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VPVnet: a velocity-pressure-vorticity neural network method for the Stokes’ equations under reduced regularity. (English) Zbl 1492.76040

Summary: We present VPVnet, a deep neural network method for the Stokes’ equations under reduced regularity. Different with recently proposed deep learning methods [J. Li et al., “The deep learning Galerkin method for the general Stokes equations”, Preprint, arXiv:2009.11701; M. Raissi et al., J. Comput. Phys. 378, 686–707 (2019; Zbl 1415.68175)] which are based on the original form of PDEs, VPVnet uses the least square functional of the first-order velocity-pressure-vorticity (VPV) formulation [B.-N. Jiang and C. L. Chang, Comput. Methods Appl. Mech. Eng. 78, No. 3, 297–311 (1990; Zbl 0706.76033)] 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.

MSC:

76D07 Stokes and related (Oseen, etc.) flows
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35B45 A priori estimates in context of PDEs
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