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 discuss the existence of solutions for the problem, and focus on 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 the regularized problem. 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 (SSN) algorithm. Therefore, we propose and discuss the main properties of a SSN algorithm, including the convergence properties. The paper finishes with two computational experiments that exhibit the main properties of the numerical approach.

本文致力于通过半光滑牛顿法对具有温度相关参数的非等温平稳Bingham流动进行数值求解。我们讨论有关此问题的主要理论方面。主要是，我们讨论该问题的解决方案的存在，并着重于乘数公式，该公式使我们得到一个包含Navier–Stokes型方程和抛物线能量PDE的PDE耦合系统。进一步，我们为偏微分方程的耦合系统提出了一个Huber正则化，并简要讨论了正则化问题的适定性。详细的有限元离散化，基于所谓的（交叉网格${\mathbb{P}}_{\mathrm{1个}}$）- ${\mathbb{问}}_0$提出了空间变量的元素，包括加权刚度和质量矩阵。在空间上离散之后，将二阶BDF方法用作时间推进技术，从而在每次时间迭代中导致方程组的非平滑系统，该系统适用于半平滑牛顿（SSN）算法求解。因此，我们提出并讨论了SSN算法的主要特性，包括收敛特性。本文完成了两个计算实验，这些实验展示了数值方法的主要特性。