The classical dual mixed finite element method for flow simulations is based on $H(\mathrm{div},\Omega )$ conforming approximation spaces for the flux, which guarantees continuous normal components on element interfaces, and discontinuous approximations in $L^2\left(\Omega \right)$ for the pressure. However, stability and convergence can only be obtained for compatible approximation spaces. Stabilized finite element methods may provide an alternative stable procedure to avoid this kind of delicate balance. The main purpose of this paper is to present a high-order finite element methodology to solve the Darcy problem based on the combination of an unconditionally stable mixed finite element method with a hierarchical methodology for the construction of finite dimensional subspaces of $H(\mathrm{div},\Omega )$ and $H^1\left(\Omega \right)$. The chosen stabilized method is free of mesh dependent stabilization parameters and allows for the use of different high order finite element approximations for the flux and the pressure variables, without requiring any compatibility constraint, as required in mixed methods for these problems. Convergence studies are presented comparing the numerical solutions obtained for different approximation orders on quadrilateral elements with the ones given by classical mixed formulation with Raviart–Thomas elements.

用于流模拟的经典双重混合有限元方法基于 $H（\mathrm{div}，\Omega ）$ 通量的一致逼近空间，保证元素界面上连续的法向分量，以及 ${\mathrm{大号}}^2（\Omega ）$对于压力。但是，仅对于兼容的近似空间才能获得稳定性和收敛性。稳定的有限元方法可以提供另一种稳定的过程，以避免这种微妙的平衡。本文的主要目的是基于无条件稳定混合有限元方法与分层方法相结合，提出一种高阶有限元方法，以解决达西问题，该方法用于构造有限维子空间。$H（\mathrm{div}，\Omega ）$ 和 $H^{\mathrm{1个}}（\Omega ）$。所选择的稳定化方法没有依赖于网格的稳定化参数，并且允许对通量和压力变量使用不同的高阶有限元近似值，而无需任何兼容性约束，就像这些问题的混合方法所要求的那样。提出了收敛性研究，比较了四边形元素的不同近似阶数与经典混合公式与Raviart–Thomas元素给出的近似解获得的数值解。