In this article we present a goal-oriented adaptive finite element method for a class of subsurface flow problems in porous media, which exhibit seepage faces. We focus on a representative case of the steady state flows governed by a nonlinear Darcy-Buckingham law with physical constraints on subsurface-atmosphere boundaries. This leads to the formulation of the problem as a variational inequality. The solutions to this problem are investigated using an adaptive finite element method based on a dual-weighted a posteriori error estimate, derived with the aim of reducing error in a specific target quantity. The quantity of interest is chosen as volumetric water flux across the seepage face, and therefore depends on an a priori unknown free boundary. We apply our method to challenging numerical examples as well as specific case studies, from which this research originates, illustrating the major difficulties that arise in practical situations. We summarise extensive numerical results that clearly demonstrate the designed method produces rapid error reduction measured against the number of degrees of freedom.
翻译:在本篇文章中,我们提出了一个面向目标的适应性有限要素方法,用于应对在多孔媒体中出现的一类地下流动问题,这些媒体显示渗漏面;我们侧重于一个具有代表性的、由非线性达西-布金汉姆法律管辖的、对地表下大气边界有实际限制的稳态流动案例;这导致将问题表述成一种变式的不平等;正在使用基于双加权的事后误差估计的适应性有限要素方法来调查这一问题的解决方案,该方法旨在减少特定目标量的误差;利息数量被选为渗漏面的体积水流,因此取决于事先未知的自由边界;我们采用我们的方法对数字实例以及这种研究产生的具体案例研究提出质疑,说明在实际情况下出现的主要困难;我们总结大量的数字结果,明确显示所设计的方法能够根据自由度的数量迅速减少误差。