We propose and analyse an augmented mixed finite element method for the Navier--Stokes equations written in terms of velocity, vorticity, and pressure with non-constant viscosity and no-slip boundary conditions. The weak formulation includes least-squares terms arising from the constitutive equation and from the incompressibility condition, and we use a fixed point strategies to show the existence and uniqueness of continuous and discrete solutions under the assumption of sufficiently small data. The method is constructed using any compatible finite element pair (conforming or non-conforming) for velocity and pressure as dictated by Stokes inf-sup stability, while for vorticity any generic discrete space (of arbitrary order) can be used. We establish optimal a priori error estimates. Finally, we provide a set of numerical tests in 2D and 3D illustrating the behaviour of the scheme as well as verifying the theoretical convergence rates.
翻译:我们提出并分析以速度、园艺和压力、非恒定粘度和无倾斜边界条件为公式的纳维埃-斯托克斯方程式的扩大混合限定要素方法。弱方程式包括构成方程式和不压缩条件产生的最小方位术语,我们使用固定点战略来显示在足够小的数据假设下持续和离散的解决方案的存在和独特性。该方法的构建采用任何兼容的最小要素对齐(符合或不符合),以适应斯托克斯内向稳定要求的速度和压力,而对于任何通用离散空间(任意顺序)则可以使用。我们建立了最理想的先验误差估计。最后,我们提供了一套2D和3D数字测试,说明该办法的行为,并核实理论汇合率。