Summary 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.

中文翻译：

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可变饱和渗流问题的自适应建模

总结 在本文中，我们提出了一种面向目标的自适应有限元方法，用于解决多孔介质中的一类地下流动问题，这些问题具有渗流面。我们专注于由非线性达西-白金汉定律控制的稳态流动的代表性案例，该定律对地下-大气边界具有物理约束。这导致将问题表述为变分不等式。使用基于双加权后验误差估计的自适应有限元方法研究了该问题的解决方案，其目的是减少特定目标量中的误差。感兴趣的量被选择为穿过渗流面的体积水通量，因此取决于先验未知的自由边界。我们将我们的方法应用于具有挑战性的数值示例以及特定案例研究，这项研究的起源，说明了在实际情况中出现的主要困难。我们总结了广泛的数值结果，这些结果清楚地表明所设计的方法可以根据自由度的数量快速减少误差。