@Article{CiCP-31-3,
author = {Yujie, Liu and Yang, Chao},
title = {VPVnet: A Velocity-Pressure-Vorticity Neural Network Method for the Stokes’ Equations under Reduced Regularity},
journal = {Communications in Computational Physics},
year = {2022},
volume = {31},
number = {3},
pages = {739--770},
abstract = {<p style="text-align: justify;">We present VPVnet, a deep neural network method for the Stokes’ equations under reduced regularity. Different with recently proposed deep learning methods [40,51] which are based on the original form of PDEs, VPVnet uses the least square
functional of the first-order velocity-pressure-vorticity (VPV) formulation ([30]) as loss
functions. As such, only first-order derivative is required in the loss functions, hence
the method is applicable to a much larger class of problems, e.g. problems with nonsmooth solutions. Despite that several methods have been proposed recently to reduce
the regularity requirement by transforming the original problem into a corresponding
variational form, while for the Stokes’ equations, the choice of approximating spaces
for the velocity and the pressure has to satisfy the LBB condition additionally. Here
by making use of the VPV formulation, lower regularity requirement is achieved with
no need for considering the LBB condition. Convergence and error estimates have
been established for the proposed method. It is worth emphasizing that the VPVnet
method is divergence-free and pressure-robust, while classical inf-sup stable mixed
finite elements for the Stokes’ equations are not pressure-robust. Various numerical
experiments including 2D and 3D lid-driven cavity test cases are conducted to demonstrate its efficiency and accuracy.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2021-0104},
url = {https://global-sci.com/article/79527/vpvnet-a-velocity-pressure-vorticity-neural-network-method-for-the-stokes-equations-under-reduced-regularity}
}