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

01/07/2021

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by   Buyang Li, et al.

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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.