论文标题
分析带有一般乘法噪声的随机Stokes方程的脉络蛋白型投影方法
Analysis of Chorin-Type Projection Methods for the Stochastic Stokes Equations with General Multiplicative Noises
论文作者
论文摘要
本文涉及对具有一般非甲状腺乘噪声的随机Stokastic Stokes方程的两种完全离散的Chorin型投影方法的数值分析。第一个方案是标准的浮雕方案，第二个方案是一种修改的绒蛋白方案，该方案是通过在每个时间步骤对噪声函数上的helmholtz分解而设计的，以在结合原始压力和分解的卷发部分后产生一个投影的无差异噪声和“伪压力”。对于这两个（半分化）Chorin方案的速度和压力近似值证明了收敛的最佳顺序速率。在适当规范中衡量错误至关重要。完全离散的有限元方法是通过通过标准有限元方法在空间中离散空间中的两个半混凝蛋白方案来制定的。对于两种完全离散的方法，得出了次优阶误差估计。事实证明，所有空间错误常数都包含一个生长因子$ k^{ - 1/2} $，其中$ k $表示时间步长大小，这解释了当$ k \ the $ k \ d 0 $且空间网格尺寸在[9]的数值测试中所观察到的，这解释了标准chorin方案的恶化性能。还提供了数值结果，以引起提出的数值方法的性能，并验证理论误差估计的清晰度。
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.
