论文标题
hellan-Herrmann-Johnson类似于两个空间维度的Stokes方程的流函数公式的方法
A Hellan-Herrmann-Johnson-like method for the stream function formulation of the Stokes equations in two and three space dimensions
论文作者
论文摘要
我们引入了一个新的离散化，以在两个和三个空间维度中的不可压缩Stokes方程的流函数公式。该方法与Hellan-Herrmann-Johnson方法密切相关，并且基于最近发现的质量保存混合应力配方[J. Gopalakrishnan，P.L。 Lederer，J.Schöberl，IMA数字分析杂志，2019年]，该速度近似于$ h（\ operatorName {div}）$ - 符合空间，并引入了一个新的压力样变量，以近似于函数空间$ h（\ operatornearnearnemaime $ h（curerornage $ h（cur）， （离散）DE RHAM复合物的属性允许将此方法扩展到两个和三个空间尺寸的流函数公式。 我们在连续和离散设置中介绍了详细的稳定性分析，其中流函数$ψ$及其近似值$ψ_H$是$ h（\ operatatorName {curl}）$的元素，以及$ h（\ operatatorName {curl}）$ - 构造nédélecturecFILITE元件空间。我们以错误分析结论，揭示了离散速度的误差$ u_h = \ operatatorName {curl}（ψ_h）$在离散$ h^1 $ -NORM中测量的最佳收敛率。我们提出数值示例，以验证我们的发现并讨论结构保留的特性，例如压力量。
We introduce a new discretization for the stream function formulation of the incompressible Stokes equations in two and three space dimensions. The method is strongly related to the Hellan-Herrmann-Johnson method and is based on the recently discovered mass conserving mixed stress formulation [J. Gopalakrishnan, P.L. Lederer, J. Schöberl, IMA Journal of numerical Analysis, 2019] that approximates the velocity in an $H(\operatorname{div})$-conforming space and introduces a new stress-like variable for the approximation of the gradient of the velocity within the function space $H(\operatorname{curl}\operatorname{div})$. The properties of the (discrete) de Rham complex allows to extend this method to a stream function formulation in two and three space dimensions. We present a detailed stability analysis in the continuous and the discrete setting where the stream function $ψ$ and its approximation $ψ_h$ are elements of $H(\operatorname{curl})$ and the $H(\operatorname{curl})$-conforming Nédélec finite element space, respectively. We conclude with an error analysis revealing optimal convergence rates for the error of the discrete velocity $u_h = \operatorname{curl}(ψ_h)$ measured in a discrete $H^1$-norm. We present numerical examples to validate our findings and discuss structure-preserving properties such as pressure-robustness.
