In this paper we present an a posteriori error analysis of a mixed-VEM discretization for a nonlinear Brinkman model of porous media flow, which has been proposed by the authors in a previous work. Therein, the system is formulated in terms of a pseudostress tensor and the velocity gradient, whereas the velocity and the pressure of the fluid are computed via postprocessing formulas. Furthermore, the well-posedness of the associated augmented formulation along with a priori error bounds for the discrete scheme also were established. We now propose reliable and efficient residual-based a posteriori error estimates for a computable approximation of the virtual solution associated to the aforementioned problem. The resulting error estimator is fully computable from the degrees of freedom of the solutions and applies on very general polygonal meshes. For the analysis we make use of a global inf–sup condition, Helmholtz decomposition, local approximation properties of interpolation operators and inverse inequalities together with localization arguments based on bubble functions. Finally, we provide some numerical results confirming the properties of our estimator and illustrating the good performance of the associated adaptive algorithm.

在本文中，我们提出了多孔介质流非线性Brinkman模型的混合VEM离散化的后验误差分析，这是作者在先前的工作中提出的。其中，该系统由伪应力张量和速度梯度表示，而流体的速度和压力是通过后处理公式计算的。此外，还建立了相关的增强公式的适定性以及离散方案的先验误差范围。现在我们为与上述问题相关的虚拟解决方案的可计算近似值，提出可靠且有效的基于残差的后验误差估计。由此产生的误差估计器可以从解的自由度完全计算出来，并适用于非常普通的多边形网格。为了进行分析，我们使用了全局inf-sup条件，Helmholtz分解，插值算子的局部逼近性质和逆不等式以及基于气泡函数的局部化参数。最后，我们提供了一些数值结果，证实了估计器的属性并说明了相关自适应算法的良好性能。