathbb ⁇ 0$ 元素,为空间变量提议了一个详细的有限离散元素(跨格 $\mathb{P ⁇ 1$) - $\ mathbb ⁇ 0$ 元素,涉及加权僵硬度和质量矩阵。在空间分解后,第二顺序BDF方法被用作一种时间推进技术,每次分解后,都导致一个非模拟方程式的方程式系统,适合通过半摩托式牛顿算法解。 因此,我们提议并讨论空间变量的主要实验特性,包括SNSN 数字演算法的主要特性。
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