Existence of smooth solutions to the 3D Navier–Stokes equations based on numerical solutions by the Crank–Nicolson finite element method

• Published in: Calcolo Vol. 61; no. 3
• Main Authors: Cai, Wentao, Zhang, Mingyan
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-09-2024; Springer Nature B.V
• Subjects: Finite element method; Fluid flow; Mathematics; Mathematics and Statistics; Navier-Stokes equations; Numerical Analysis; Smoothness; Theory of Computation; Finite element method; 65M15; Navier–Stokes equations; Linearized Crank–Nicolson scheme; 35B65; 35Q30; Uniqueness and existence
• ISSN: 0008-0624; 1126-5434
• DOI: 10.1007/s10092-024-00590-4

A Crank–Nicolson finite element method is proposed to solve the time-dependent Navier–Stokes equations. We prove that for a given smooth initial value, if a fully discrete finite element solution of the three-dimensional Navier–Stokes equations is bounded in a certain norm, then the strong solution of the Navier–Stokes equations satisfies uniqueness and smoothness. Further, the fully discrete solution converges to the strong solution as temporal step size
τ
→
0
and spatial step size
h
→
0
.