论文标题
随机Stokes方程的最佳收敛混合有限元方法
Optimally Convergent Mixed Finite Element Methods for the Stochastic Stokes Equations
论文作者
论文摘要
我们为带有乘法噪声的随机随机stokes方程提出了一些新的混合有限元方法，这些方程使用了驱动乘法噪声的Helmholtz分解。众所周知，压力溶液的规律性较低，在数值模拟中，众所周知的INF-SUP稳定混合有限元方法的次优收敛速率表现出来，请参见[10]。我们表明，从数值方案中的噪声中消除这一梯度部分会导致最佳收敛的混合有限元方法，并且这种概念性想法可用于重新修复确定性设置中众所周知的数值方法，包括压力稳定方法，因此其最佳逆转性能仍然可以在随机设置中维持。还提供了计算实验来验证理论结果并说明所提出的数值方法的概念性实用性。
We propose some new mixed finite element methods for the time dependent stochastic Stokes equations with multiplicative noise, which use the Helmholtz decomposition of the driving multiplicative noise. It is known [16] that the pressure solution has a low regularity, which manifests in sub-optimal convergence rates for well-known inf-sup stable mixed finite element methods in numerical simulations, see [10]. We show that eliminating this gradient part from the noise in the numerical scheme leads to optimally convergent mixed finite element methods, and that this conceptual idea may be used to retool numerical methods that are well-known in the deterministic setting, including pressure stabilization methods, so that their optimal convergence properties can still be maintained in the stochastic setting. Computational experiments are also provided to validate the theoretical results and to illustrate the conceptional usefulness of the proposed numerical approach.
