In this work, we develop a posteriori error control for a generalized Boussinesq model in which thermal conductivity and viscosity are temperature-dependent. Therein, the stationary Navier-Stokes equations are coupled with a stationary heat equation. The coupled problem is modeled and solved in a monolithic fashion. The focus is on multigoal-oriented error estimation with the dual-weighted residual method in which an adjoint problem is utilized to obtain sensitivity measures with respect to several goal functionals. The error localization is achieved with the help of a partition-of-unity in a weak formulation, which is specifically convenient for coupled problems as we have at hand. The error indicators are used to employ adaptive algorithms, which are substantiated with several numerical tests such as one benchmark and two further experiments that are motivated from laser material processing. Therein, error reductions and effectivity indices are consulted to establish the robustness and efficiency of our framework.
翻译:在这项工作中,我们为一个通用的Boussinesq模型开发了事后误差控制,该模型的热传导率和粘度取决于温度。在这种模型中,固定式Navier-Stokes方程式与固定式热方程式相伴而生。同时,问题是以单体方式建模和解决的。重点是多目标误差估计,同时采用双重加权剩余方法,利用共同问题获得对若干目标功能的敏感度度度。差错本地化的实现是在弱方程式中求得统一分法的帮助下实现的,这特别方便于我们手头的混合问题。误差指标用于采用适应性算法,这些算法得到若干数字测试的证实,例如一个基准和两个由激光材料处理驱动的进一步实验。在其中,参考了差错减少和效果指数,以确定我们框架的稳健性和效率。