This paper is concerned with numerical analysis of two fully discrete Chorin-type projection methods for the stochastic Stokes equations with general non-solenoidal multiplicative noise. The first scheme is the standard Chorin scheme and the second one is a modified Chorin scheme which is designed by employing the Helmholtz decomposition on the noise function at each time step to produce a projected divergence-free noise and a "pseudo pressure" after combining the original pressure and the curl-free part of the decomposition. Optimal order rates of the convergence are proved for both velocity and pressure approximations of these two (semi-discrete) Chorin schemes. It is crucial to measure the errors in appropriate norms. The fully discrete finite element methods are formulated by discretizing both semi-discrete Chorin schemes in space by the standard finite element method. Suboptimal order error estimates are derived for both fully discrete methods. It is proved that all spatial error constants contain a growth factor $k^{-1/2}$, where $k$ denotes the time step size, which explains the deteriorating performance of the standard Chorin scheme when $k\to 0$ and the space mesh size is fixed as observed earlier in the numerical tests of [9]. Numerical results are also provided to guage the performance of the proposed numerical methods and to validate the sharpness of the theoretical error estimates.
翻译:本文涉及对具有一般非溶性多复制噪音的Stochacistic Stokes方程式的两种完全离散的Chorin型预测方法的数值分析。第一个方案是标准的Chorin方案,第二个方案是修改的Chorin方案,其设计方法是在每一步的噪音功能上采用Helmholtz分解法,以产生一个预测的无差异噪音和“假冒压力”,在将原压力和分解分解部分结合起来之后,得出一个预测的无差异噪音和“假冒压力”的“假体压”。对于这两种(半分解-分解)Chorin方案的速度和压力近似方案,都证明趋同的最佳顺序速度,这两类(半分解-分解-分解-)Chorin方案的速度和压力近似接近率。衡量适当规范中的错误至关重要。完全离散的元素方法是通过标准的限定元素法将空间的半分解办法分解,以完全分解的方法计算出各种偏差的方法。所有空间差常数常数都包含一个增长系数 $-1/2美元 美元 美元,其中,美元表示时间步骤的精确度的精确度,在所观察到的缩缩缩缩缩度中,计算方法中,其精确度为0.值为0.值为0.值为0.值。