瞬态UCM粘弹性流体的最小二乘有限元算法

Least-Squares Finite Element Algorithm for the Time-Dependent UCM Viscoelastic Flow

【目的】建立求解瞬态UCM粘弹性流体的最小二乘有限元算法。【方法】利用具有一阶精度的差分格式对模型中的时间导数进行离散,得到了线性的半离散近似模型,采用最小二乘有限元方法对该近似模型进行求解。【结果】证明了最小二乘有限元解的存在唯一性,分析了最小二乘有限元解的先验误差估计,指出该估计依赖于时间步长Δt和空间步长h。通过一个三维空间的流动问题,验证了算法的有效性和收敛性。指出在实际计算中,相对于空间步长h,时间步长Δt对计算结果的影响较大。【结论】本文算法在数值精度方面,优于基于SUPG的混合有限元方法。

[Purposes]A least squares finite element algorithm was presented for the time-dependent UCM viscoelastic flow.[Methods]The finite difference with first-order accuracy was used to discretize the time derivatives.The linear semidiscrete system was obtained and a weighted least squares finite element was used to solve the system.The existence and uniqueness of the numerical approximations were proved.[Findings]A priori estimate that was dependent on time step size Δt and grid sizehwas analyzed.[Conclusions]The 3D numerical results illustrated the effectiveness and the convergence of the method.It was pointed out that the errors were mainly influenced by the time step sizeΔt.The least-squares finite element method was better than the mixed finite element method based on SUPG approximation in numerical accuracy.