A Galerkin Linear Boundary Element Method For Solving Plane Stationary Stokes Problem

Steady flow of incompressible viscous fluid is expressed by stationary Stokes equation at small Reynolds number. In order to solve the problem with boundary element method, the way of turning Stokes boundary problem into boundary integral equation is not unique. Zhu Jialin[23]used a simple layer distribution expression based on velocity-pressure formula for the Dirichlet problem of Stokes equation with a Galerkin boundary element method and presented some numerical examples calculated by constant boundary element. The solution expression is suitable both for the inner and outer problem with a close boundary, and the calculation of velocity and pressure can be carried out separately, which is an advantage of boundary element scheme. To obtain the unknown intermediate vector function needs to solve a Fredholm integral equation of the first kind, the solution of which can only be determined with a difference of a vector proportional to the normal of the boundary and its integration on the boundary must zero. In the paper, we use the same idea to solve the variational equation,which equivalent to the Fredholm integral equation of the first kind with constraint. To avoid the trouble to construct the basis functions, we introduce a Lagrange multiplier into the variational formulation to replace the constraint condition for the basis. The variational formulation is solved by linear boundary elements in this paper. After the solution of the discretized equation, one obtain the intermediate vector on the boundary and the behavior at infinity of velocity at the same time. For calculation of the double integration on the boundary, the analytical formula of the first integral is deduced in detail with linear basic function, and a Gauss integration formula is used in the second integral. The results of numerical examples with closed boundary demonstrate that the scheme presented is practical and effective. Moreover we extend this method to the unbounded Stokes problem, the boundary of which is an open straight or an open arc segment. Boundary integral expression and its equivalent variation formulation are deduced. We plan to use a singular basis function to simulate the singular behavior at the tips of the segment, so we deduced an analytical formula for the first integral with a singular basis function on the elements with a tip.