no. 6
Analysis of fully discrete finite element methods for 2D Navier–Stokes equations with critical initial data
Li, Buyang orcid ; Ma, Shu orcid ; Ueda, Yuki orcid
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 6, pp. 2105-2139

First-order convergence in time and space is proved for a fully discrete semi-implicit finite element method for the two-dimensional Navier–Stokes equations with L2 initial data in convex polygonal domains, without extra regularity assumptions or grid-ratio conditions. The proof utilises the smoothing properties of the Navier–Stokes equations in the analysis of the consistency errors, an appropriate duality argument, and the smallness of the numerical solution in the discrete L2(0, t$$; H1) norm when t$$ is smaller than some constant. Numerical examples are provided to support the theoretical analysis.

DOI : 10.1051/m2an/2022073
Classification : 65M12, 65M15, 76D05
Keywords: Navier–Stokes equations, $$2 initial data, semi-implicit Euler scheme, finite element method, error estimate 
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     author = {Li, Buyang and Ma, Shu and Ueda, Yuki},
     title = {Analysis of fully discrete finite element methods for {2D} {Navier{\textendash}Stokes} equations with critical initial data},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {2105--2139},
     year = {2022},
     publisher = {EDP-Sciences},
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     url = {https://www.numdam.org/articles/10.1051/m2an/2022073/}
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Li, Buyang; Ma, Shu; Ueda, Yuki. Analysis of fully discrete finite element methods for 2D Navier–Stokes equations with critical initial data. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 6, pp. 2105-2139. doi: 10.1051/m2an/2022073

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