求解定常不可压Stokes方程的两层罚函数方法

TWO-LEVEL PENALTY METHOD FOR THE STEADY INCOMPRESSIBLE STOKES EQUATIONS

借助于两套有限元网格空间提出了一种求解定常不可压Stokes方程的两层罚函数方法.该方法只需要求解粗网格空间上的Stokes方程和细网格空间上的两个易于求解的罚参数方程(离散后的线性方程组具有相同的对称正定系数矩阵).收敛性分析表明粗网格空间相对于细网格空间可以选择很小,并且罚参数的选取只与粗网格步长和问题的正则性有关.因此罚参数不必选择很小仍能够得到最优解.最后通过数值算例验证了上述理论结果,并且数值对比可知两层罚函数方法对于求解定常不可压Stokes方程具有很好的效果.

In this paper,we present a two-level penalty method for the steady incompressible Stokes equations by employing two finite element spaces.This method involves solving one small Stokes equation on the coarse space and two penalty equations on the fine space(the linear systems with same symetric and positive coefficient matrices).The convergence shows that the coarse space can be chosen very small.Moreover,the penalty parameter is only dependent on the coarse mesh size and the regularity of the problem.Therefore,the resulting solution still achieves asymptotically optimal accuracy when the penalty parameter is chosen'not very small'.The numerical results confirm the convergence analysis,and the numerical comparison also shows that this method is efficient for solving the steady incompressible Stokes equations.