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. The theoretical and practical implications of using augmentation is discussed in detail. In addition, we use 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 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写出的纳维埃-斯托克斯方程式的扩大混合有限要素方法。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。详细讨论了使用扩增的理论和实际影响。此外,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、由斯托克斯内侧稳定决定的速度和压力的有限要素配对,而对于任何普通离散空间(任意秩序)则可以使用。我们建立了最佳的先行误差估计。最后,我们提供了一套2D和3D数字测试,以说明该办法的行为,并核实理论趋同率。