In this paper we consider the Navier–Stokes–Brinkman equations, which constitute one of the most common nonlinear models utilized to simulate viscous fluids through porous media, and propose and analyze a Banach spaces-based approach yielding new mixed finite element methods for its numerical solution. In addition to the velocity and pressure, the strain rate tensor, the vorticity, and the stress tensor are introduced as auxiliary unknowns, and then the incompressibility condition is used to eliminate the pressure, which is computed afterwards by a postprocessing formula depending on the stress and the velocity. The resulting continuous formulation becomes a nonlinear perturbation of, in turn, a perturbed saddle point linear system, which is then rewritten as an equivalent fixed-point equation whose operator involved maps the velocity space into itself. The well-posedness of it is then analyzed by applying the classical Banach fixed point theorem, along with a smallness assumption on the data, the Babuška–Brezzi theory in Banach spaces, and a slight variant of a recently obtained solvability result for perturbed saddle point formulations in Banach spaces as well. The resulting Galerkin scheme is momentum-conservative. Its unique solvability is analyzed, under suitable hypotheses on the finite element subspaces, using a similar fixed-point strategy as in the continuous problem. A priori error estimates are rigorously derived, including also that for the pressure. We show that PEERS and AFW elements for the stress, the velocity and the rotation, together with piecewise polynomials of a proper degree for the strain rate tensor, yield stable discrete schemes. Then, the approximation properties of these subspaces and the Céa estimate imply the respective rates of convergence. Finally, we include two and three dimensional numerical experiments that serve to corroborate the theoretical findings, and these tests illustrate the performance of the proposed mixed finite element methods.

中文翻译：

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使用 Banach 空间的 Navier-Stokes-Brinkman 方程的新非增广混合有限元方法

在本文中，我们考虑 Navier-Stokes-Brinkman 方程，它构成了最常见的非线性模型之一，用于模拟通过多孔介质的粘性流体，并提出和分析了一种基于 Banach 空间的方法，产生了新的混合有限元方法，用于计算其数值解决方案。除了速度和压力外，还引入应变率张量、涡量和应力张量作为辅助未知数，然后使用不可压缩条件消除压力，然后根据应力通过后处理公式计算压力和速度。由此产生的连续公式变成了一个非线性扰动，反过来，一个扰动的鞍点线性系统，然后将其重写为等效的不动点方程，其中涉及的运算符将速度空间映射到自身。然后通过应用经典 Banach 不动点定理、数据的小假设、Banach 空间中的 Babuška-Brezzi 理论以及最近获得的扰动鞍点的可解性结果的轻微变体来分析它的适定性Banach 空间中的公式也是如此。由此产生的 Galerkin 方案是动量保守的。在有限元子空间的适当假设下，使用与连续问题中类似的不动点策略，分析了其独特的可解性。先验误差估计是严格导出的，也包括压力的估计。我们展示了用于应力、速度和旋转的 PEERS 和 AFW 元素，与应变率张量的适当次数的分段多项式一起，产生稳定的离散方案。然后，这些子空间的近似特性和 Céa 估计暗示了各自的收敛速度。最后，我们包括用于证实理论发现的二维和三维数值实验，这些测试说明了所提出的混合有限元方法的性能。