无界区域上Stokes问题的自然边界元与Mini元耦合法

The Coupling of Natural Boundary Element and Mini Element for the Stokes Problem on Unbounded Domains

本文讨论用自然边界元与mini元耦合法求解描述平面无界区域上不可压缩粘滞低速流动的定常Stokes问题。首先以圆为人工边界,利用自然边界归化将原问题转化为耦合变分问题,并证明该变分问题的存在唯一性,然后在人工边界上采用分段线性边界元,在有界区域上应用mini元分别进行离散化,合成总刚度矩阵,从而建立耦合法的线性方程组,最后,证明其收敛性和误差估计,并通过数值实验以表现该方法的实际有效性及其理论分析的正确性。

This paper investigates the coupling of natural boundary element and mini element for the steady Stokes problem,which describes the impressible viscous slow flow on the planar unbounded domains. The original problem is turned into a variational coupling problem by constructing an artificial boundary and using a natural boundary reduction,and its uniqueness and existence are proved. The piecewise linear boundary element on the artificial boundary and the mini element in the bounded domain are applied. The total stiffness matrix is combined. The coupling system of the linear equations is established, and its convergence and error estimate are proved. Finally, the Uzawa algorithm is used to solve this indefinite system of the linear equations, and the corresponding numerical experiments are carried out to show the practical effectiveness of the proposed method and the correctness of its numerical analysis.