Velocity-vorticity-pressure formulation for the Oseen problem with variable viscosity
Velocity-vorticity-pressure formulation for the Oseen problem with variable viscosity
复制标题
变粘度 Oseen 问题的速度-涡量-压力公式
作者：
Verónica Anaya;Ruben Caraballo;Bryan Gomez;D. Mora;R. Ruiz
我们提出并分析了一种用于用速度、涡度和压力编写的 Oseen 方程的增强混合有限元方法，具有非恒定粘度和速度的齐次狄利克雷边界条件。弱公式包括由本构方程和不可压缩条件产生的最小二乘项，我们证明它满足 Babuška-Brezzi 理论的假设。重复连续分析的论证，建立了离散问题的稳定性和可解性。该方法适用于速度和压力的任何 Stokes inf-sup 稳定有限元对，而对于涡度，可以使用任何通用离散空间（任意阶）。先验和后验误差估计是使用两个特定的离散子空间族得出的。最后，我们提供了一组数值测试来说明该方案的行为，验证理论收敛速度，并显示由残余后验误差估计引导的自适应算法的性能。
We propose and analyse an augmented mixed finite element method for the Oseen equations written in terms of velocity, vorticity, and pressure with non-constant viscosity and homogeneous Dirichlet boundary condition for the velocity. The weak formulation includes least-squares terms arising from the constitutive equation and from the incompressibility condition, and we show that it satisfies the hypotheses of the Babuška-Brezzi theory. Repeating the arguments of the continuous analysis, the stability and solvability of the discrete problem are established. The method is suited for any Stokes inf-sup stable finite element pair for velocity and pressure, while for vorticity any generic discrete space (of arbitrary order) can be used. A priori and a posteriori error estimates are derived using two specific families of discrete subspaces. Finally, we provide a set of numerical tests illustrating the behaviour of the scheme, verifying the theoretical convergence rates, and showing the performance of the adaptive algorithm guided by residual a posteriori error estimation.