Analysis of fully discrete finite element methods for 2D Navier--Stokes equations with critical initial data

Buyang Li; Shu Ma; Yuki Ueda

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arXiv · 2101.02444

Analysis of fully discrete finite element methods for 2D Navier--Stokes equations with critical initial data

Buyang Li·Shu Ma·Yuki Ueda

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Abstract

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 $L^2$ 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, an appropriate duality argument, and the smallness of the numerical solution in the discrete $L^2(0,t_m;H^1)$ norm when $t_m$ is smaller than some constant. Numerical examples are provided to support the theoretical analysis.

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Buyang Li, Shu Ma, Yuki Ueda. 2021-01-07. Analysis of fully discrete finite element methods for 2D Navier--Stokes equations with critical initial data. https://arxiv.org/abs/2101.02444

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