On Smooth Solution to Three-Dimensional Incompressible Navier–Stokes Equations Based on Numerical Solutions by Finite Element Approximation

Fengnan Liu (), Junpeng Cao and Ziqiu Zhang
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Fengnan Liu: Leicester International Institute, Dalian University of Technology, Panjin 124221, China
Junpeng Cao: Leicester International Institute, Dalian University of Technology, Panjin 124221, China
Ziqiu Zhang: College of Sciences, Northeastern University, Shenyang 110004, China

Mathematics, 2025, vol. 13, issue 19, 1-13

Abstract: In this paper, we develop a fully discrete finite element scheme, based on a second-order backward differentiation formula (BDF2), for numerically solving the three-dimensional incompressible Navier–Stokes equations. Under the assumption that the fully discrete solution remains bounded in a certain norm, we establish that any smooth initial data necessarily gives rise to a unique strong solution that remains smooth. Moreover, we demonstrate that the fully discrete numerical solution converges strongly to this exact solution as the temporal and spatial discretization parameters approach zero.

Keywords: incompressible Navier–Stokes equations; BDF2 scheme; finite element method; smooth solution (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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