A FINITE ELEMENT METHOD FOR THE SURFACE STOKES PROBLEM
A FINITE ELEMENT METHOD FOR THE SURFACE STOKES PROBLEM
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DOI：
10.1137/18m1166183
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发表时间：
2018-01-01
影响因子：
3.1
通讯作者：
Yushutin, Vladimir
中科院分区：
文献类型：
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作者：
Olshanskii, Maxim A.;Quaini, Annalisa;Yushutin, Vladimir
我们考虑一个Stokes问题，它是在一个嵌入在一个3D域中的2D表面上提出的。该方程描述了一个平衡，面积保持切向流动的粘性表面流体，并作为一个模型问题，在材料界面的动力学。在本文中，我们开发和分析了这样的表面Stokes问题的迹有限元方法（TraceFEM）。TraceFEM依赖于在固定的、表面独立的背景网格上定义的有限元空间，该背景网格由形状规则的四面体组成。因此，不需要曲面参数化或曲面拟合网格。这里处理的TraceFEM是基于P-1体积有限元的速度和压力。为了使速度矢量场切向于曲面，我们引入了一个惩罚项。该方法是简单的实现，并有一个O（h（2））的几何一致性误差，这是由于P-1 P-1对速度和压力的近似误差相同的顺序。我们证明了稳定性和最佳阶离散误差界的表面H-1和L-2的范数。一系列的数值实验来说明建议TraceFEM的某些功能。
We consider a Stokes problem posed on a 2D surface embedded in a 3D domain. The equations describe an equilibrium, area-preserving tangential flow of a viscous surface fluid and serve as a model problem in the dynamics of material interfaces. In this paper, we develop and analyze a trace finite element method (TraceFEM) for such a surface Stokes problem. A TraceFEM relies on finite element spaces de fined on a fixed, surface-independent background mesh which consists of shape-regular tetrahedra. Thus, there is no need for surface parametrization or surface fitting with the mesh. The TraceFEM treated here is based on P-1 bulk finite elements for both the velocity and the pressure. In order to enforce the velocity vector field to be tangential to the surface, we introduce a penalty term. The method is straightforward to implement and has an O(h(2)) geometric consistency error, which is of the same order as the approximation error due to the P-1 P-1 pair for velocity and pressure. We prove stability and optimal order discretization error bounds in the surface H-1 and L-2 norms. A series of numerical experiments is presented to illustrate certain features of the proposed TraceFEM.