Resumen
Considering the interaction between the large and small eddy components of solutions and using the idea of the Newton iteration, a small eddy correction method is proposed for approximating and numerically solving nonlinear dissipative PDEs of parabolic type, in particular the Navier--Stokes equations (NSE). We assume that the large eddy approximation to the solution is known. Formally applying the Newton iterative procedure to the small eddy equation, we then generate approximate systems. It is shown that the first two steps in fact lead to the standard Galerkin method (SGM) and the so-called optimum nonlinear Galerkin method (ONG), and therefore the small eddy correction method is actually a certain generalization of SGM and ONG. The boundedness and convergence analysis are presented in the framework of the two-dimensional NSE. The results show that the small eddy correction method can greatly improve the accuracy of SGM approximate solutions.