A Novel Mesh‐Free Approach for Solving Incompressible Fluid Flow Problems

ABSTRACT In this study, we present a novel mesh‐free approach for solving incompressible fluid flow problems, which is introduced here for the first time. Our approach solves the steady‐state Navier–Stokes equations without requiring traditional mesh generation. For this purpose, we adopt a discrete framework in which variables are defined at specific points within the domain, thereby eliminating the need for numerical integration. The proposed approach combines a weighted least squares (WLS) approximation with a high‐order continuation method (HOCM). This approach significantly enhances the accuracy of steady‐state incompressible flow simulations, offering both improved precision and reduced computation time compared to classical methods. Our results indicate that this approach holds substantial potential for expanding practical applications across various engineering fields. In contrast to the coupling of the moving least squares (MLS) method with the HOCM, our approach avoids computing derivatives of the weight function within the influence domain, which reduces the computational cost and enhances accuracy. This original combination highlights the novelty of our work compared to research conducted in recent years. A comparison is presented between the results obtained using the HOCM with MLS approximation and those reported in the literature.