Viscosity-uniform error estimates for a semi-implicit Fourier method in Besov spaces

Abstract

We prove error estimates for a semi-implicit Fourier--Galerkin method for two-dimensional incompressible flow on the torus, with viscosity $0\le \nu \le 1$. The temporal error is first order in $H^s$ for $H^{s+2}$ initial data and integer $s\ge 1$. For data in $B_{2,1}^m$, with integer $m\ge 3$, the fully discrete error is of order $\tau +N^{-(m-2)}$ in $B_{2,1}^1$ and $B_{\infty ,1}^0$. These estimates hold on a common local interval, with constants and time-step thresholds independent of viscosity and Fourier resolution. For data in $B_{2,1}^{2+\epsilon }$, $0<\epsilon <1$, the error relative to Euler flow is bounded by $C({\tau }^{(1+\epsilon )/2}+N^{-\epsilon }+{\nu }^{(1+\epsilon )/2})$. The consistency bounds use viscosity-weighted regularity. Orthogonality of the Leray and Fourier projectors in the block energies gives the endpoint estimates, and stability with respect to the initial data yields the fractional rate. An $L^2$ residual tolerance $\delta$ contributes $CN\delta$ to the endpoint error of computed iterates. We report refinement studies for smooth and fixed multiscale data, together with qualitative flow comparisons. An exactly solvable rough shear has fractional errors, up to logarithms, in $B_{2,1}^1$ and first-order errors in $B_{\infty ,1}^0$.

arXiv comment44 pages, 6 figures. Combined and substantially revised version of arXiv:2406.12320v1 and arXiv:2509.23192v1; title changed; expanded analysis and numerical experiments