Chorin Projection Methods for Stochastic Stokes Equations

Dr. Liet Vo
Dr. Liet Vo

Abstract: In this talk, I will discuss the 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. An $O(k^\frac14)$ rate of convergence is proved for the standard Chorin scheme, which is sharp but not optimal due to the use of non-solenoidal noise, where $k$ denotes the time mesh size. On the other hand, an optimal convergence rate  $O(k^\frac12)$ is established for the modified Chorin scheme. 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 $k^{-\frac12}$, 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 numerical tests.

Speaker’s Bio: Liet Vo is a fifth year PhD candidate at the Department of Mathematics, University of Tennessee Knoxville, under the supervision of Dr. Xiaobing Feng. His dissertation is Numerical Methods for Stochastic Stokes and Navier-Stokes Equations. He received a MS in Applied Mathematics from University of Alabama in 2017, studying fluid dynamics. His current research interests include stochastic Navier-Stokes equations with multiplicative noise, stochastic total variation flow that has applications in image processing and material science, stochastic two-phase fluid flow Navier-Stokes-Cahn-Hilliard equations, efficient methods for random partial differential equations, and neural network and machine learning.

Last Updated: January 10, 2022 - 7:56 am