Convergence analysis of an efficient scheme for the steady state second grade fluid model

Abstract

We are interested in studying the stationary second grade fluid model in a bounded domain in $R^2$. To approximate the solution of the continuous model, we propose a fully decoupled numerical scheme based on a splitting method combined with the use of the Grad–Div operator. This approach allows the complete decoupling of the three variables: velocity, pressure and vorticity. Each variable is computed using an iterative procedure, with the pressure step involving a simple $L^2$-projection.

We provide a proof of the convergence of the scheme to the continuous problem under smallness assumptions on the data. This theoretical analysis ensures the reliability of our method in approximating the behavior of the stationary second grade fluid model.

Finally, we present several numerical tests to validate our approach. These tests illustrate the effectiveness and efficiency of our scheme in various scenarios, highlighting its potential applicability to a wide range of problems involving second grade fluids.