不可压缩流问题低次元稳定有限体积数值方法研究

Numerical Study of Stabilization of the Lower Order Finite Volume Methods for the Incompressible Flows

分析了R^d,d=2,3维不可压缩流Stokes问题低次元稳定有限体积方法,它主要利用局部压力投影方法对两种流行但不满足inf-sup条件的有限元配对(P_1-P_0和P_1-P_1)在有限体积方法的框架下进行稳定;利用有限元与有限体积方法的等价性进行有限体积方法理论分析.结果表明不可压缩流Stokes问题在f∈H^1情况下,本文方法得到的解与稳定有限元方法解之间具有O(h^2)阶超收敛阶结果,且稳定有限体积方法取得了与稳定有限元方法相同的收敛速度,与稳定有限元方法比较,稳定有限体积方法计算简单高效,同时保持物理守恒,因此在实际应用中具有很好的潜力。

In this paper,the stabilized finite volume method is considered for the R^d,d = 2,3 Stokes equations approximated by the lower order finite element pairs （P_1 - Po and P_1 - P_1）,which do not satisfy the so-called inf-sup condition.This method applies the local pressure projection to stabilize the lower order finite element. The convergence analysis also shows an important superclose result O（h^2） between the conforming mixed finite element solution and the finite volume solution using the same finite element pair for the incompressible flow.Based on the relationship between the finite element method and finite volume method,optimal estimate and superconvergenc result are obtained.Moreover,the stabilized finite volume method has a convergence rate of the same order as that of the usual stabilized finite element method of the stationary Stokes equations with additional regular assumption on the body force f∈H^1.