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

TL;DR

This paper proves first-order convergence of a fully discrete semi-implicit finite element method for 2D Navier--Stokes equations with initial data in L^2, using smoothing properties and duality arguments, supported by numerical tests.

Contribution

It establishes convergence without extra regularity or grid-ratio conditions for the first time in this setting.

Findings

First-order convergence proved theoretically.

Numerical examples confirm the theoretical results.

Method works with minimal regularity assumptions.

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.